Network Working Group M. Abdalla Internet-Draft DFINITY - Zurich Intended status: Informational B. Haase Expires: 26 March 2024 Endress + Hauser Liquid Analysis - Gerlingen J. Hesse IBM Research Europe - Zurich 23 September 2023 CPace, a balanced composable PAKE draft-irtf-cfrg-cpace-09 Abstract This document describes CPace which is a protocol that allows two parties that share a low-entropy secret (password) to derive a strong shared key without disclosing the secret to offline dictionary attacks. The CPace protocol was tailored for constrained devices, is compatible with any cyclic group of prime- and non-prime order. Discussion Venues This note is to be removed before publishing as an RFC. Discussion of this document takes place on the Crypto Forum Research Group mailing list (cfrg@ietf.org), which is archived at https://mailarchive.ietf.org/arch/search/?email_list=cfrg. Source for this draft and an issue tracker can be found at https://github.com/cfrg/draft-irtf-cfrg-cpace. Status of This Memo This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79. Internet-Drafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as Internet-Drafts. The list of current Internet- Drafts is at https://datatracker.ietf.org/drafts/current/. Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress." This Internet-Draft will expire on 26 March 2024. Abdalla, et al. Expires 26 March 2024 [Page 1] Internet-Draft CPace September 2023 Copyright Notice Copyright (c) 2023 IETF Trust and the persons identified as the document authors. All rights reserved. This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (https://trustee.ietf.org/ license-info) in effect on the date of publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect to this document. Code Components extracted from this document must include Revised BSD License text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the Revised BSD License. Table of Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1. Outline of this document . . . . . . . . . . . . . . . . 6 2. Requirements Notation . . . . . . . . . . . . . . . . . . . . 6 3. High-level application perspective . . . . . . . . . . . . . 7 3.1. Optional CPace inputs . . . . . . . . . . . . . . . . . . 7 3.2. Responsibilities of the application layer . . . . . . . . 8 4. CPace cipher suites . . . . . . . . . . . . . . . . . . . . . 9 5. Definitions and notation . . . . . . . . . . . . . . . . . . 10 5.1. Hash function H . . . . . . . . . . . . . . . . . . . . . 10 5.2. Group environment G . . . . . . . . . . . . . . . . . . . 11 5.3. Notation for string operations . . . . . . . . . . . . . 11 5.4. Notation for group operations . . . . . . . . . . . . . . 13 6. The CPace protocol . . . . . . . . . . . . . . . . . . . . . 13 6.1. Protocol flow . . . . . . . . . . . . . . . . . . . . . . 13 6.2. CPace protocol instructions . . . . . . . . . . . . . . . 13 7. Implementation of recommended CPace cipher suites . . . . . . 14 7.1. Common function for computing generators . . . . . . . . 14 7.2. CPace group objects G_X25519 and G_X448 for single-coordinate Ladders on Montgomery curves . . . . . 15 7.2.1. Verification tests . . . . . . . . . . . . . . . . . 16 7.3. CPace group objects G_Ristretto255 and G_Decaf448 for prime-order group abstractions . . . . . . . . . . . . . 17 7.3.1. Verification tests . . . . . . . . . . . . . . . . . 19 7.4. CPace group objects for curves in Short-Weierstrass representation . . . . . . . . . . . . . . . . . . . . . 19 7.4.1. Curves and associated functions . . . . . . . . . . . 19 7.4.2. Suitable encode_to_curve methods . . . . . . . . . . 19 7.4.3. Definition of the group environment G for Short-Weierstrass curves . . . . . . . . . . . . . . 20 7.4.4. Verification tests . . . . . . . . . . . . . . . . . 21 8. Implementation verification . . . . . . . . . . . . . . . . . 22 9. Security Considerations . . . . . . . . . . . . . . . . . . . 22 Abdalla, et al. Expires 26 March 2024 [Page 2] Internet-Draft CPace September 2023 9.1. Party identifiers and relay attacks . . . . . . . . . . . 22 9.2. Network message encoding and hashing protocol transcripts . . . . . . . . . . . . . . . . . . . . . . . 23 9.3. Key derivation . . . . . . . . . . . . . . . . . . . . . 23 9.4. Key confirmation . . . . . . . . . . . . . . . . . . . . 23 9.5. Sampling of scalars . . . . . . . . . . . . . . . . . . . 24 9.6. Single-coordinate CPace on Montgomery curves . . . . . . 24 9.7. Nonce values . . . . . . . . . . . . . . . . . . . . . . 25 9.8. Side channel attacks . . . . . . . . . . . . . . . . . . 25 9.9. Quantum computers . . . . . . . . . . . . . . . . . . . . 26 10. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 26 11. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 26 12. References . . . . . . . . . . . . . . . . . . . . . . . . . 26 12.1. Normative References . . . . . . . . . . . . . . . . . . 26 12.2. Informative References . . . . . . . . . . . . . . . . . 27 Appendix A. CPace function definitions . . . . . . . . . . . . . 29 A.1. Definition and test vectors for string utility functions . . . . . . . . . . . . . . . . . . . . . . . . 29 A.1.1. prepend_len function . . . . . . . . . . . . . . . . 29 A.1.2. prepend_len test vectors . . . . . . . . . . . . . . 29 A.1.3. lv_cat function . . . . . . . . . . . . . . . . . . . 29 A.1.4. Testvector for lv_cat() . . . . . . . . . . . . . . . 30 A.1.5. Examples for messages not obtained from a lv_cat-based encoding . . . . . . . . . . . . . . . . . . . . . . 30 A.2. Definition of generator_string function. . . . . . . . . 30 A.3. Definitions and test vector ordered concatenation . . . . 30 A.3.1. Definitions for lexiographical ordering . . . . . . . 30 A.3.2. Definitions for ordered concatenation . . . . . . . . 31 A.3.3. Test vectors ordered concatenation . . . . . . . . . 31 A.4. Decoding and Encoding functions according to RFC7748 . . 31 A.5. Elligator 2 reference implementation . . . . . . . . . . 32 Appendix B. Test vectors . . . . . . . . . . . . . . . . . . . . 33 B.1. Test vector for CPace using group X25519 and hash SHA-512 . . . . . . . . . . . . . . . . . . . . . . . . . 33 B.1.1. Test vectors for calculate_generator with group X25519 . . . . . . . . . . . . . . . . . . . . . . . 33 B.1.2. Test vector for MSGa . . . . . . . . . . . . . . . . 33 B.1.3. Test vector for MSGb . . . . . . . . . . . . . . . . 34 B.1.4. Test vector for secret points K . . . . . . . . . . . 34 B.1.5. Test vector for ISK calculation initiator/ responder . . . . . . . . . . . . . . . . . . . . . . 34 B.1.6. Test vector for ISK calculation parallel execution . 35 B.1.7. Corresponding ANSI-C initializers . . . . . . . . . . 35 B.1.8. Test vectors for G_X25519.scalar_mult_vfy: low order points . . . . . . . . . . . . . . . . . . . . . . . 37 B.2. Test vector for CPace using group X448 and hash SHAKE-256 . . . . . . . . . . . . . . . . . . . . . . . . 38 Abdalla, et al. Expires 26 March 2024 [Page 3] Internet-Draft CPace September 2023 B.2.1. Test vectors for calculate_generator with group X448 . . . . . . . . . . . . . . . . . . . . . . . . 38 B.2.2. Test vector for MSGa . . . . . . . . . . . . . . . . 38 B.2.3. Test vector for MSGb . . . . . . . . . . . . . . . . 38 B.2.4. Test vector for secret points K . . . . . . . . . . . 39 B.2.5. Test vector for ISK calculation initiator/ responder . . . . . . . . . . . . . . . . . . . . . . 39 B.2.6. Test vector for ISK calculation parallel execution . 40 B.2.7. Corresponding ANSI-C initializers . . . . . . . . . . 40 B.2.8. Test vectors for G_X448.scalar_mult_vfy: low order points . . . . . . . . . . . . . . . . . . . . . . . 42 B.3. Test vector for CPace using group ristretto255 and hash SHA-512 . . . . . . . . . . . . . . . . . . . . . . . . . 43 B.3.1. Test vectors for calculate_generator with group ristretto255 . . . . . . . . . . . . . . . . . . . . 43 B.3.2. Test vector for MSGa . . . . . . . . . . . . . . . . 44 B.3.3. Test vector for MSGb . . . . . . . . . . . . . . . . 44 B.3.4. Test vector for secret points K . . . . . . . . . . . 45 B.3.5. Test vector for ISK calculation initiator/ responder . . . . . . . . . . . . . . . . . . . . . . 45 B.3.6. Test vector for ISK calculation parallel execution . 45 B.3.7. Corresponding ANSI-C initializers . . . . . . . . . . 46 B.3.8. Test case for scalar_mult with valid inputs . . . . . 47 B.3.9. Invalid inputs for scalar_mult_vfy . . . . . . . . . 48 B.4. Test vector for CPace using group decaf448 and hash SHAKE-256 . . . . . . . . . . . . . . . . . . . . . . . . 48 B.4.1. Test vectors for calculate_generator with group decaf448 . . . . . . . . . . . . . . . . . . . . . . 48 B.4.2. Test vector for MSGa . . . . . . . . . . . . . . . . 49 B.4.3. Test vector for MSGb . . . . . . . . . . . . . . . . 49 B.4.4. Test vector for secret points K . . . . . . . . . . . 50 B.4.5. Test vector for ISK calculation initiator/ responder . . . . . . . . . . . . . . . . . . . . . . 50 B.4.6. Test vector for ISK calculation parallel execution . 51 B.4.7. Corresponding ANSI-C initializers . . . . . . . . . . 51 B.4.8. Test case for scalar_mult with valid inputs . . . . . 53 B.4.9. Invalid inputs for scalar_mult_vfy . . . . . . . . . 53 B.5. Test vector for CPace using group NIST P-256 and hash SHA-256 . . . . . . . . . . . . . . . . . . . . . . . . . 53 B.5.1. Test vectors for calculate_generator with group NIST P-256 . . . . . . . . . . . . . . . . . . . . . . . . 53 B.5.2. Test vector for MSGa . . . . . . . . . . . . . . . . 54 B.5.3. Test vector for MSGb . . . . . . . . . . . . . . . . 54 B.5.4. Test vector for secret points K . . . . . . . . . . . 55 B.5.5. Test vector for ISK calculation initiator/ responder . . . . . . . . . . . . . . . . . . . . . . 55 B.5.6. Test vector for ISK calculation parallel execution . 56 B.5.7. Corresponding ANSI-C initializers . . . . . . . . . . 57 Abdalla, et al. Expires 26 March 2024 [Page 4] Internet-Draft CPace September 2023 B.5.8. Test case for scalar_mult_vfy with correct inputs . . 58 B.5.9. Invalid inputs for scalar_mult_vfy . . . . . . . . . 59 B.6. Test vector for CPace using group NIST P-384 and hash SHA-384 . . . . . . . . . . . . . . . . . . . . . . . . . 59 B.6.1. Test vectors for calculate_generator with group NIST P-384 . . . . . . . . . . . . . . . . . . . . . . . . 59 B.6.2. Test vector for MSGa . . . . . . . . . . . . . . . . 60 B.6.3. Test vector for MSGb . . . . . . . . . . . . . . . . 61 B.6.4. Test vector for secret points K . . . . . . . . . . . 61 B.6.5. Test vector for ISK calculation initiator/ responder . . . . . . . . . . . . . . . . . . . . . . 61 B.6.6. Test vector for ISK calculation parallel execution . 62 B.6.7. Corresponding ANSI-C initializers . . . . . . . . . . 63 B.6.8. Test case for scalar_mult_vfy with correct inputs . . 65 B.6.9. Invalid inputs for scalar_mult_vfy . . . . . . . . . 65 B.7. Test vector for CPace using group NIST P-521 and hash SHA-512 . . . . . . . . . . . . . . . . . . . . . . . . . 66 B.7.1. Test vectors for calculate_generator with group NIST P-521 . . . . . . . . . . . . . . . . . . . . . . . . 66 B.7.2. Test vector for MSGa . . . . . . . . . . . . . . . . 66 B.7.3. Test vector for MSGb . . . . . . . . . . . . . . . . 67 B.7.4. Test vector for secret points K . . . . . . . . . . . 68 B.7.5. Test vector for ISK calculation initiator/ responder . . . . . . . . . . . . . . . . . . . . . . 68 B.7.6. Test vector for ISK calculation parallel execution . 69 B.7.7. Corresponding ANSI-C initializers . . . . . . . . . . 70 B.7.8. Test case for scalar_mult_vfy with correct inputs . . 72 B.7.9. Invalid inputs for scalar_mult_vfy . . . . . . . . . 73 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 73 1. Introduction This document describes CPace which is a balanced Password- Authenticated-Key-Establishment (PAKE) protocol for two parties where both parties derive a cryptographic key of high entropy from a shared secret of low-entropy. CPace protects the passwords against offline dictionary attacks by requiring adversaries to actively interact with a protocol party and by allowing for at most one single password guess per active interaction. The CPace design was tailored considering the following main objectives: * Efficiency: Deployment of CPace is feasible on resource- constrained devices. Abdalla, et al. Expires 26 March 2024 [Page 5] Internet-Draft CPace September 2023 * Versatility: CPace supports different application scenarios via versatile input formats, and by supporting applications with and without clear initiator and responder roles. * Implementation error resistance: CPace aims at avoiding common implementation pitfalls already by-design, such as avoiding incentives for insecure execution-time speed optimizations. For smooth integration into different cryptographic library ecosystems, this document provides a variety of cipher suites. * Post-quantum annoyance: CPace comes with mitigations with respect to adversaries that become capable of breaking the discrete logarithm problem on elliptic curves. 1.1. Outline of this document * Section 3 describes the expected properties of an application using CPace, and discusses in particular which application-level aspects are relevant for CPace's security. * Section 4 gives an overview over the recommended cipher suites for CPace which were optimized for different types of cryptographic library ecosystems. * Section 5 introduces the notation used throughout this document. * Section 6 specifies the CPace protocol. * The final section provides explicit reference implementations and test vectors of all of the functions defined for CPace in the appendix. As this document is primarily written for implementers and application designers, we would like to refer the theory-inclined reader to the scientific paper [AHH21] which covers the detailed security analysis of the different CPace instantiations as defined in this document via the cipher suites. 2. Requirements Notation The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all capitals, as shown here. Abdalla, et al. Expires 26 March 2024 [Page 6] Internet-Draft CPace September 2023 3. High-level application perspective CPace enables balanced password-authenticated key establishment. CPace requires a shared secret octet string, the password-related string (PRS), is available for both parties A and B. PRS can be a low-entropy secret itself, for instance a clear-text password encoded according to [RFC8265], or any string derived from a common secret, for instance by use of a password-based key derivation function. Applications with clients and servers where the server side is storing account and password information in its persistent memory are recommended to use augmented PAKE protocols such as OPAQUE [I-D.irtf-cfrg-opaque]. In the course of the CPace protocol, A sends one message MSGa to B and B sends one message MSGb to A. CPace does not mandate any ordering of these two messages. We use the term "initiator- responder" for CPace where A always speaks first, and the term "symmetric" setting where anyone can speak first. CPace's output is an intermediate session key (ISK), but any party might abort in case of an invalid received message. A and B will produce the same ISK value only if both sides did initiate the protocol using the same protocol inputs, specifically the same PRS string and the same value for the optional input parameters CI, ADa, ADb and sid that will be specified in the upcoming sections. The naming of ISK key as "intermediate" session key highlights the fact that it is RECOMMENDED that applications process ISK by use of a suitable strong key derivation function KDF (such as defined in [RFC5869]) before using the key in a higher-level protocol. 3.1. Optional CPace inputs For accomodating different application settings, CPace offers the following OPTIONAL inputs, i.e. inputs which MAY also be the empty string: * Channel identifier (CI). CI can be used to bind a session key exchanged with CPace to a specific networking channel which interconnects the protocol parties. Both parties are required to have the same view of CI. CI will not be publicly sent on the wire and may also include confidential information. * Associated data fields (ADa and ADb). These fields can be used to authenticate public associated data alongside the CPace protocol. The values ADa (and ADb, respectively) are guaranteed to be authenticated in case both parties agree on a key. Abdalla, et al. Expires 26 March 2024 [Page 7] Internet-Draft CPace September 2023 ADa and ADb can for instance include party identities or protocol version information of an application protocol (e.g. to avoid downgrade attacks). If party identities are not encoded as part of CI, party identities SHOULD be included in ADa and ADb (see Section 9.1). In a setting with clear initiator and responder roles, identity information in ADa sent by the initiator can be used by the responder for choosing the right PRS string (respectively password) for this identity. * Session identifier (sid). CPace comes with a security analysis [AHH21] in the framework of universal composability. This framework allows for modular analysis of a larger application protocol which uses CPace as a building block. For such analysis the CPace protocol is bound to a specific session of the larger protocol by use of a sid string that is globally unique. As a result, when used with a unique sid, CPace instances remain secure when running concurrently with other CPace instances, and even arbitrary other protocols. For this reason, it is RECOMMENDED that applications establish a unique session identifier sid prior to running the CPace protocol. This can be implemented by concatenating random bytes produced by A with random bytes produced by B. If such preceding round is not an option but parties are assigned clear initiator-responder roles, it is RECOMMENDED to let the initiator A choose a fresh random sid and send it to B together with the first message. If a sid string is used it SHOULD HAVE a length of at least 8 bytes. 3.2. Responsibilities of the application layer The following tasks are out of the scope of this document and left to the application layer * Setup phase: - The application layer is responsible for the handshake that makes parties agree on a common CPace cipher suite. Abdalla, et al. Expires 26 March 2024 [Page 8] Internet-Draft CPace September 2023 - The application layer needs to specify how to encode the CPace byte strings Ya / Yb and ADa / ADb defined in section Section 6 for transfer over the network. For CPace it is RECOMMENDED to encode network messages by using MSGa = lv_cat(Ya,ADa) and MSGb = lv_cat(Yb,ADb) using the length-value concatenation function lv_cat speficied in Section 5.3. This document provides test vectors for lv_cat-encoded messages. Alternative network encodings, e.g., the encoding method used for the client hello and server hello messages of the TLS protocol, MAY be used when considering the guidance given in Section 9. * This document does not specify which encodings applications use for the mandatory PRS input and the optional inputs CI, sid, ADa and ADb. If PRS is a clear-text password or an octet string derived from a clear-text password, e.g. by use of a key- derivation function, the clear-text password SHOULD BE encoded according to [RFC8265]. * The application needs to settle whether CPace is used in the initiator-responder or the symmetric setting, as in the symmetric setting transcripts must be generated using ordered string concatenation. In this document we will provide test vectors for both, initiator-responder and symmetric settings. 4. CPace cipher suites In the setup phase of CPace, both communication partners need to agree on a common cipher suite. Cipher suites consist of a combination of a hash function H and an elliptic curve environment G. For naming cipher suites we use the convention "CPACE-G-H". We RECOMMEND the following cipher suites: * CPACE-X25519-SHA512. This suite uses the group environment G_X25519 defined in Section 7.2 and SHA-512 as hash function. This cipher suite comes with the smallest messages on the wire and a low computational cost. * CPACE-P256_XMD:SHA-256_SSWU_NU_-SHA256. This suite instantiates the group environment G as specified in Section 7.4 using the encode_to_curve function P256_XMD:SHA-256_SSWU_NU_ from [RFC9380] on curve NIST-P256, and hash function SHA-256. The following RECOMMENDED cipher suites provide higher security margins. * CPACE-X448-SHAKE256. This suite uses the group environment G_X448 defined in Section 7.2 and SHAKE-256 as hash function. Abdalla, et al. Expires 26 March 2024 [Page 9] Internet-Draft CPace September 2023 * CPACE-P384_XMD:SHA-384_SSWU_NU_-SHA384. This suite instantiates G as specified in Section 7.4 using the encode_to_curve function P384_XMD:SHA-384_SSWU_NU_ from [RFC9380] on curve NIST-P384 with H = SHA-384. * CPACE-P521_XMD:SHA-512_SSWU_NU_-SHA512. This suite instantiates G as specified in Section 7.4 using the encode_to_curve function P521_XMD:SHA-512_SSWU_NU_ from [RFC9380] on curve NIST-P521 with H = SHA-512. CPace can also securely be implemented using the cipher suites CPACE- RISTR255-SHA512 and CPACE-DECAF448-SHAKE256 defined in Section 7.3. Section 9 gives guidance on how to implement CPace on further elliptic curves. 5. Definitions and notation 5.1. Hash function H Common choices for H are SHA-512 [RFC6234] or SHAKE-256 [FIPS202]. (I.e. the hash function outputs octet strings, and not group elements.) For considering both variable-output-length hashes and fixed-output-length hashes, we use the following convention. In case that the hash function is specified for a fixed-size output, we define H.hash(m,l) such that it returns the first l octets of the output. We use the following notation for referring to the specific properties of a hash function H: * H.hash(m,l) is a function that operates on an input octet string m and returns a hashing result of l octets. * H.b_in_bytes denotes the default output size in bytes corresponding to the symmetric security level of the hash function. E.g. H.b_in_bytes = 64 for SHA-512 and SHAKE-256 and H.b_in_bytes = 32 for SHA-256 and SHAKE-128. We use the notation H.hash(m) = H.hash(m, H.b_in_bytes) and let the hash operation output the default length if no explicit length parameter is given. * H.bmax_in_bytes denotes the _maximum_ output size in octets supported by the hash function. In case of fixed-size hashes such as SHA-256, this is the same as H.b_in_bytes, while there is no such limit for hash functions such as SHAKE-256. Abdalla, et al. Expires 26 March 2024 [Page 10] Internet-Draft CPace September 2023 * H.s_in_bytes denotes the _input block size_ used by H. For instance, for SHA-512 the input block size s_in_bytes is 128, while for SHAKE-256 the input block size amounts to 136 bytes. 5.2. Group environment G The group environment G specifies an elliptic curve group (also denoted G for convenience) and associated constants and functions as detailed below. In this document we use multiplicative notation for the group operation. * G.calculate_generator(H,PRS,CI,sid) denotes a function that outputs a representation of a generator (referred to as "generator" from now on) of the group which is derived from input octet strings PRS, CI, and sid and with the help of the hash function H. * G.sample_scalar() is a function returning a representation of a scalar (referred to as "scalar" from now on) appropriate as a private Diffie-Hellman key for the group. * G.scalar_mult(y,g) is a function operating on a scalar y and a group element g. It returns an octet string representation of the group element Y = g^y. * G.I denotes a unique octet string representation of the neutral element of the group. G.I is used for detecting and signaling certain error conditions. * G.scalar_mult_vfy(y,g) is a function operating on a scalar y and a group element g. It returns an octet string representation of the group element g^y. Additionally, scalar_mult_vfy specifies validity conditions for y,g and g^y and outputs G.I in case they are not met. * G.DSI denotes a domain-separation identifier string which SHALL be uniquely identifying the group environment G. 5.3. Notation for string operations * bytes1 || bytes2 and denotes concatenation of octet strings. * len(S) denotes the number of octets in a string S. * nil denotes an empty octet string, i.e., len(nil) = 0. Abdalla, et al. Expires 26 March 2024 [Page 11] Internet-Draft CPace September 2023 * prepend_len(octet_string) denotes the octet sequence that is obtained from prepending the length of the octet string to the string itself. The length shall be prepended by using an LEB128 encoding of the length. This will result in a single-byte encoding for values below 128. (Test vectors and reference implementations for prepend_len and the LEB128 encodings are given in the appendix.) * lv_cat(a0,a1, ...) is the "length-value" encoding function which returns the concatenation of the input strings with an encoding of their respective length prepended. E.g. lv_cat(a0,a1) returns prepend_len(a0) || prepend_len(a1). The detailed specification of lv_cat and a reference implementations are given in the appendix. * network_encode(Y,AD) denotes the function specified by the application layer that outputs an octet string encoding of the input octet strings Y and AD for transfer on the network. The implementation of MSG = network_encode(Y,AD) SHALL allow the receiver party to parse MSG for the individual subcomponents Y and AD. For CPace we RECOMMEND to implement network_encode(Y,AD) as network_encode(Y,AD) = lv_cat(Y,AD). Other encodings, such as the network encoding used for the client- hello messages in TLS MAY also be used when following the guidance given in the security consideration section. * sample_random_bytes(n) denotes a function that returns n octets uniformly distributed between 0 and 255. * zero_bytes(n) denotes a function that returns n octets with value 0. * oCat(bytes1,bytes2) denotes ordered concatenation of octet strings, which places the lexiographically larger octet string first. (Explicit reference code for this function is given in the appendix.) * transcript(MSGa,MSGb) denotes function outputing a string for the protocol transcript with messages MSGa and MSGb. In applications where CPace is used without clear initiator and responder roles, i.e. where the ordering of messages is not enforced by the protocol flow, transcript(MSGa,MSGb) = oCat(MSGa,MSGb) SHOULD be used. In the initiator-responder setting transcript(MSGa,MSGb) SHOULD BE implemented such that the later message is appended to the earlier message, i.e., transcript(MSGa,MSGb) = MSGa||MSGb if MSGa is sent first. Abdalla, et al. Expires 26 March 2024 [Page 12] Internet-Draft CPace September 2023 5.4. Notation for group operations We use multiplicative notation for the group, i.e., X^2 denotes the element that is obtained by computing X*X, for group element X and group operation *. 6. The CPace protocol CPace is a one round protocol between two parties, A and B. At invocation, A and B are provisioned with PRS,G,H and OPTIONAL CI,sid,ADa (for A) and CI,sid,ADb (for B). A sends a message MSGa to B. MSGa contains the public share Ya and OPTIONAL associated data ADa (i.e. an ADa field that MAY have a length of 0 bytes). Likewise, B sends a message MSGb to A. MSGb contains the public share Yb and OPTIONAL associated data ADb (i.e. an ADb field that MAY have a length of 0 bytes). Both A and B use the received messages for deriving a shared intermediate session key, ISK. 6.1. Protocol flow Optional parameters and messages are denoted with []. public: G, H, [CI], [sid] A: PRS,[ADa] B: PRS,[ADb] --------------------------------------- compute Ya | Ya,[ADa] | compute Yb |----------------->| | Yb,[ADb] | verify inputs |<-----------------| verify inputs derive ISK | | derive ISK --------------------------------------- output ISK output ISK 6.2. CPace protocol instructions A computes a generator g = G.calculate_generator(H,PRS,CI,sid), scalar ya = G.sample_scalar() and group element Ya = G.scalar_mult (ya,g). A then transmits MSGa = network_encode(Ya, ADa) with optional associated data ADa to B. B computes a generator g = G.calculate_generator(H,PRS,CI,sid), scalar yb = G.sample_scalar() and group element Yb = G.scalar_mult(yb,g). B sends MSGb = network_encode(Yb, ADb) with optional associated data ADb to A. Abdalla, et al. Expires 26 March 2024 [Page 13] Internet-Draft CPace September 2023 Upon reception of MSGa, B checks that MSGa was properly generated conform with the chosen encoding of network messages (notably correct length fields). If this parsing fails, then B MUST abort. (Testvectors of examples for invalid messages when using lv_cat() as network_encode function for CPace are given in the appendix.) B then computes K = G.scalar_mult_vfy(yb,Ya). B MUST abort if K=G.I. Otherwise B returns ISK = H.hash(prefix_free_cat(G.DSI || "_ISK", sid, K)||transcript(MSGa, MSGb)). B returns ISK and terminates. Likewise upon reception of MSGb, A parses MSGb for Yb and ADb and checks for a valid encoding. If this parsing fails, then A MUST abort. A then computes K = G.scalar_mult_vfy(ya,Yb). A MUST abort if K=G.I. Otherwise A returns ISK = H.hash(lv_cat(G.DSI || "_ISK", sid, K) || transcript(MSGa, MSGb)). A returns ISK and terminates. The session key ISK returned by A and B is identical if and only if the supplied input parameters PRS, CI and sid match on both sides and transcript view (containing of MSGa and MSGb) of both parties match. Note that in case of a symmetric protocol execution without clear initiator/responder roles, ordered concatenation needs to be used for generating a matching view of the transcript by both parties. 7. Implementation of recommended CPace cipher suites 7.1. Common function for computing generators The different cipher suites for CPace defined in the upcoming sections share the same method for deterministically combining the individual strings PRS, CI, sid and the domain-separation identifier DSI to a generator string that we describe here. * generator_string(DSI, PRS, CI, sid, s_in_bytes) denotes a function that returns the string lv_cat(DSI, PRS, zero_bytes(len_zpad), CI, sid). * len_zpad = MAX(0, s_in_bytes - len(prepend_len(PRS)) - len(prepend_len(G.DSI)) - 1) The zero padding of length len_zpad is designed such that the encoding of DSI and PRS together with the zero padding field completely fills the first input block (of length s_in_bytes) of the hash. As a result for the common case of short PRS the number of bytes to hash becomes independent of the actual length of the password (PRS). (A reference implementation and test vectors are provided in the appendix.) Abdalla, et al. Expires 26 March 2024 [Page 14] Internet-Draft CPace September 2023 The introduction of a zero-padding within the generator string also helps mitigating attacks of a side-channel adversary that analyzes correlations between publicly known variable information with the low-entropy PRS string. Note that the hash of the first block is intentionally made independent of session-specific inputs, such as sid or CI. 7.2. CPace group objects G_X25519 and G_X448 for single-coordinate Ladders on Montgomery curves In this section we consider the case of CPace when using the X25519 and X448 Diffie-Hellman functions from [RFC7748] operating on the Montgomery curves Curve25519 and Curve448 [RFC7748]. CPace implementations using single-coordinate ladders on further Montgomery curves SHALL use the definitions in line with the specifications for X25519 and X448 and review the guidance given in Section 9. For the group environment G_X25519 the following definitions apply: * G_X25519.field_size_bytes = 32 * G_X25519.field_size_bits = 255 * G_X25519.sample_scalar() = sample_random_bytes(G.field_size_bytes) * G_X25519.scalar_mult(y,g) = G.scalar_mult_vfy(y,g) = X25519(y,g) * G_X25519.I = zero_bytes(G.field_size_bytes) * G_X25519.DSI = "CPace255" CPace cipher suites using G_X25519 MUST use a hash function producing at least H.b_max_in_bytes >= 32 bytes of output. It is RECOMMENDED to use G_X25519 in combination with SHA-512. For X448 the following definitions apply: * G_X448.field_size_bytes = 56 * G_X448.field_size_bits = 448 * G_X448.sample_scalar() = sample_random_bytes(G.field_size_bytes) * G_X448.scalar_mult(y,g) = G.scalar_mult_vfy(y,g) = X448(y,g) * G_X448.I = zero_bytes(G.field_size_bytes) * G_X448.DSI = "CPace448" Abdalla, et al. Expires 26 March 2024 [Page 15] Internet-Draft CPace September 2023 CPace cipher suites using G_X448 MUST use a hash function producing at least H.b_max_in_bytes >= 56 bytes of output. It is RECOMMENDED to use G_X448 in combination with SHAKE-256. For both G_X448 and G_X25519 the G.calculate_generator(H, PRS,sid,CI) function shall be implemented as follows. * First gen_str = generator_string(G.DSI,PRS,CI,sid, H.s_in_bytes) SHALL BE calculated using the input block size of the chosen hash function. * This string SHALL then BE hashed to the required length gen_str_hash = H.hash(gen_str, G.field_size_bytes). Note that this implies that the permissible output length H.maxb_in_bytes MUST BE larger or equal to the field size of the group G for making a hashing function suitable. * This result is then considered as a field coordinate using the u = decodeUCoordinate(gen_str_hash, G.field_size_bits) function from [RFC7748] which we repeat in the appendix for convenience. * The result point g is then calculated as (g,v) = map_to_curve_elligator2(u) using the function from [RFC9380]. Note that the v coordinate produced by the map_to_curve_elligator2 function is not required for CPace and discarded. The appendix repeats the definitions from [RFC9380] for convenience. In the appendix we show sage code that can be used as reference implementation. 7.2.1. Verification tests For single-coordinate Montgomery ladders on Montgomery curves verification tests according to Section 8 SHALL consider the u coordinate values that encode a low-order point on either, the curve or the quadratic twist. In addition to that in case of G_X25519 the tests SHALL also verify that the implementation of G.scalar_mult_vfy(y,g) produces the expected results for non-canonical u coordinate values with bit #255 set, which also encode low-order points. Corresponding test vectors are provided in the appendix. Abdalla, et al. Expires 26 March 2024 [Page 16] Internet-Draft CPace September 2023 7.3. CPace group objects G_Ristretto255 and G_Decaf448 for prime-order group abstractions In this section we consider the case of CPace using the Ristretto255 and Decaf448 group abstractions [I-D.draft-irtf-cfrg-ristretto255-decaf448]. These abstractions define an encode and decode function, group operations using an internal encoding and a one-way-map. With the group abstractions there is a distinction between an internal representation of group elements and an external encoding of the same group element. In order to distinguish between these different representations, we prepend an underscore before values using the internal representation within this section. For Ristretto255 the following definitions apply: * G_Ristretto255.DSI = "CPaceRistretto255" * G_Ristretto255.field_size_bytes = 32 * G_Ristretto255.group_size_bits = 252 * G_Ristretto255.group_order = 2^252 + 27742317777372353535851937790883648493 CPace cipher suites using G_Ristretto255 MUST use a hash function producing at least H.b_max_in_bytes >= 64 bytes of output. It is RECOMMENDED to use G_Ristretto255 in combination with SHA-512. For decaf448 the following definitions apply: * G_Decaf448.DSI = "CPaceDecaf448" * G_Decaf448.field_size_bytes = 56 * G_Decaf448.group_size_bits = 445 * G_Decaf448.group_order = l = 2^446 - 1381806680989511535200738674851542 6880336692474882178609894547503885 CPace cipher suites using G_Decaf448 MUST use a hash function producing at least H.b_max_in_bytes >= 112 bytes of output. It is RECOMMENDED to use G_Decaf448 in combination with SHAKE-256. For both abstractions the following definitions apply: * It is RECOMMENDED to implement G.sample_scalar() as follows. Abdalla, et al. Expires 26 March 2024 [Page 17] Internet-Draft CPace September 2023 - Set scalar = sample_random_bytes(G.group_size_bytes). - Then clear the most significant bits larger than G.group_size_bits. - Interpret the result as the little-endian encoding of an integer value and return the result. * Alternatively, if G.sample_scalar() is not implemented according to the above recommendation, it SHALL be implemented using uniform sampling between 1 and (G.group_order - 1). Note that the more complex uniform sampling process can provide a larger side-channel attack surface for embedded systems in hostile environments. * G.scalar_mult(y,_g) SHALL operate on a scalar y and a group element _g in the internal representation of the group abstraction environment. It returns the value Y = encode((_g)^y), i.e. it returns a value using the public encoding. * G.I = is the public encoding representation of the identity element. * G.scalar_mult_vfy(y,X) operates on a value using the public encoding and a scalar and is implemented as follows. If the decode(X) function fails, it returns G.I. Otherwise it returns encode( decode(X)^y ). * The G.calculate_generator(H, PRS,sid,CI) function SHALL return a decoded point and SHALL BE implemented as follows. - First gen_str = generator_string(G.DSI,PRS,CI,sid, H.s_in_bytes) is calculated using the input block size of the chosen hash function. - This string is then hashed to the required length gen_str_hash = H.hash(gen_str, 2 * G.field_size_bytes). Note that this implies that the permissible output length H.maxb_in_bytes MUST BE larger or equal to twice the field size of the group G for making a hash function suitable. - Finally the internal representation of the generator _g is calculated as _g = one_way_map(gen_str_hash) using the element derivation function from the abstraction. Note that with these definitions the scalar_mult function operates on a decoded point _g and returns an encoded point, while the scalar_mult_vfy(y,X) function operates on an encoded point X (and also returns an encoded point). Abdalla, et al. Expires 26 March 2024 [Page 18] Internet-Draft CPace September 2023 7.3.1. Verification tests For group abstractions verification tests according to Section 8 SHALL consider encodings of the neutral element and an octet string that does not decode to a valid group element. 7.4. CPace group objects for curves in Short-Weierstrass representation The group environment objects G defined in this section for use with Short-Weierstrass curves, are parametrized by the choice of an elliptic curve and by choice of a suitable encode_to_curve function. encode_to_curve must map an octet string to a point on the curve. 7.4.1. Curves and associated functions Elliptic curves in Short-Weierstrass form are considered in [IEEE1363]. [IEEE1363] allows for both, curves of prime and non- prime order. However, for the procedures described in this section any suitable group MUST BE of prime order. The specification for the group environment objects specified in this section closely follow the ECKAS-DH1 method from [IEEE1363]. I.e. we use the same methods and encodings and protocol substeps as employed in the TLS [RFC5246] [RFC8446] protocol family. For CPace only the uncompressed full-coordinate encodings from [SEC1] (x and y coordinate) SHOULD be used. Commonly used curve groups are specified in [SEC2] and [RFC5639]. A typical representative of such a Short-Weierstrass curve is NIST-P256. Point verification as used in ECKAS-DH1 is described in Annex A.16.10. of [IEEE1363]. For deriving Diffie-Hellman shared secrets ECKAS-DH1 from [IEEE1363] specifies the use of an ECSVDP-DH method. We use ECSVDP-DH in combination with the identy map such that it either returns "error" or the x-coordinate of the Diffie-Hellman result point as shared secret in big endian format (fixed length output by FE2OSP without truncating leading zeros). 7.4.2. Suitable encode_to_curve methods All the encode_to_curve methods specified in [RFC9380] are suitable for CPace. For Short-Weierstrass curves it is RECOMMENDED to use the non-uniform variant of the SSWU mapping primitive from [RFC9380] if a SSWU mapping is available for the chosen curve. (We recommend non- uniform maps in order to give implementations the flexibility to opt for x-coordinate-only scalar multiplication algorithms.) Abdalla, et al. Expires 26 March 2024 [Page 19] Internet-Draft CPace September 2023 7.4.3. Definition of the group environment G for Short-Weierstrass curves In this paragraph we use the following notation for defining the group object G for a selected curve and encode_to_curve method: * With G.group_order we denote the order of the elliptic curve which MUST BE a prime. * With is_valid(X) we denote a method which operates on an octet stream according to [SEC1] of a point on the group and returns true if the point is valid and returns false otherwise. This is_valid(X) method SHALL be implemented according to Annex A.16.10. of [IEEE1363]. I.e. it shall return false if X encodes either the neutral element on the group or does not form a valid encoding of a point on the group. * With encode_to_curve(str,DST) we denote mapping function from [RFC9380]. I.e. a function that maps octet string str to a point on the group using the domain separation tag DST. [RFC9380] considers both, uniform and non-uniform mappings based on several different strategies. It is RECOMMENDED to use the nonuniform variant of the SSWU mapping primitive within [RFC9380]. * G.DSI denotes a domain-separation identifier string. G.DSI which SHALL BE obtained by the concatenation of "CPace" and the associated name of the cipher suite used for the encode_to_curve function as specified in [RFC9380]. E.g. when using the map with the name "P384_XMD:SHA-384_SSWU_NU_" on curve NIST-P384 the resulting value SHALL BE G.DSI = "CPaceP384_XMD:SHA-384_SSWU_NU_". Using the above definitions, the CPace functions required for the group object G are defined as follows. * G.DST denotes the domain-separation tag value to use in conjunction with the encode_to_curve function from [RFC9380]. G.DST shall be obtained by concatenating G.DSI and "_DST". * G.sample_scalar() SHALL return a value between 1 and (G.group_order - 1). The value sampling MUST BE uniformly random. It is RECOMMENDED to use rejection sampling for converting a uniform bitstring to a uniform value between 1 and (G.group_order - 1). * G.calculate_generator(H, PRS,sid,CI) function SHALL be implemented as follows. Abdalla, et al. Expires 26 March 2024 [Page 20] Internet-Draft CPace September 2023 - First gen_str = generator_string(G.DSI,PRS,CI,sid, H.s_in_bytes) is calculated. - Then the output of a call to encode_to_curve(gen_str, G.DST) is returned, using the selected suite from [RFC9380]. * G.scalar_mult(s,X) is a function that operates on a scalar s and an input point X. The input X shall use the same encoding as produced by the G.calculate_generator method above. G.scalar_mult(s,X) SHALL return an encoding of either the point X^s or the point X^(-s) according to [SEC1]. Implementations SHOULD use the full-coordinate format without compression, as important protocols such as TLS 1.3 removed support for compression. Implementations of scalar_mult(s,X) MAY output either X^s or X^(-s) as both points X^s and X^(-s) have the same x-coordinate and result in the same Diffie-Hellman shared secrets K. (This allows implementations to opt for x-coordinate-only scalar multiplication algorithms.) * G.scalar_mult_vfy(s,X) merges verification of point X according to [IEEE1363] A.16.10. and the the ECSVDP-DH procedure from [IEEE1363]. It SHALL BE implemented as follows: - If is_valid(X) = False then G.scalar_mult_vfy(s,X) SHALL return "error" as specified in [IEEE1363] A.16.10 and 7.2.1. - Otherwise G.scalar_mult_vfy(s,X) SHALL return the result of the ECSVDP-DH procedure from [IEEE1363] (section 7.2.1). I.e. it shall either return "error" (in case that X^s is the neutral element) or the secret shared value "z" (otherwise). "z" SHALL be encoded by using the big-endian encoding of the x-coordinate of the result point X^s according to [SEC1]. * We represent the neutral element G.I by using the representation of the "error" result case from [IEEE1363] as used in the G.scalar_mult_vfy method above. 7.4.4. Verification tests For Short-Weierstrass curves verification tests according to Section 8 SHALL consider encodings of the point at infinity and an encoding of a point not on the group. Abdalla, et al. Expires 26 March 2024 [Page 21] Internet-Draft CPace September 2023 8. Implementation verification Any CPace implementation MUST be tested against invalid or weak point attacks. Implementation MUST be verified to abort upon conditions where G.scalar_mult_vfy functions outputs G.I. For testing an implementation it is RECOMMENDED to include weak or invalid point encodings within MSGa and MSGb and introduce this in a protocol run. It SHALL be verified that the abort condition is properly handled. Moreover any implementation MUST be tested with respect invalid encodings of MSGa and MSGb. E.g. when lv_cat is used for encoding MSGa and MSGb, the sum of the prepended lengths of the fields must be verified to match the actual length of the message. Corresponding test vectors are given in the appendix for all recommended cipher suites. 9. Security Considerations A security proof of CPace is found in [AHH21]. This proof covers all recommended cipher suites included in this document. In the following sections we describe how to protect CPace against several attack families, such as relay-, length extension- or side channel attacks. We also describe aspects to consider when deviating from recommended cipher suites. 9.1. Party identifiers and relay attacks If unique strings identifying the protocol partners are included either as part of the channel identifier CI, the session id sid or the associated data fields ADa, ADb, the ISK will provide implicit authentication also regarding the party identities. Incorporating party identifier strings is important for fending off relay attacks. Such attacks become relevant in a setting where several parties, say, A, B and C, share the same password PRS. An adversary might relay messages from a honest user A, who aims at interacting with user B, to a party C instead. If no party identifier strings are used, and B and C use the same PRS value, A might be establishing a common ISK key with C while assuming to interact with party B. Including and checking party identifiers can fend off such relay attacks. Abdalla, et al. Expires 26 March 2024 [Page 22] Internet-Draft CPace September 2023 9.2. Network message encoding and hashing protocol transcripts It is RECOMMENDED to encode the (Ya,ADa) and (Yb,ADb) fields on the network by using network_encode(Y,AD) = lv_cat(Y,AD). I.e. we RECOMMEND to prepend an encoding of the length of the subfields. Prepending the length of of all variable-size input strings results in a so-called prefix-free encoding of transcript strings, using terminology introduced in [CDMP05]. This property allows for disregarding length-extension imperfections that come with the commonly used Merkle-Damgard hash function constructions such as SHA256 and SHA512. Other alternative network encoding formats which prepend an encoding of the length of variable-size data fields in the protocol messages are equally suitable. This includes, e.g., the type-length-value format specified in the DER encoding standard (X.690) or the protocol message encoding used in the TLS protocol family for the TLS client- hello or server-hello messages. In case that an application uses another form of network message encoding which is not prefix-free, the guidance given in [CDMP05] SHOULD BE considered (e.g. by replacing hash functions with the HMAC constructions from[RFC2104]). 9.3. Key derivation Although already K is a shared value, it MUST NOT itself be used as an application key. Instead, ISK MUST BE used. Leakage of K to an adversary can lead to offline dictionary attacks. As noted already in Section 6 it is RECOMMENDED to process ISK by use of a suitable strong key derivation function KDF (such as defined in [RFC5869]) first, before using the key in a higher-level protocol. 9.4. Key confirmation In many applications it is advisable to add an explicit key confirmation round after the CPace protocol flow. However, as some applications might only require implicit authentication and as explicit authentication messages are already a built-in feature in many higher-level protocols (e.g. TLS 1.3) the CPace protocol described here does not mandate use of a key confirmation on the level of the CPace sub-protocol. Already without explicit key confirmation, CPace enjoys weak forward security under the sCDH and sSDH assumptions [AHH21]. With added explicit confirmation, CPace enjoys perfect forward security also under the strong sCDH and sSDH assumptions [AHH21]. Abdalla, et al. Expires 26 March 2024 [Page 23] Internet-Draft CPace September 2023 Note that in [ABKLX21] it was shown that an idealized variant of CPace also enjoys perfect forward security without explicit key confirmation. However this proof does not explicitly cover the recommended cipher suites in this document and requires the stronger assumption of an algebraic adversary model. For this reason, we recommend adding explicit key confirmation if perfect forward security is required. When implementing explicit key confirmation, it is recommended to use an appropriate message-authentication code (MAC) such as HMAC [RFC2104] or CMAC [RFC4493] using a key mac_key derived from ISK. One suitable option that works also in the parallel setting without message ordering is to proceed as follows. * First calculate mac_key as as mac_key = H.hash(b"CPaceMac" || ISK). * Then let each party send an authenticator tag Ta, Tb that is calculated over the protocol message that it has sent previously. I.e. let party A calculate its transmitted authentication code Ta as Ta = MAC(mac_key, MSGa) and let party B calculate its transmitted authentication code Tb as Tb = MAC(mac_key, MSGb). * Let the receiving party check the remote authentication tag for the correct value and abort in case that it's incorrect. 9.5. Sampling of scalars For curves over fields F_p where p is a prime close to a power of two, we recommend sampling scalars as a uniform bit string of length field_size_bits. We do so in order to reduce both, complexity of the implementation and reducing the attack surface with respect to side- channels for embedded systems in hostile environments. The effect of non-uniform sampling on security was demonstrated to be begning in [AHH21] for the case of Curve25519 and Curve448. This analysis however does not transfer to most curves in Short-Weierstrass form. As a result, we recommend rejection sampling if G is as in Section 7.4. 9.6. Single-coordinate CPace on Montgomery curves The recommended cipher suites for the Montgomery curves Curve25519 and Curve448 in Section 7.2 rely on the following properties [AHH21]: * The curve has order (p * c) with p prime and c a small cofactor. Also the curve's quadratic twist must be of order (p' * c') with p' prime and c' a cofactor. Abdalla, et al. Expires 26 March 2024 [Page 24] Internet-Draft CPace September 2023 * The cofactor c' of the twist MUST BE EQUAL to or an integer multiple of the cofactor c of the curve. * Both field order q and group order p MUST BE close to a power of two along the lines of [AHH21], Appendix E. * The representation of the neutral element G.I MUST BE the same for both, the curve and its twist. * The implementation of G.scalar_mult_vfy(y,X) MUST map all c low- order points on the curve and all c' low-order points on the twist to G.I. Montgomery curves other than the ones recommended here can use the specifications given in Section 7.2, given that the above properties hold. 9.7. Nonce values Secret scalars ya and yb MUST NOT be reused. Values for sid SHOULD NOT be reused since the composability guarantees established by the simulation-based proof rely on the uniqueness of session ids [AHH21]. If CPace is used in a concurrent system, it is RECOMMENDED that a unique sid is generated by the higher-level protocol and passed to CPace. One suitable option is that sid is generated by concatenating ephemeral random strings contributed by both parties. 9.8. Side channel attacks All state-of-the art methods for realizing constant-time execution SHOULD be used. In case that side channel attacks are to be considered practical for a given application, it is RECOMMENDED to pay special attention on computing the secret generator G.calculate_generator(PRS,CI,sid). The most critical substep to consider might be the processing of the first block of the hash that includes the PRS string. The zero-padding introduced when hashing the sensitive PRS string can be expected to make the task for a side- channel attack somewhat more complex. Still this feature alone is not sufficient for ruling out power analysis attacks. Abdalla, et al. Expires 26 March 2024 [Page 25] Internet-Draft CPace September 2023 9.9. Quantum computers CPace is proven secure under the hardness of the strong computational Simultaneous Diffie-Hellmann (sSDH) and strong computational Diffie- Hellmann (sCDH) assumptions in the group G (as defined in [AHH21]). These assumptions are not expected to hold any longer when large- scale quantum computers (LSQC) are available. Still, even in case that LSQC emerge, it is reasonable to assume that discrete-logarithm computations will remain costly. CPace with ephemeral session id values sid forces the adversary to solve one computational Diffie- Hellman problem per password guess [ES21]. In this sense, using the wording suggested by Steve Thomas on the CFRG mailing list, CPace is "quantum-annoying". 10. IANA Considerations No IANA action is required. 11. Acknowledgements We would like to thank the participants on the CFRG list for comments and advice. Any comment and advice is appreciated. 12. References 12.1. Normative References [I-D.draft-irtf-cfrg-ristretto255-decaf448] de Valence, H., Grigg, J., Hamburg, M., Lovecruft, I., Tankersley, G., and F. Valsorda, "The ristretto255 and decaf448 Groups", Work in Progress, Internet-Draft, draft- irtf-cfrg-ristretto255-decaf448-08, 5 September 2023, . [I-D.irtf-cfrg-opaque] Bourdrez, D., Krawczyk, H., Lewi, K., and C. A. Wood, "The OPAQUE Asymmetric PAKE Protocol", Work in Progress, Internet-Draft, draft-irtf-cfrg-opaque-11, 8 June 2023, . [IEEE1363] "Standard Specifications for Public Key Cryptography, IEEE 1363", 2000. Abdalla, et al. Expires 26 March 2024 [Page 26] Internet-Draft CPace September 2023 [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, March 1997, . [RFC7748] Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves for Security", RFC 7748, DOI 10.17487/RFC7748, January 2016, . [RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, May 2017, . [SEC1] Standards for Efficient Cryptography Group (SECG), "SEC 1: Elliptic Curve Cryptography", May 2009, . 12.2. Informative References [ABKLX21] Abdalla, M., Barbosa, M., Katz, J., Loss, J., and J. Xu, "Algebraic Adversaries in the Universal Composability Framework.", n.d., . [AHH21] Abdalla, M., Haase, B., and J. Hesse, "Security analysis of CPace", n.d., . [CDMP05] Coron, J.-S., Dodis, Y., Malinaud, C., and P. Puniya, "Merkle-Damgaard Revisited: How to Construct a Hash Function", In Advances in Cryptology - CRYPTO 2005, pages 430-448, DOI 10.1007/11535218_26, 2005, . [ES21] Eaton, E. and D. Stebila, "The 'quantum annoying' property of password-authenticated key exchange protocols.", n.d., . [FIPS202] National Institute of Standards and Technology (NIST), "SHA-3 Standard: Permutation-Based Hash and Extendable- Output Functions", August 2015, . [RFC2104] Krawczyk, H., Bellare, M., and R. Canetti, "HMAC: Keyed- Hashing for Message Authentication", RFC 2104, DOI 10.17487/RFC2104, February 1997, . Abdalla, et al. Expires 26 March 2024 [Page 27] Internet-Draft CPace September 2023 [RFC4493] Song, JH., Poovendran, R., Lee, J., and T. Iwata, "The AES-CMAC Algorithm", RFC 4493, DOI 10.17487/RFC4493, June 2006, . [RFC5246] Dierks, T. and E. Rescorla, "The Transport Layer Security (TLS) Protocol Version 1.2", RFC 5246, DOI 10.17487/RFC5246, August 2008, . [RFC5639] Lochter, M. and J. Merkle, "Elliptic Curve Cryptography (ECC) Brainpool Standard Curves and Curve Generation", RFC 5639, DOI 10.17487/RFC5639, March 2010, . [RFC5869] Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand Key Derivation Function (HKDF)", RFC 5869, DOI 10.17487/RFC5869, May 2010, . [RFC6234] Eastlake 3rd, D. and T. Hansen, "US Secure Hash Algorithms (SHA and SHA-based HMAC and HKDF)", RFC 6234, DOI 10.17487/RFC6234, May 2011, . [RFC8265] Saint-Andre, P. and A. Melnikov, "Preparation, Enforcement, and Comparison of Internationalized Strings Representing Usernames and Passwords", RFC 8265, DOI 10.17487/RFC8265, October 2017, . [RFC8446] Rescorla, E., "The Transport Layer Security (TLS) Protocol Version 1.3", RFC 8446, DOI 10.17487/RFC8446, August 2018, . [RFC9380] Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R. S., and C. A. Wood, "Hashing to Elliptic Curves", RFC 9380, DOI 10.17487/RFC9380, August 2023, . [SEC2] Standards for Efficient Cryptography Group (SECG), "SEC 2: Recommended Elliptic Curve Domain Parameters", January 2010, . Abdalla, et al. Expires 26 March 2024 [Page 28] Internet-Draft CPace September 2023 Appendix A. CPace function definitions A.1. Definition and test vectors for string utility functions A.1.1. prepend_len function def prepend_len(data): "prepend LEB128 encoding of length" length = len(data) length_encoded = b"" while True: if length < 128: length_encoded += bytes([length]) else: length_encoded += bytes([(length & 0x7f) + 0x80]) length = int(length >> 7) if length == 0: break; return length_encoded + data A.1.2. prepend_len test vectors prepend_len(b""): (length: 1 bytes) 00 prepend_len(b"1234"): (length: 5 bytes) 0431323334 prepend_len(bytes(range(127))): (length: 128 bytes) 7f000102030405060708090a0b0c0d0e0f101112131415161718191a1b 1c1d1e1f202122232425262728292a2b2c2d2e2f303132333435363738 393a3b3c3d3e3f404142434445464748494a4b4c4d4e4f505152535455 565758595a5b5c5d5e5f606162636465666768696a6b6c6d6e6f707172 737475767778797a7b7c7d7e prepend_len(bytes(range(128))): (length: 130 bytes) 8001000102030405060708090a0b0c0d0e0f101112131415161718191a 1b1c1d1e1f202122232425262728292a2b2c2d2e2f3031323334353637 38393a3b3c3d3e3f404142434445464748494a4b4c4d4e4f5051525354 55565758595a5b5c5d5e5f606162636465666768696a6b6c6d6e6f7071 72737475767778797a7b7c7d7e7f A.1.3. lv_cat function def lv_cat(*args): result = b"" for arg in args: result += prepend_len(arg) return result Abdalla, et al. Expires 26 March 2024 [Page 29] Internet-Draft CPace September 2023 A.1.4. Testvector for lv_cat() lv_cat(b"1234",b"5",b"",b"6789"): (length: 13 bytes) 04313233340135000436373839 A.1.5. Examples for messages not obtained from a lv_cat-based encoding The following messages are examples which have invalid encoded length fields. I.e. they are examples where parsing for the sum of the length of subfields as expected for a message generated using lv_cat(Y,AD) does not give the correct length of the message. Parties MUST abort upon reception of such invalid messages as MSGa or MSGb. Inv_MSG1 not encoded by lv_cat: (length: 3 bytes) ffffff Inv_MSG2 not encoded by lv_cat: (length: 3 bytes) ffff03 Inv_MSG3 not encoded by lv_cat: (length: 4 bytes) 00ffff03 Inv_MSG4 not encoded by lv_cat: (length: 4 bytes) 00ffffff A.2. Definition of generator_string function. def generator_string(DSI,PRS,CI,sid,s_in_bytes): # Concat all input fields with prepended length information. # Add zero padding in the first hash block after DSI and PRS. len_zpad = max(0,s_in_bytes - 1 - len(prepend_len(PRS)) - len(prepend_len(DSI))) return lv_cat(DSI, PRS, zero_bytes(len_zpad), CI, sid) A.3. Definitions and test vector ordered concatenation A.3.1. Definitions for lexiographical ordering For ordered concatenation lexiographical ordering of byte sequences is used: Abdalla, et al. Expires 26 March 2024 [Page 30] Internet-Draft CPace September 2023 def lexiographically_larger(bytes1,bytes2): "Returns True if bytes1 > bytes2 using lexiographical ordering." min_len = min (len(bytes1), len(bytes2)) for m in range(min_len): if bytes1[m] > bytes2[m]: return True; elif bytes1[m] < bytes2[m]: return False; return len(bytes1) > len(bytes2) A.3.2. Definitions for ordered concatenation With the above definition of lexiographical ordering ordered concatenation is specified as follows. def oCAT(bytes1,bytes2): if lexiographically_larger(bytes1,bytes2): return bytes1 + bytes2 else: return bytes2 + bytes1 A.3.3. Test vectors ordered concatenation string comparison for oCAT: lexiographically_larger(b"\0", b"\0\0") == False lexiographically_larger(b"\1", b"\0\0") == True lexiographically_larger(b"\0\0", b"\0") == True lexiographically_larger(b"\0\0", b"\1") == False lexiographically_larger(b"\0\1", b"\1") == False lexiographically_larger(b"ABCD", b"BCD") == False oCAT(b"ABCD",b"BCD"): (length: 7 bytes) 42434441424344 oCAT(b"BCD",b"ABCDE"): (length: 8 bytes) 4243444142434445 A.4. Decoding and Encoding functions according to RFC7748 Abdalla, et al. Expires 26 March 2024 [Page 31] Internet-Draft CPace September 2023 def decodeLittleEndian(b, bits): return sum([b[i] << 8*i for i in range((bits+7)/8)]) def decodeUCoordinate(u, bits): u_list = [ord(b) for b in u] # Ignore any unused bits. if bits % 8: u_list[-1] &= (1<<(bits%8))-1 return decodeLittleEndian(u_list, bits) def encodeUCoordinate(u, bits): return ''.join([chr((u >> 8*i) & 0xff) for i in range((bits+7)/8)]) A.5. Elligator 2 reference implementation The Elligator 2 map requires a non-square field element Z which shall be calculated as follows. def find_z_ell2(F): # Find nonsquare for Elligator2 # Argument: F, a field object, e.g., F = GF(2^255 - 19) ctr = F.gen() while True: for Z_cand in (F(ctr), F(-ctr)): # Z must be a non-square in F. if is_square(Z_cand): continue return Z_cand ctr += 1 The values of the non-square Z only depend on the curve. The algorithm above results in a value of Z = 2 for Curve25519 and Z=-1 for Ed448. The following code maps a field element r to an encoded field element which is a valid u-coordinate of a Montgomery curve with curve parameter A. Abdalla, et al. Expires 26 March 2024 [Page 32] Internet-Draft CPace September 2023 def elligator2(r, q, A, field_size_bits): # Inputs: field element r, field order q, # curve parameter A and field size in bits Fq = GF(q); A = Fq(A); B = Fq(1); # get non-square z as specified in the hash2curve draft. z = Fq(find_z_ell2(Fq)) powerForLegendreSymbol = floor((q-1)/2) v = - A / (1 + z * r^2) epsilon = (v^3 + A * v^2 + B * v)^powerForLegendreSymbol x = epsilon * v - (1 - epsilon) * A/2 return encodeUCoordinate(Integer(x), field_size_bits) Appendix B. Test vectors B.1. Test vector for CPace using group X25519 and hash SHA-512 B.1.1. Test vectors for calculate_generator with group X25519 Inputs H = SHA-512 with input block size 128 bytes. PRS = b'Password' ; ZPAD length: 109 ; DSI = b'CPace255' CI = b'\nAinitiator\nBresponder' CI = 0a41696e69746961746f720a42726573706f6e646572 sid = 7e4b4791d6a8ef019b936c79fb7f2c57 Outputs generator_string(G.DSI,PRS,CI,sid,H.s_in_bytes): (length: 168 bytes) 0843506163653235350850617373776f72646d000000000000000000 00000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000160a41696e69746961746f72 0a42726573706f6e646572107e4b4791d6a8ef019b936c79fb7f2c57 hash generator string: (length: 32 bytes) 10047198e8c4cacf0ab8a6d0ac337b8ae497209d042f7f3a50945863 94e821fc decoded field element of 255 bits: (length: 32 bytes) 10047198e8c4cacf0ab8a6d0ac337b8ae497209d042f7f3a50945863 94e8217c generator g: (length: 32 bytes) 4e6098733061c0e8486611a904fe5edb049804d26130a44131a6229e 55c5c321 B.1.2. Test vector for MSGa Abdalla, et al. Expires 26 March 2024 [Page 33] Internet-Draft CPace September 2023 Inputs ADa = b'ADa' ya (little endian): (length: 32 bytes) 45acf93116ae5d3dae995a7c627df2924321a8e857d9a200807131e3 8839b0c2 Outputs Ya: (length: 32 bytes) 6f7fd31863b18b0cc9830fc842c60dea80120ccf2fd375498225e45a 52065361 MSGa = lv_cat(Ya,ADa): (length: 37 bytes) 206f7fd31863b18b0cc9830fc842c60dea80120ccf2fd375498225e4 5a5206536103414461 B.1.3. Test vector for MSGb Inputs ADb = b'ADb' yb (little endian): (length: 32 bytes) a145e914b347002d298ce2051394f0ed68cf3623dfe5db082c78ffa5 a667acdc Outputs Yb: (length: 32 bytes) e1b730a4956c0f853d96c5d125cebeeea46952c07c6f66da65bd9ffd 2f71a462 MSGb = lv_cat(Yb,ADb): (length: 37 bytes) 20e1b730a4956c0f853d96c5d125cebeeea46952c07c6f66da65bd9f fd2f71a46203414462 B.1.4. Test vector for secret points K scalar_mult_vfy(ya,Yb): (length: 32 bytes) 2a905bc5f0b93ee72ac4b6ea8723520941adfc892935bf6f86d9e199 befa6024 scalar_mult_vfy(yb,Ya): (length: 32 bytes) 2a905bc5f0b93ee72ac4b6ea8723520941adfc892935bf6f86d9e199 befa6024 B.1.5. Test vector for ISK calculation initiator/responder Abdalla, et al. Expires 26 March 2024 [Page 34] Internet-Draft CPace September 2023 unordered cat of transcript : (length: 74 bytes) 206f7fd31863b18b0cc9830fc842c60dea80120ccf2fd375498225e4 5a520653610341446120e1b730a4956c0f853d96c5d125cebeeea469 52c07c6f66da65bd9ffd2f71a46203414462 DSI = G.DSI_ISK, b'CPace255_ISK': (length: 12 bytes) 43506163653235355f49534b lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 137 bytes) 0c43506163653235355f49534b107e4b4791d6a8ef019b936c79fb7f 2c57202a905bc5f0b93ee72ac4b6ea8723520941adfc892935bf6f86 d9e199befa6024206f7fd31863b18b0cc9830fc842c60dea80120ccf 2fd375498225e45a520653610341446120e1b730a4956c0f853d96c5 d125cebeeea46952c07c6f66da65bd9ffd2f71a46203414462 ISK result: (length: 64 bytes) 99a9e0ff35acb94ad8af1cd6b32ac409dc7d00557ccd9a7d19d3b462 9e5f1f084f9332096162438c7ecc78331b4eda17e1a229a47182eccc 9ea58cd9cdcd8e9a B.1.6. Test vector for ISK calculation parallel execution ordered cat of transcript : (length: 74 bytes) 20e1b730a4956c0f853d96c5d125cebeeea46952c07c6f66da65bd9f fd2f71a46203414462206f7fd31863b18b0cc9830fc842c60dea8012 0ccf2fd375498225e45a5206536103414461 DSI = G.DSI_ISK, b'CPace255_ISK': (length: 12 bytes) 43506163653235355f49534b lv_cat(DSI,sid,K)||oCAT(MSGa,MSGb): (length: 137 bytes) 0c43506163653235355f49534b107e4b4791d6a8ef019b936c79fb7f 2c57202a905bc5f0b93ee72ac4b6ea8723520941adfc892935bf6f86 d9e199befa602420e1b730a4956c0f853d96c5d125cebeeea46952c0 7c6f66da65bd9ffd2f71a46203414462206f7fd31863b18b0cc9830f c842c60dea80120ccf2fd375498225e45a5206536103414461 ISK result: (length: 64 bytes) 3cd6a9670fa3ff211d829b845baa0f5ba9ad580c3ba0ee790bd0e9cd 556290a8ffce44419fbf94e4cb8e7fe9f454fd25dc13e689e4d6ab0a c2211c70a8ac0062 B.1.7. Corresponding ANSI-C initializers const uint8_t tc_PRS[] = { 0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64, }; const uint8_t tc_CI[] = { 0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a, 0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72, }; const uint8_t tc_sid[] = { 0x7e,0x4b,0x47,0x91,0xd6,0xa8,0xef,0x01,0x9b,0x93,0x6c,0x79, 0xfb,0x7f,0x2c,0x57, Abdalla, et al. Expires 26 March 2024 [Page 35] Internet-Draft CPace September 2023 }; const uint8_t tc_g[] = { 0x4e,0x60,0x98,0x73,0x30,0x61,0xc0,0xe8,0x48,0x66,0x11,0xa9, 0x04,0xfe,0x5e,0xdb,0x04,0x98,0x04,0xd2,0x61,0x30,0xa4,0x41, 0x31,0xa6,0x22,0x9e,0x55,0xc5,0xc3,0x21, }; const uint8_t tc_ya[] = { 0x45,0xac,0xf9,0x31,0x16,0xae,0x5d,0x3d,0xae,0x99,0x5a,0x7c, 0x62,0x7d,0xf2,0x92,0x43,0x21,0xa8,0xe8,0x57,0xd9,0xa2,0x00, 0x80,0x71,0x31,0xe3,0x88,0x39,0xb0,0xc2, }; const uint8_t tc_ADa[] = { 0x41,0x44,0x61, }; const uint8_t tc_Ya[] = { 0x6f,0x7f,0xd3,0x18,0x63,0xb1,0x8b,0x0c,0xc9,0x83,0x0f,0xc8, 0x42,0xc6,0x0d,0xea,0x80,0x12,0x0c,0xcf,0x2f,0xd3,0x75,0x49, 0x82,0x25,0xe4,0x5a,0x52,0x06,0x53,0x61, }; const uint8_t tc_yb[] = { 0xa1,0x45,0xe9,0x14,0xb3,0x47,0x00,0x2d,0x29,0x8c,0xe2,0x05, 0x13,0x94,0xf0,0xed,0x68,0xcf,0x36,0x23,0xdf,0xe5,0xdb,0x08, 0x2c,0x78,0xff,0xa5,0xa6,0x67,0xac,0xdc, }; const uint8_t tc_ADb[] = { 0x41,0x44,0x62, }; const uint8_t tc_Yb[] = { 0xe1,0xb7,0x30,0xa4,0x95,0x6c,0x0f,0x85,0x3d,0x96,0xc5,0xd1, 0x25,0xce,0xbe,0xee,0xa4,0x69,0x52,0xc0,0x7c,0x6f,0x66,0xda, 0x65,0xbd,0x9f,0xfd,0x2f,0x71,0xa4,0x62, }; const uint8_t tc_K[] = { 0x2a,0x90,0x5b,0xc5,0xf0,0xb9,0x3e,0xe7,0x2a,0xc4,0xb6,0xea, 0x87,0x23,0x52,0x09,0x41,0xad,0xfc,0x89,0x29,0x35,0xbf,0x6f, 0x86,0xd9,0xe1,0x99,0xbe,0xfa,0x60,0x24, }; const uint8_t tc_ISK_IR[] = { 0x99,0xa9,0xe0,0xff,0x35,0xac,0xb9,0x4a,0xd8,0xaf,0x1c,0xd6, 0xb3,0x2a,0xc4,0x09,0xdc,0x7d,0x00,0x55,0x7c,0xcd,0x9a,0x7d, 0x19,0xd3,0xb4,0x62,0x9e,0x5f,0x1f,0x08,0x4f,0x93,0x32,0x09, 0x61,0x62,0x43,0x8c,0x7e,0xcc,0x78,0x33,0x1b,0x4e,0xda,0x17, 0xe1,0xa2,0x29,0xa4,0x71,0x82,0xec,0xcc,0x9e,0xa5,0x8c,0xd9, 0xcd,0xcd,0x8e,0x9a, }; const uint8_t tc_ISK_SY[] = { 0x3c,0xd6,0xa9,0x67,0x0f,0xa3,0xff,0x21,0x1d,0x82,0x9b,0x84, 0x5b,0xaa,0x0f,0x5b,0xa9,0xad,0x58,0x0c,0x3b,0xa0,0xee,0x79, Abdalla, et al. Expires 26 March 2024 [Page 36] Internet-Draft CPace September 2023 0x0b,0xd0,0xe9,0xcd,0x55,0x62,0x90,0xa8,0xff,0xce,0x44,0x41, 0x9f,0xbf,0x94,0xe4,0xcb,0x8e,0x7f,0xe9,0xf4,0x54,0xfd,0x25, 0xdc,0x13,0xe6,0x89,0xe4,0xd6,0xab,0x0a,0xc2,0x21,0x1c,0x70, 0xa8,0xac,0x00,0x62, }; B.1.8. Test vectors for G_X25519.scalar_mult_vfy: low order points Test vectors for which G_X25519.scalar_mult_vfy(s_in,ux) must return the neutral element or would return the neutral element if bit #255 of field element representation was not correctly cleared. (The decodeUCoordinate function from RFC7748 mandates clearing bit #255 for field element representations for use in the X25519 function.). u0: 0000000000000000000000000000000000000000000000000000000000000000 u1: 0100000000000000000000000000000000000000000000000000000000000000 u2: ecffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f u3: e0eb7a7c3b41b8ae1656e3faf19fc46ada098deb9c32b1fd866205165f49b800 u4: 5f9c95bca3508c24b1d0b1559c83ef5b04445cc4581c8e86d8224eddd09f1157 u5: edffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f u6: daffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff u7: eeffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f u8: dbffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff u9: d9ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ua: cdeb7a7c3b41b8ae1656e3faf19fc46ada098deb9c32b1fd866205165f49b880 ub: 4c9c95bca3508c24b1d0b1559c83ef5b04445cc4581c8e86d8224eddd09f11d7 u0 ... ub MUST be verified to produce the correct results q0 ... qb: Additionally, u0,u1,u2,u3,u4,u5 and u7 MUST trigger the abort case when included in MSGa or MSGb. s = af46e36bf0527c9d3b16154b82465edd62144c0ac1fc5a18506a2244ba449aff qN = G_X25519.scalar_mult_vfy(s, uX) q0: 0000000000000000000000000000000000000000000000000000000000000000 q1: 0000000000000000000000000000000000000000000000000000000000000000 q2: 0000000000000000000000000000000000000000000000000000000000000000 q3: 0000000000000000000000000000000000000000000000000000000000000000 q4: 0000000000000000000000000000000000000000000000000000000000000000 q5: 0000000000000000000000000000000000000000000000000000000000000000 q6: d8e2c776bbacd510d09fd9278b7edcd25fc5ae9adfba3b6e040e8d3b71b21806 q7: 0000000000000000000000000000000000000000000000000000000000000000 q8: c85c655ebe8be44ba9c0ffde69f2fe10194458d137f09bbff725ce58803cdb38 q9: db64dafa9b8fdd136914e61461935fe92aa372cb056314e1231bc4ec12417456 qa: e062dcd5376d58297be2618c7498f55baa07d7e03184e8aada20bca28888bf7a qb: 993c6ad11c4c29da9a56f7691fd0ff8d732e49de6250b6c2e80003ff4629a175 Abdalla, et al. Expires 26 March 2024 [Page 37] Internet-Draft CPace September 2023 B.2. Test vector for CPace using group X448 and hash SHAKE-256 B.2.1. Test vectors for calculate_generator with group X448 Inputs H = SHAKE-256 with input block size 136 bytes. PRS = b'Password' ; ZPAD length: 117 ; DSI = b'CPace448' CI = b'\nAinitiator\nBresponder' CI = 0a41696e69746961746f720a42726573706f6e646572 sid = 5223e0cdc45d6575668d64c552004124 Outputs generator_string(G.DSI,PRS,CI,sid,H.s_in_bytes): (length: 176 bytes) 0843506163653434380850617373776f726475000000000000000000 00000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000160a4169 6e69746961746f720a42726573706f6e646572105223e0cdc45d6575 668d64c552004124 hash generator string: (length: 56 bytes) 769e06d6c41c8cf1c87aa3df8e687167f6d0a2e41821e856276a0221 d88272359d0b43204b546174c9179c83c107b707f296eafaa1c5a293 decoded field element of 448 bits: (length: 56 bytes) 769e06d6c41c8cf1c87aa3df8e687167f6d0a2e41821e856276a0221 d88272359d0b43204b546174c9179c83c107b707f296eafaa1c5a293 generator g: (length: 56 bytes) 6fdae14718eb7506dd96e3f7797896efdb8db9ec0797485c9c48a192 2e44961da097f2908b084a5de33ab671630660d27d79ffd6ee8ec846 B.2.2. Test vector for MSGa Inputs ADa = b'ADa' ya (little endian): (length: 56 bytes) 21b4f4bd9e64ed355c3eb676a28ebedaf6d8f17bdc365995b3190971 53044080516bd083bfcce66121a3072646994c8430cc382b8dc543e8 Outputs Ya: (length: 56 bytes) 396bd11daf223711e575cac6021e3fa31558012048a1cec7876292b9 6c61eda353fe04f33028d2352779668a934084da776c1c51a58ce4b5 MSGa = lv_cat(Ya,ADa): (length: 61 bytes) 38396bd11daf223711e575cac6021e3fa31558012048a1cec7876292 b96c61eda353fe04f33028d2352779668a934084da776c1c51a58ce4 b503414461 B.2.3. Test vector for MSGb Abdalla, et al. Expires 26 March 2024 [Page 38] Internet-Draft CPace September 2023 Inputs ADb = b'ADb' yb (little endian): (length: 56 bytes) 848b0779ff415f0af4ea14df9dd1d3c29ac41d836c7808896c4eba19 c51ac40a439caf5e61ec88c307c7d619195229412eaa73fb2a5ea20d Outputs Yb: (length: 56 bytes) 53c519fb490fde5a04bda8c18b327d0fc1a9391d19e0ac00c59df9c6 0422284e593d6b092eac94f5aa644ed883f39bd4f04e4beb6af86d58 MSGb = lv_cat(Yb,ADb): (length: 61 bytes) 3853c519fb490fde5a04bda8c18b327d0fc1a9391d19e0ac00c59df9 c60422284e593d6b092eac94f5aa644ed883f39bd4f04e4beb6af86d 5803414462 B.2.4. Test vector for secret points K scalar_mult_vfy(ya,Yb): (length: 56 bytes) e00af217556a40ccbc9822cc27a43542e45166a653aa4df746d5f8e1 e8df483e9baff71c9eb03ee20a688ad4e4d359f70ac9ec3f6a659997 scalar_mult_vfy(yb,Ya): (length: 56 bytes) e00af217556a40ccbc9822cc27a43542e45166a653aa4df746d5f8e1 e8df483e9baff71c9eb03ee20a688ad4e4d359f70ac9ec3f6a659997 B.2.5. Test vector for ISK calculation initiator/responder unordered cat of transcript : (length: 122 bytes) 38396bd11daf223711e575cac6021e3fa31558012048a1cec7876292 b96c61eda353fe04f33028d2352779668a934084da776c1c51a58ce4 b5034144613853c519fb490fde5a04bda8c18b327d0fc1a9391d19e0 ac00c59df9c60422284e593d6b092eac94f5aa644ed883f39bd4f04e 4beb6af86d5803414462 DSI = G.DSI_ISK, b'CPace448_ISK': (length: 12 bytes) 43506163653434385f49534b lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 209 bytes) 0c43506163653434385f49534b105223e0cdc45d6575668d64c55200 412438e00af217556a40ccbc9822cc27a43542e45166a653aa4df746 d5f8e1e8df483e9baff71c9eb03ee20a688ad4e4d359f70ac9ec3f6a 65999738396bd11daf223711e575cac6021e3fa31558012048a1cec7 876292b96c61eda353fe04f33028d2352779668a934084da776c1c51 a58ce4b5034144613853c519fb490fde5a04bda8c18b327d0fc1a939 1d19e0ac00c59df9c60422284e593d6b092eac94f5aa644ed883f39b d4f04e4beb6af86d5803414462 ISK result: (length: 64 bytes) 4030297722c1914711da6b2a224a44b53b30c05ab02c2a3d3ccc7272 a3333ce3a4564c17031b634e89f65681f52d5c3d1df7baeb88523d2e 481b3858aed86315 Abdalla, et al. Expires 26 March 2024 [Page 39] Internet-Draft CPace September 2023 B.2.6. Test vector for ISK calculation parallel execution ordered cat of transcript : (length: 122 bytes) 3853c519fb490fde5a04bda8c18b327d0fc1a9391d19e0ac00c59df9 c60422284e593d6b092eac94f5aa644ed883f39bd4f04e4beb6af86d 580341446238396bd11daf223711e575cac6021e3fa31558012048a1 cec7876292b96c61eda353fe04f33028d2352779668a934084da776c 1c51a58ce4b503414461 DSI = G.DSI_ISK, b'CPace448_ISK': (length: 12 bytes) 43506163653434385f49534b lv_cat(DSI,sid,K)||oCAT(MSGa,MSGb): (length: 209 bytes) 0c43506163653434385f49534b105223e0cdc45d6575668d64c55200 412438e00af217556a40ccbc9822cc27a43542e45166a653aa4df746 d5f8e1e8df483e9baff71c9eb03ee20a688ad4e4d359f70ac9ec3f6a 6599973853c519fb490fde5a04bda8c18b327d0fc1a9391d19e0ac00 c59df9c60422284e593d6b092eac94f5aa644ed883f39bd4f04e4beb 6af86d580341446238396bd11daf223711e575cac6021e3fa3155801 2048a1cec7876292b96c61eda353fe04f33028d2352779668a934084 da776c1c51a58ce4b503414461 ISK result: (length: 64 bytes) 925e95d1095dad1af6378d5ef8b9a998bd3855bfc7d36cb5ca05b0a7 a93346abcb8cef04bceb28c38fdaf0cc608fd1dcd462ab523f3b7f75 2c77c411be3ac8fb B.2.7. Corresponding ANSI-C initializers const uint8_t tc_PRS[] = { 0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64, }; const uint8_t tc_CI[] = { 0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a, 0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72, }; const uint8_t tc_sid[] = { 0x52,0x23,0xe0,0xcd,0xc4,0x5d,0x65,0x75,0x66,0x8d,0x64,0xc5, 0x52,0x00,0x41,0x24, }; const uint8_t tc_g[] = { 0x6f,0xda,0xe1,0x47,0x18,0xeb,0x75,0x06,0xdd,0x96,0xe3,0xf7, 0x79,0x78,0x96,0xef,0xdb,0x8d,0xb9,0xec,0x07,0x97,0x48,0x5c, 0x9c,0x48,0xa1,0x92,0x2e,0x44,0x96,0x1d,0xa0,0x97,0xf2,0x90, 0x8b,0x08,0x4a,0x5d,0xe3,0x3a,0xb6,0x71,0x63,0x06,0x60,0xd2, 0x7d,0x79,0xff,0xd6,0xee,0x8e,0xc8,0x46, }; const uint8_t tc_ya[] = { 0x21,0xb4,0xf4,0xbd,0x9e,0x64,0xed,0x35,0x5c,0x3e,0xb6,0x76, 0xa2,0x8e,0xbe,0xda,0xf6,0xd8,0xf1,0x7b,0xdc,0x36,0x59,0x95, 0xb3,0x19,0x09,0x71,0x53,0x04,0x40,0x80,0x51,0x6b,0xd0,0x83, Abdalla, et al. Expires 26 March 2024 [Page 40] Internet-Draft CPace September 2023 0xbf,0xcc,0xe6,0x61,0x21,0xa3,0x07,0x26,0x46,0x99,0x4c,0x84, 0x30,0xcc,0x38,0x2b,0x8d,0xc5,0x43,0xe8, }; const uint8_t tc_ADa[] = { 0x41,0x44,0x61, }; const uint8_t tc_Ya[] = { 0x39,0x6b,0xd1,0x1d,0xaf,0x22,0x37,0x11,0xe5,0x75,0xca,0xc6, 0x02,0x1e,0x3f,0xa3,0x15,0x58,0x01,0x20,0x48,0xa1,0xce,0xc7, 0x87,0x62,0x92,0xb9,0x6c,0x61,0xed,0xa3,0x53,0xfe,0x04,0xf3, 0x30,0x28,0xd2,0x35,0x27,0x79,0x66,0x8a,0x93,0x40,0x84,0xda, 0x77,0x6c,0x1c,0x51,0xa5,0x8c,0xe4,0xb5, }; const uint8_t tc_yb[] = { 0x84,0x8b,0x07,0x79,0xff,0x41,0x5f,0x0a,0xf4,0xea,0x14,0xdf, 0x9d,0xd1,0xd3,0xc2,0x9a,0xc4,0x1d,0x83,0x6c,0x78,0x08,0x89, 0x6c,0x4e,0xba,0x19,0xc5,0x1a,0xc4,0x0a,0x43,0x9c,0xaf,0x5e, 0x61,0xec,0x88,0xc3,0x07,0xc7,0xd6,0x19,0x19,0x52,0x29,0x41, 0x2e,0xaa,0x73,0xfb,0x2a,0x5e,0xa2,0x0d, }; const uint8_t tc_ADb[] = { 0x41,0x44,0x62, }; const uint8_t tc_Yb[] = { 0x53,0xc5,0x19,0xfb,0x49,0x0f,0xde,0x5a,0x04,0xbd,0xa8,0xc1, 0x8b,0x32,0x7d,0x0f,0xc1,0xa9,0x39,0x1d,0x19,0xe0,0xac,0x00, 0xc5,0x9d,0xf9,0xc6,0x04,0x22,0x28,0x4e,0x59,0x3d,0x6b,0x09, 0x2e,0xac,0x94,0xf5,0xaa,0x64,0x4e,0xd8,0x83,0xf3,0x9b,0xd4, 0xf0,0x4e,0x4b,0xeb,0x6a,0xf8,0x6d,0x58, }; const uint8_t tc_K[] = { 0xe0,0x0a,0xf2,0x17,0x55,0x6a,0x40,0xcc,0xbc,0x98,0x22,0xcc, 0x27,0xa4,0x35,0x42,0xe4,0x51,0x66,0xa6,0x53,0xaa,0x4d,0xf7, 0x46,0xd5,0xf8,0xe1,0xe8,0xdf,0x48,0x3e,0x9b,0xaf,0xf7,0x1c, 0x9e,0xb0,0x3e,0xe2,0x0a,0x68,0x8a,0xd4,0xe4,0xd3,0x59,0xf7, 0x0a,0xc9,0xec,0x3f,0x6a,0x65,0x99,0x97, }; const uint8_t tc_ISK_IR[] = { 0x40,0x30,0x29,0x77,0x22,0xc1,0x91,0x47,0x11,0xda,0x6b,0x2a, 0x22,0x4a,0x44,0xb5,0x3b,0x30,0xc0,0x5a,0xb0,0x2c,0x2a,0x3d, 0x3c,0xcc,0x72,0x72,0xa3,0x33,0x3c,0xe3,0xa4,0x56,0x4c,0x17, 0x03,0x1b,0x63,0x4e,0x89,0xf6,0x56,0x81,0xf5,0x2d,0x5c,0x3d, 0x1d,0xf7,0xba,0xeb,0x88,0x52,0x3d,0x2e,0x48,0x1b,0x38,0x58, 0xae,0xd8,0x63,0x15, }; const uint8_t tc_ISK_SY[] = { 0x92,0x5e,0x95,0xd1,0x09,0x5d,0xad,0x1a,0xf6,0x37,0x8d,0x5e, 0xf8,0xb9,0xa9,0x98,0xbd,0x38,0x55,0xbf,0xc7,0xd3,0x6c,0xb5, Abdalla, et al. Expires 26 March 2024 [Page 41] Internet-Draft CPace September 2023 0xca,0x05,0xb0,0xa7,0xa9,0x33,0x46,0xab,0xcb,0x8c,0xef,0x04, 0xbc,0xeb,0x28,0xc3,0x8f,0xda,0xf0,0xcc,0x60,0x8f,0xd1,0xdc, 0xd4,0x62,0xab,0x52,0x3f,0x3b,0x7f,0x75,0x2c,0x77,0xc4,0x11, 0xbe,0x3a,0xc8,0xfb, }; B.2.8. Test vectors for G_X448.scalar_mult_vfy: low order points Test vectors for which G_X448.scalar_mult_vfy(s_in,ux) must return the neutral element. This includes points that are non-canonicaly encoded, i.e. have coordinate values larger than the field prime. Weak points for X448 smaller than the field prime (canonical) u0: (length: 56 bytes) 0000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 u1: (length: 56 bytes) 0100000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 u2: (length: 56 bytes) fefffffffffffffffffffffffffffffffffffffffffffffffffffffffe ffffffffffffffffffffffffffffffffffffffffffffffffffffff Weak points for X448 larger or equal to the field prime (non- canonical) u3: (length: 56 bytes) fffffffffffffffffffffffffffffffffffffffffffffffffffffffffe ffffffffffffffffffffffffffffffffffffffffffffffffffffff u4: (length: 56 bytes) 00000000000000000000000000000000000000000000000000000000ff ffffffffffffffffffffffffffffffffffffffffffffffffffffff All of the above points u0 ... u4 MUST trigger the abort case when included in the protocol messages MSGa or MSGb. Expected results for X448 resp. G_X448.scalar_mult_vfy Abdalla, et al. Expires 26 March 2024 [Page 42] Internet-Draft CPace September 2023 scalar s: (length: 56 bytes) af8a14218bf2a2062926d2ea9b8fe4e8b6817349b6ed2feb1e5d64d7a4 523f15fceec70fb111e870dc58d191e66a14d3e9d482d04432cadd G_X448.scalar_mult_vfy(s,u0): (length: 56 bytes) 0000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 G_X448.scalar_mult_vfy(s,u1): (length: 56 bytes) 0000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 G_X448.scalar_mult_vfy(s,u2): (length: 56 bytes) 0000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 G_X448.scalar_mult_vfy(s,u3): (length: 56 bytes) 0000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 G_X448.scalar_mult_vfy(s,u4): (length: 56 bytes) 0000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 Test vectors for scalar_mult with nonzero outputs scalar s: (length: 56 bytes) af8a14218bf2a2062926d2ea9b8fe4e8b6817349b6ed2feb1e5d64d7a4 523f15fceec70fb111e870dc58d191e66a14d3e9d482d04432cadd point coordinate u_curve on the curve: (length: 56 bytes) ab0c68d772ec2eb9de25c49700e46d6325e66d6aa39d7b65eb84a68c55 69d47bd71b41f3e0d210f44e146dec8926b174acb3f940a0b82cab G_X448.scalar_mult_vfy(s,u_curve): (length: 56 bytes) 3b0fa9bc40a6fdc78c9e06ff7a54c143c5d52f365607053bf0656f5142 0496295f910a101b38edc1acd3bd240fd55dcb7a360553b8a7627e point coordinate u_twist on the twist: (length: 56 bytes) c981cd1e1f72d9c35c7d7cf6be426757c0dc8206a2fcfa564a8e7618c0 3c0e61f9a2eb1c3e0dd97d6e9b1010f5edd03397a83f5a914cb3ff G_X448.scalar_mult_vfy(s,u_twist): (length: 56 bytes) d0a2bb7e9c5c2c627793d8342f23b759fe7d9e3320a85ca4fd61376331 50ffd9a9148a9b75c349fac43d64bec49a6e126cc92cbfbf353961 B.3. Test vector for CPace using group ristretto255 and hash SHA-512 B.3.1. Test vectors for calculate_generator with group ristretto255 Abdalla, et al. Expires 26 March 2024 [Page 43] Internet-Draft CPace September 2023 Inputs H = SHA-512 with input block size 128 bytes. PRS = b'Password' ; ZPAD length: 100 ; DSI = b'CPaceRistretto255' CI = b'\nAinitiator\nBresponder' CI = 0a41696e69746961746f720a42726573706f6e646572 sid = 7e4b4791d6a8ef019b936c79fb7f2c57 Outputs generator_string(G.DSI,PRS,CI,sid,H.s_in_bytes): (length: 168 bytes) 11435061636552697374726574746f3235350850617373776f726464 00000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000160a41696e69746961746f72 0a42726573706f6e646572107e4b4791d6a8ef019b936c79fb7f2c57 hash result: (length: 64 bytes) a5ce446f63a1ae6d1fee80fa67d0b4004a4b1283ec5549a462bf33a6 c1ae06a0871f9bf48545f49b2a792eed255ac04f52758c9c60448306 810b44e986e3dcbb encoded generator g: (length: 32 bytes) 5e25411ca1ad7c9debfd0b33ad987a95cefef2d3f15dcc8bd26415a5 dfe2e15a B.3.2. Test vector for MSGa Inputs ADa = b'ADa' ya (little endian): (length: 32 bytes) da3d23700a9e5699258aef94dc060dfda5ebb61f02a5ea77fad53f4f f0976d08 Outputs Ya: (length: 32 bytes) 383a85dd236978f17f8c8545b50dabc52a39fcdab2cf8bc531ce040f f77ca82d MSGa = lv_cat(Ya,ADa): (length: 37 bytes) 20383a85dd236978f17f8c8545b50dabc52a39fcdab2cf8bc531ce04 0ff77ca82d03414461 B.3.3. Test vector for MSGb Abdalla, et al. Expires 26 March 2024 [Page 44] Internet-Draft CPace September 2023 Inputs ADb = b'ADb' yb (little endian): (length: 32 bytes) d2316b454718c35362d83d69df6320f38578ed5984651435e2949762 d900b80d Outputs Yb: (length: 32 bytes) a6206309c0e8e5f579295e35997ac4300ab3fecec3c17f7b604f3e69 8fa1383c MSGb = lv_cat(Yb,ADb): (length: 37 bytes) 20a6206309c0e8e5f579295e35997ac4300ab3fecec3c17f7b604f3e 698fa1383c03414462 B.3.4. Test vector for secret points K scalar_mult_vfy(ya,Yb): (length: 32 bytes) fa1d0318864e2cacb26875f1b791c9ae83204fe8359addb53e95a2e9 8893853f scalar_mult_vfy(yb,Ya): (length: 32 bytes) fa1d0318864e2cacb26875f1b791c9ae83204fe8359addb53e95a2e9 8893853f B.3.5. Test vector for ISK calculation initiator/responder unordered cat of transcript : (length: 74 bytes) 20383a85dd236978f17f8c8545b50dabc52a39fcdab2cf8bc531ce04 0ff77ca82d0341446120a6206309c0e8e5f579295e35997ac4300ab3 fecec3c17f7b604f3e698fa1383c03414462 DSI = G.DSI_ISK, b'CPaceRistretto255_ISK': (length: 21 bytes) 435061636552697374726574746f3235355f49534b lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 146 bytes) 15435061636552697374726574746f3235355f49534b107e4b4791d6 a8ef019b936c79fb7f2c5720fa1d0318864e2cacb26875f1b791c9ae 83204fe8359addb53e95a2e98893853f20383a85dd236978f17f8c85 45b50dabc52a39fcdab2cf8bc531ce040ff77ca82d0341446120a620 6309c0e8e5f579295e35997ac4300ab3fecec3c17f7b604f3e698fa1 383c03414462 ISK result: (length: 64 bytes) e91ccb2c0f5e0d0993a33956e3be59754f3f2b07db57631f5394452e a2e7b4354674eb1f5686c078462bf83bec72e8743df440108e638f35 26d9b90e85be096f B.3.6. Test vector for ISK calculation parallel execution Abdalla, et al. Expires 26 March 2024 [Page 45] Internet-Draft CPace September 2023 ordered cat of transcript : (length: 74 bytes) 20a6206309c0e8e5f579295e35997ac4300ab3fecec3c17f7b604f3e 698fa1383c0341446220383a85dd236978f17f8c8545b50dabc52a39 fcdab2cf8bc531ce040ff77ca82d03414461 DSI = G.DSI_ISK, b'CPaceRistretto255_ISK': (length: 21 bytes) 435061636552697374726574746f3235355f49534b lv_cat(DSI,sid,K)||oCAT(MSGa,MSGb): (length: 146 bytes) 15435061636552697374726574746f3235355f49534b107e4b4791d6 a8ef019b936c79fb7f2c5720fa1d0318864e2cacb26875f1b791c9ae 83204fe8359addb53e95a2e98893853f20a6206309c0e8e5f579295e 35997ac4300ab3fecec3c17f7b604f3e698fa1383c0341446220383a 85dd236978f17f8c8545b50dabc52a39fcdab2cf8bc531ce040ff77c a82d03414461 ISK result: (length: 64 bytes) 2472dedbff868bfc12b4c256f790539af0e2bab7efc28d1a995d18a1 a58e5bec639273d4604512669ab7953153d437eb90314dcba7539724 02b0d9c5ec5283f8 B.3.7. Corresponding ANSI-C initializers const uint8_t tc_PRS[] = { 0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64, }; const uint8_t tc_CI[] = { 0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a, 0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72, }; const uint8_t tc_sid[] = { 0x7e,0x4b,0x47,0x91,0xd6,0xa8,0xef,0x01,0x9b,0x93,0x6c,0x79, 0xfb,0x7f,0x2c,0x57, }; const uint8_t tc_g[] = { 0x5e,0x25,0x41,0x1c,0xa1,0xad,0x7c,0x9d,0xeb,0xfd,0x0b,0x33, 0xad,0x98,0x7a,0x95,0xce,0xfe,0xf2,0xd3,0xf1,0x5d,0xcc,0x8b, 0xd2,0x64,0x15,0xa5,0xdf,0xe2,0xe1,0x5a, }; const uint8_t tc_ya[] = { 0xda,0x3d,0x23,0x70,0x0a,0x9e,0x56,0x99,0x25,0x8a,0xef,0x94, 0xdc,0x06,0x0d,0xfd,0xa5,0xeb,0xb6,0x1f,0x02,0xa5,0xea,0x77, 0xfa,0xd5,0x3f,0x4f,0xf0,0x97,0x6d,0x08, }; const uint8_t tc_ADa[] = { 0x41,0x44,0x61, }; const uint8_t tc_Ya[] = { 0x38,0x3a,0x85,0xdd,0x23,0x69,0x78,0xf1,0x7f,0x8c,0x85,0x45, 0xb5,0x0d,0xab,0xc5,0x2a,0x39,0xfc,0xda,0xb2,0xcf,0x8b,0xc5, Abdalla, et al. Expires 26 March 2024 [Page 46] Internet-Draft CPace September 2023 0x31,0xce,0x04,0x0f,0xf7,0x7c,0xa8,0x2d, }; const uint8_t tc_yb[] = { 0xd2,0x31,0x6b,0x45,0x47,0x18,0xc3,0x53,0x62,0xd8,0x3d,0x69, 0xdf,0x63,0x20,0xf3,0x85,0x78,0xed,0x59,0x84,0x65,0x14,0x35, 0xe2,0x94,0x97,0x62,0xd9,0x00,0xb8,0x0d, }; const uint8_t tc_ADb[] = { 0x41,0x44,0x62, }; const uint8_t tc_Yb[] = { 0xa6,0x20,0x63,0x09,0xc0,0xe8,0xe5,0xf5,0x79,0x29,0x5e,0x35, 0x99,0x7a,0xc4,0x30,0x0a,0xb3,0xfe,0xce,0xc3,0xc1,0x7f,0x7b, 0x60,0x4f,0x3e,0x69,0x8f,0xa1,0x38,0x3c, }; const uint8_t tc_K[] = { 0xfa,0x1d,0x03,0x18,0x86,0x4e,0x2c,0xac,0xb2,0x68,0x75,0xf1, 0xb7,0x91,0xc9,0xae,0x83,0x20,0x4f,0xe8,0x35,0x9a,0xdd,0xb5, 0x3e,0x95,0xa2,0xe9,0x88,0x93,0x85,0x3f, }; const uint8_t tc_ISK_IR[] = { 0xe9,0x1c,0xcb,0x2c,0x0f,0x5e,0x0d,0x09,0x93,0xa3,0x39,0x56, 0xe3,0xbe,0x59,0x75,0x4f,0x3f,0x2b,0x07,0xdb,0x57,0x63,0x1f, 0x53,0x94,0x45,0x2e,0xa2,0xe7,0xb4,0x35,0x46,0x74,0xeb,0x1f, 0x56,0x86,0xc0,0x78,0x46,0x2b,0xf8,0x3b,0xec,0x72,0xe8,0x74, 0x3d,0xf4,0x40,0x10,0x8e,0x63,0x8f,0x35,0x26,0xd9,0xb9,0x0e, 0x85,0xbe,0x09,0x6f, }; const uint8_t tc_ISK_SY[] = { 0x24,0x72,0xde,0xdb,0xff,0x86,0x8b,0xfc,0x12,0xb4,0xc2,0x56, 0xf7,0x90,0x53,0x9a,0xf0,0xe2,0xba,0xb7,0xef,0xc2,0x8d,0x1a, 0x99,0x5d,0x18,0xa1,0xa5,0x8e,0x5b,0xec,0x63,0x92,0x73,0xd4, 0x60,0x45,0x12,0x66,0x9a,0xb7,0x95,0x31,0x53,0xd4,0x37,0xeb, 0x90,0x31,0x4d,0xcb,0xa7,0x53,0x97,0x24,0x02,0xb0,0xd9,0xc5, 0xec,0x52,0x83,0xf8, }; B.3.8. Test case for scalar_mult with valid inputs Abdalla, et al. Expires 26 March 2024 [Page 47] Internet-Draft CPace September 2023 s: (length: 32 bytes) 7cd0e075fa7955ba52c02759a6c90dbbfc10e6d40aea8d283e407d88 cf538a05 X: (length: 32 bytes) 2c3c6b8c4f3800e7aef6864025b4ed79bd599117e427c41bd47d93d6 54b4a51c G.scalar_mult(s,decode(X)): (length: 32 bytes) 7c13645fe790a468f62c39beb7388e541d8405d1ade69d1778c5fe3e 7f6b600e G.scalar_mult_vfy(s,X): (length: 32 bytes) 7c13645fe790a468f62c39beb7388e541d8405d1ade69d1778c5fe3e 7f6b600e B.3.9. Invalid inputs for scalar_mult_vfy For these test cases scalar_mult_vfy(y,.) MUST return the representation of the neutral element G.I. When points Y_i1 or Y_i2 are included in MSGa or MSGb the protocol MUST abort. s: (length: 32 bytes) 7cd0e075fa7955ba52c02759a6c90dbbfc10e6d40aea8d283e407d88 cf538a05 Y_i1: (length: 32 bytes) 2b3c6b8c4f3800e7aef6864025b4ed79bd599117e427c41bd47d93d6 54b4a51c Y_i2 == G.I: (length: 32 bytes) 00000000000000000000000000000000000000000000000000000000 00000000 G.scalar_mult_vfy(s,Y_i1) = G.scalar_mult_vfy(s,Y_i2) = G.I B.4. Test vector for CPace using group decaf448 and hash SHAKE-256 B.4.1. Test vectors for calculate_generator with group decaf448 Abdalla, et al. Expires 26 March 2024 [Page 48] Internet-Draft CPace September 2023 Inputs H = SHAKE-256 with input block size 136 bytes. PRS = b'Password' ; ZPAD length: 112 ; DSI = b'CPaceDecaf448' CI = b'\nAinitiator\nBresponder' CI = 0a41696e69746961746f720a42726573706f6e646572 sid = 5223e0cdc45d6575668d64c552004124 Outputs generator_string(G.DSI,PRS,CI,sid,H.s_in_bytes): (length: 176 bytes) 0d435061636544656361663434380850617373776f72647000000000 00000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000160a4169 6e69746961746f720a42726573706f6e646572105223e0cdc45d6575 668d64c552004124 hash result: (length: 112 bytes) 8955b426ff1d3a22032d21c013cf94134cee9a4235e93261a4911edb f68f2945f0267c983954262c7f59badb9caf468ebe21b7e9885657af b8f1a3b783c2047ba519e113ecf81b2b580dd481f499beabd401cc77 1d28915fb750011209040f5f03b2ceb5e5eb259c96b478382d5a5c57 encoded generator g: (length: 56 bytes) 682d1a4f49fc2a4834356ae4d7f58636bc9481521c845e66e6fb0b29 69341df45fbaeaea9e2221b3f5babc54c5f8ce456988ffc519defaeb B.4.2. Test vector for MSGa Inputs ADa = b'ADa' ya (little endian): (length: 56 bytes) d8d2e26c821a12d7f59a8dee023d3f6155976152e16c73cbf68c303d f0404399f0a7b614a65df50a9788f00b410586b443f738ad7ff03930 Outputs Ya: (length: 56 bytes) d4b87d2fcdcac1096dba1898361f27e29dc1e019f74f84a71199bfd3 dc8d09d2b823038f579f517591474be366968e2fb599bf14e55704f4 MSGa = lv_cat(Ya,ADa): (length: 61 bytes) 38d4b87d2fcdcac1096dba1898361f27e29dc1e019f74f84a71199bf d3dc8d09d2b823038f579f517591474be366968e2fb599bf14e55704 f403414461 B.4.3. Test vector for MSGb Abdalla, et al. Expires 26 March 2024 [Page 49] Internet-Draft CPace September 2023 Inputs ADb = b'ADb' yb (little endian): (length: 56 bytes) 91bae9793f4a8aceb1b5c54375a7ed1858a79a6e72dab959c8bdf3a7 5ac9bb4de2a25af4d4a9a5c5bc5441d19b8e3f6fcce7196c6afc2236 Outputs Yb: (length: 56 bytes) d61c6c039c01560e8b19b8299fb39513f39302eebd4c462694a33155 a3a387e44aa613647fcf6303f918bad598aaab53bea849b9fd14da74 MSGb = lv_cat(Yb,ADb): (length: 61 bytes) 38d61c6c039c01560e8b19b8299fb39513f39302eebd4c462694a331 55a3a387e44aa613647fcf6303f918bad598aaab53bea849b9fd14da 7403414462 B.4.4. Test vector for secret points K scalar_mult_vfy(ya,Yb): (length: 56 bytes) e434cda1783ddaaef08fc1d5f2201f1540fbc295fe2dd7cc38f20385 64824c98dbbe1978f121bdfead8e1a638913a6952cbec54867eb770a scalar_mult_vfy(yb,Ya): (length: 56 bytes) e434cda1783ddaaef08fc1d5f2201f1540fbc295fe2dd7cc38f20385 64824c98dbbe1978f121bdfead8e1a638913a6952cbec54867eb770a B.4.5. Test vector for ISK calculation initiator/responder unordered cat of transcript : (length: 122 bytes) 38d4b87d2fcdcac1096dba1898361f27e29dc1e019f74f84a71199bf d3dc8d09d2b823038f579f517591474be366968e2fb599bf14e55704 f40341446138d61c6c039c01560e8b19b8299fb39513f39302eebd4c 462694a33155a3a387e44aa613647fcf6303f918bad598aaab53bea8 49b9fd14da7403414462 DSI = G.DSI_ISK, b'CPaceDecaf448_ISK': (length: 17 bytes) 435061636544656361663434385f49534b lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 214 bytes) 11435061636544656361663434385f49534b105223e0cdc45d657566 8d64c55200412438e434cda1783ddaaef08fc1d5f2201f1540fbc295 fe2dd7cc38f2038564824c98dbbe1978f121bdfead8e1a638913a695 2cbec54867eb770a38d4b87d2fcdcac1096dba1898361f27e29dc1e0 19f74f84a71199bfd3dc8d09d2b823038f579f517591474be366968e 2fb599bf14e55704f40341446138d61c6c039c01560e8b19b8299fb3 9513f39302eebd4c462694a33155a3a387e44aa613647fcf6303f918 bad598aaab53bea849b9fd14da7403414462 ISK result: (length: 64 bytes) 13636dc9b7d233ac24a2d5c4a85a72fe20145f7a47ad51cab40e087c 057831b69ee59b9c828732bde171cfca99afda4852bcaf04fe9f0a97 592cdf5e2c9a5948 Abdalla, et al. Expires 26 March 2024 [Page 50] Internet-Draft CPace September 2023 B.4.6. Test vector for ISK calculation parallel execution ordered cat of transcript : (length: 122 bytes) 38d61c6c039c01560e8b19b8299fb39513f39302eebd4c462694a331 55a3a387e44aa613647fcf6303f918bad598aaab53bea849b9fd14da 740341446238d4b87d2fcdcac1096dba1898361f27e29dc1e019f74f 84a71199bfd3dc8d09d2b823038f579f517591474be366968e2fb599 bf14e55704f403414461 DSI = G.DSI_ISK, b'CPaceDecaf448_ISK': (length: 17 bytes) 435061636544656361663434385f49534b lv_cat(DSI,sid,K)||oCAT(MSGa,MSGb): (length: 214 bytes) 11435061636544656361663434385f49534b105223e0cdc45d657566 8d64c55200412438e434cda1783ddaaef08fc1d5f2201f1540fbc295 fe2dd7cc38f2038564824c98dbbe1978f121bdfead8e1a638913a695 2cbec54867eb770a38d61c6c039c01560e8b19b8299fb39513f39302 eebd4c462694a33155a3a387e44aa613647fcf6303f918bad598aaab 53bea849b9fd14da740341446238d4b87d2fcdcac1096dba1898361f 27e29dc1e019f74f84a71199bfd3dc8d09d2b823038f579f51759147 4be366968e2fb599bf14e55704f403414461 ISK result: (length: 64 bytes) 999e8f8486670bc1bf874a4d8f1496b9ebd8909eb01cf46b275ec942 2f22593064b272ba9e9e201a4a34a18729e48859a2d038c7c8cf0a0f e8a90ddcbdde1126 B.4.7. Corresponding ANSI-C initializers const uint8_t tc_PRS[] = { 0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64, }; const uint8_t tc_CI[] = { 0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a, 0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72, }; const uint8_t tc_sid[] = { 0x52,0x23,0xe0,0xcd,0xc4,0x5d,0x65,0x75,0x66,0x8d,0x64,0xc5, 0x52,0x00,0x41,0x24, }; const uint8_t tc_g[] = { 0x68,0x2d,0x1a,0x4f,0x49,0xfc,0x2a,0x48,0x34,0x35,0x6a,0xe4, 0xd7,0xf5,0x86,0x36,0xbc,0x94,0x81,0x52,0x1c,0x84,0x5e,0x66, 0xe6,0xfb,0x0b,0x29,0x69,0x34,0x1d,0xf4,0x5f,0xba,0xea,0xea, 0x9e,0x22,0x21,0xb3,0xf5,0xba,0xbc,0x54,0xc5,0xf8,0xce,0x45, 0x69,0x88,0xff,0xc5,0x19,0xde,0xfa,0xeb, }; const uint8_t tc_ya[] = { 0xd8,0xd2,0xe2,0x6c,0x82,0x1a,0x12,0xd7,0xf5,0x9a,0x8d,0xee, 0x02,0x3d,0x3f,0x61,0x55,0x97,0x61,0x52,0xe1,0x6c,0x73,0xcb, 0xf6,0x8c,0x30,0x3d,0xf0,0x40,0x43,0x99,0xf0,0xa7,0xb6,0x14, Abdalla, et al. Expires 26 March 2024 [Page 51] Internet-Draft CPace September 2023 0xa6,0x5d,0xf5,0x0a,0x97,0x88,0xf0,0x0b,0x41,0x05,0x86,0xb4, 0x43,0xf7,0x38,0xad,0x7f,0xf0,0x39,0x30, }; const uint8_t tc_ADa[] = { 0x41,0x44,0x61, }; const uint8_t tc_Ya[] = { 0xd4,0xb8,0x7d,0x2f,0xcd,0xca,0xc1,0x09,0x6d,0xba,0x18,0x98, 0x36,0x1f,0x27,0xe2,0x9d,0xc1,0xe0,0x19,0xf7,0x4f,0x84,0xa7, 0x11,0x99,0xbf,0xd3,0xdc,0x8d,0x09,0xd2,0xb8,0x23,0x03,0x8f, 0x57,0x9f,0x51,0x75,0x91,0x47,0x4b,0xe3,0x66,0x96,0x8e,0x2f, 0xb5,0x99,0xbf,0x14,0xe5,0x57,0x04,0xf4, }; const uint8_t tc_yb[] = { 0x91,0xba,0xe9,0x79,0x3f,0x4a,0x8a,0xce,0xb1,0xb5,0xc5,0x43, 0x75,0xa7,0xed,0x18,0x58,0xa7,0x9a,0x6e,0x72,0xda,0xb9,0x59, 0xc8,0xbd,0xf3,0xa7,0x5a,0xc9,0xbb,0x4d,0xe2,0xa2,0x5a,0xf4, 0xd4,0xa9,0xa5,0xc5,0xbc,0x54,0x41,0xd1,0x9b,0x8e,0x3f,0x6f, 0xcc,0xe7,0x19,0x6c,0x6a,0xfc,0x22,0x36, }; const uint8_t tc_ADb[] = { 0x41,0x44,0x62, }; const uint8_t tc_Yb[] = { 0xd6,0x1c,0x6c,0x03,0x9c,0x01,0x56,0x0e,0x8b,0x19,0xb8,0x29, 0x9f,0xb3,0x95,0x13,0xf3,0x93,0x02,0xee,0xbd,0x4c,0x46,0x26, 0x94,0xa3,0x31,0x55,0xa3,0xa3,0x87,0xe4,0x4a,0xa6,0x13,0x64, 0x7f,0xcf,0x63,0x03,0xf9,0x18,0xba,0xd5,0x98,0xaa,0xab,0x53, 0xbe,0xa8,0x49,0xb9,0xfd,0x14,0xda,0x74, }; const uint8_t tc_K[] = { 0xe4,0x34,0xcd,0xa1,0x78,0x3d,0xda,0xae,0xf0,0x8f,0xc1,0xd5, 0xf2,0x20,0x1f,0x15,0x40,0xfb,0xc2,0x95,0xfe,0x2d,0xd7,0xcc, 0x38,0xf2,0x03,0x85,0x64,0x82,0x4c,0x98,0xdb,0xbe,0x19,0x78, 0xf1,0x21,0xbd,0xfe,0xad,0x8e,0x1a,0x63,0x89,0x13,0xa6,0x95, 0x2c,0xbe,0xc5,0x48,0x67,0xeb,0x77,0x0a, }; const uint8_t tc_ISK_IR[] = { 0x13,0x63,0x6d,0xc9,0xb7,0xd2,0x33,0xac,0x24,0xa2,0xd5,0xc4, 0xa8,0x5a,0x72,0xfe,0x20,0x14,0x5f,0x7a,0x47,0xad,0x51,0xca, 0xb4,0x0e,0x08,0x7c,0x05,0x78,0x31,0xb6,0x9e,0xe5,0x9b,0x9c, 0x82,0x87,0x32,0xbd,0xe1,0x71,0xcf,0xca,0x99,0xaf,0xda,0x48, 0x52,0xbc,0xaf,0x04,0xfe,0x9f,0x0a,0x97,0x59,0x2c,0xdf,0x5e, 0x2c,0x9a,0x59,0x48, }; const uint8_t tc_ISK_SY[] = { 0x99,0x9e,0x8f,0x84,0x86,0x67,0x0b,0xc1,0xbf,0x87,0x4a,0x4d, 0x8f,0x14,0x96,0xb9,0xeb,0xd8,0x90,0x9e,0xb0,0x1c,0xf4,0x6b, Abdalla, et al. Expires 26 March 2024 [Page 52] Internet-Draft CPace September 2023 0x27,0x5e,0xc9,0x42,0x2f,0x22,0x59,0x30,0x64,0xb2,0x72,0xba, 0x9e,0x9e,0x20,0x1a,0x4a,0x34,0xa1,0x87,0x29,0xe4,0x88,0x59, 0xa2,0xd0,0x38,0xc7,0xc8,0xcf,0x0a,0x0f,0xe8,0xa9,0x0d,0xdc, 0xbd,0xde,0x11,0x26, }; B.4.8. Test case for scalar_mult with valid inputs s: (length: 56 bytes) dd1bc7015daabb7672129cc35a3ba815486b139deff9bdeca7a4fc61 34323d34658761e90ff079972a7ca8aa5606498f4f4f0ebc0933a819 X: (length: 56 bytes) 601431d5e51f43d422a92d3fb2373bde28217aab42524c341aa404ea ba5aa5541f7042dbb3253ce4c90f772b038a413dcb3a0f6bf3ae9e21 G.scalar_mult(s,decode(X)): (length: 56 bytes) 388b35c60eb41b66085a2118316218681d78979d667702de105fdc1f 21ffe884a577d795f45691781390a229a3bd7b527e831380f2f585a4 G.scalar_mult_vfy(s,X): (length: 56 bytes) 388b35c60eb41b66085a2118316218681d78979d667702de105fdc1f 21ffe884a577d795f45691781390a229a3bd7b527e831380f2f585a4 B.4.9. Invalid inputs for scalar_mult_vfy For these test cases scalar_mult_vfy(y,.) MUST return the representation of the neutral element G.I. When points Y_i1 or Y_i2 are included in MSGa or MSGb the protocol MUST abort. s: (length: 56 bytes) dd1bc7015daabb7672129cc35a3ba815486b139deff9bdeca7a4fc61 34323d34658761e90ff079972a7ca8aa5606498f4f4f0ebc0933a819 Y_i1: (length: 56 bytes) 5f1431d5e51f43d422a92d3fb2373bde28217aab42524c341aa404ea ba5aa5541f7042dbb3253ce4c90f772b038a413dcb3a0f6bf3ae9e21 Y_i2 == G.I: (length: 56 bytes) 00000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000 G.scalar_mult_vfy(s,Y_i1) = G.scalar_mult_vfy(s,Y_i2) = G.I B.5. Test vector for CPace using group NIST P-256 and hash SHA-256 B.5.1. Test vectors for calculate_generator with group NIST P-256 Abdalla, et al. Expires 26 March 2024 [Page 53] Internet-Draft CPace September 2023 Inputs H = SHA-256 with input block size 64 bytes. PRS = b'Password' ; ZPAD length: 23 ; DSI = b'CPaceP256_XMD:SHA-256_SSWU_NU_' DST = b'CPaceP256_XMD:SHA-256_SSWU_NU__DST' CI = b'\nAinitiator\nBresponder' CI = 0a41696e69746961746f720a42726573706f6e646572 sid = 34b36454cab2e7842c389f7d88ecb7df Outputs generator_string(PRS,G.DSI,CI,sid,H.s_in_bytes): (length: 104 bytes) 1e4350616365503235365f584d443a5348412d3235365f535357555f 4e555f0850617373776f726417000000000000000000000000000000 0000000000000000160a41696e69746961746f720a42726573706f6e 6465721034b36454cab2e7842c389f7d88ecb7df generator g: (length: 65 bytes) 041b51433114e096c9d595f0955f5717a75169afb95557f4a6f51155 035dee19c76887bce5c7c054fa1fe48a4a62c7fb96dc75e34259d2f7 2b8d41f31b8e586bcd B.5.2. Test vector for MSGa Inputs ADa = b'ADa' ya (big endian): (length: 32 bytes) 37574cfbf1b95ff6a8e2d7be462d4d01e6dde2618f34f4de9df869b2 4f532c5d Outputs Ya: (length: 65 bytes) 04b75c1bcda84a0f324aabb7f25cf853ed7fb327c33f23db6aeb320d 81df014649c2ac691925fce0eceac7dbc75eca25e6a1558066a610b4 021488279e3b989d52 Alternative correct value for Ya: g^(-ya): (length: 65 bytes) 04b75c1bcda84a0f324aabb7f25cf853ed7fb327c33f23db6aeb320d 81df0146493d5396e5da031f1415382438a135da195eaa7f9a59ef4b fdeb77d861c46762ad MSGa = lv_cat(Ya,ADa): (length: 70 bytes) 4104b75c1bcda84a0f324aabb7f25cf853ed7fb327c33f23db6aeb32 0d81df014649c2ac691925fce0eceac7dbc75eca25e6a1558066a610 b4021488279e3b989d5203414461 B.5.3. Test vector for MSGb Abdalla, et al. Expires 26 March 2024 [Page 54] Internet-Draft CPace September 2023 Inputs ADb = b'ADb' yb (big endian): (length: 32 bytes) e5672fc9eb4e721f41d80181ec4c9fd9886668acc48024d33c82bb10 2aecba52 Outputs Yb: (length: 65 bytes) 04bb2783a57337e74671f76452876b27839c0ea9e044e3aadaad2e64 777ed27a90e80a99438e2f1c072462f2895c6dadf1b43867b92ffb65 562b78c793947dcada Alternative correct value for Yb: g^(-yb): (length: 65 bytes) 04bb2783a57337e74671f76452876b27839c0ea9e044e3aadaad2e64 777ed27a9017f566bb71d0e3f9db9d0d76a392520e4bc79847d0049a a9d487386c6b823525 MSGb = lv_cat(Yb,ADb): (length: 70 bytes) 4104bb2783a57337e74671f76452876b27839c0ea9e044e3aadaad2e 64777ed27a90e80a99438e2f1c072462f2895c6dadf1b43867b92ffb 65562b78c793947dcada03414462 B.5.4. Test vector for secret points K scalar_mult_vfy(ya,Yb): (length: 32 bytes) 8fd12b283805750aeee6151bcd4211a6b71019e8fc416293ade24ed2 bce12c39 scalar_mult_vfy(yb,Ya): (length: 32 bytes) 8fd12b283805750aeee6151bcd4211a6b71019e8fc416293ade24ed2 bce12c39 B.5.5. Test vector for ISK calculation initiator/responder Abdalla, et al. Expires 26 March 2024 [Page 55] Internet-Draft CPace September 2023 unordered cat of transcript : (length: 140 bytes) 4104b75c1bcda84a0f324aabb7f25cf853ed7fb327c33f23db6aeb32 0d81df014649c2ac691925fce0eceac7dbc75eca25e6a1558066a610 b4021488279e3b989d52034144614104bb2783a57337e74671f76452 876b27839c0ea9e044e3aadaad2e64777ed27a90e80a99438e2f1c07 2462f2895c6dadf1b43867b92ffb65562b78c793947dcada03414462 DSI = G.DSI_ISK, b'CPaceP256_XMD:SHA-256_SSWU_NU__ISK': (length: 34 bytes) 4350616365503235365f584d443a5348412d3235365f535357555f4e 555f5f49534b lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 225 bytes) 224350616365503235365f584d443a5348412d3235365f535357555f 4e555f5f49534b1034b36454cab2e7842c389f7d88ecb7df208fd12b 283805750aeee6151bcd4211a6b71019e8fc416293ade24ed2bce12c 394104b75c1bcda84a0f324aabb7f25cf853ed7fb327c33f23db6aeb 320d81df014649c2ac691925fce0eceac7dbc75eca25e6a1558066a6 10b4021488279e3b989d52034144614104bb2783a57337e74671f764 52876b27839c0ea9e044e3aadaad2e64777ed27a90e80a99438e2f1c 072462f2895c6dadf1b43867b92ffb65562b78c793947dcada034144 62 ISK result: (length: 32 bytes) 7ae1e916606e44652e3c0d7231198af6519226339c241e546afd0bbf 48e1c96a B.5.6. Test vector for ISK calculation parallel execution Abdalla, et al. Expires 26 March 2024 [Page 56] Internet-Draft CPace September 2023 ordered cat of transcript : (length: 140 bytes) 4104bb2783a57337e74671f76452876b27839c0ea9e044e3aadaad2e 64777ed27a90e80a99438e2f1c072462f2895c6dadf1b43867b92ffb 65562b78c793947dcada034144624104b75c1bcda84a0f324aabb7f2 5cf853ed7fb327c33f23db6aeb320d81df014649c2ac691925fce0ec eac7dbc75eca25e6a1558066a610b4021488279e3b989d5203414461 DSI = G.DSI_ISK, b'CPaceP256_XMD:SHA-256_SSWU_NU__ISK': (length: 34 bytes) 4350616365503235365f584d443a5348412d3235365f535357555f4e 555f5f49534b lv_cat(DSI,sid,K)||oCAT(MSGa,MSGb): (length: 225 bytes) 224350616365503235365f584d443a5348412d3235365f535357555f 4e555f5f49534b1034b36454cab2e7842c389f7d88ecb7df208fd12b 283805750aeee6151bcd4211a6b71019e8fc416293ade24ed2bce12c 394104bb2783a57337e74671f76452876b27839c0ea9e044e3aadaad 2e64777ed27a90e80a99438e2f1c072462f2895c6dadf1b43867b92f fb65562b78c793947dcada034144624104b75c1bcda84a0f324aabb7 f25cf853ed7fb327c33f23db6aeb320d81df014649c2ac691925fce0 eceac7dbc75eca25e6a1558066a610b4021488279e3b989d52034144 61 ISK result: (length: 32 bytes) c5b4e6d44f5bbb7637a77ec67afd768a1343c410f7e1f76f6549eb00 2623c0f1 B.5.7. Corresponding ANSI-C initializers const uint8_t tc_PRS[] = { 0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64, }; const uint8_t tc_CI[] = { 0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a, 0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72, }; const uint8_t tc_sid[] = { 0x34,0xb3,0x64,0x54,0xca,0xb2,0xe7,0x84,0x2c,0x38,0x9f,0x7d, 0x88,0xec,0xb7,0xdf, }; const uint8_t tc_g[] = { 0x04,0x1b,0x51,0x43,0x31,0x14,0xe0,0x96,0xc9,0xd5,0x95,0xf0, 0x95,0x5f,0x57,0x17,0xa7,0x51,0x69,0xaf,0xb9,0x55,0x57,0xf4, 0xa6,0xf5,0x11,0x55,0x03,0x5d,0xee,0x19,0xc7,0x68,0x87,0xbc, 0xe5,0xc7,0xc0,0x54,0xfa,0x1f,0xe4,0x8a,0x4a,0x62,0xc7,0xfb, 0x96,0xdc,0x75,0xe3,0x42,0x59,0xd2,0xf7,0x2b,0x8d,0x41,0xf3, 0x1b,0x8e,0x58,0x6b,0xcd, }; const uint8_t tc_ya[] = { 0x37,0x57,0x4c,0xfb,0xf1,0xb9,0x5f,0xf6,0xa8,0xe2,0xd7,0xbe, 0x46,0x2d,0x4d,0x01,0xe6,0xdd,0xe2,0x61,0x8f,0x34,0xf4,0xde, Abdalla, et al. Expires 26 March 2024 [Page 57] Internet-Draft CPace September 2023 0x9d,0xf8,0x69,0xb2,0x4f,0x53,0x2c,0x5d, }; const uint8_t tc_ADa[] = { 0x41,0x44,0x61, }; const uint8_t tc_Ya[] = { 0x04,0xb7,0x5c,0x1b,0xcd,0xa8,0x4a,0x0f,0x32,0x4a,0xab,0xb7, 0xf2,0x5c,0xf8,0x53,0xed,0x7f,0xb3,0x27,0xc3,0x3f,0x23,0xdb, 0x6a,0xeb,0x32,0x0d,0x81,0xdf,0x01,0x46,0x49,0xc2,0xac,0x69, 0x19,0x25,0xfc,0xe0,0xec,0xea,0xc7,0xdb,0xc7,0x5e,0xca,0x25, 0xe6,0xa1,0x55,0x80,0x66,0xa6,0x10,0xb4,0x02,0x14,0x88,0x27, 0x9e,0x3b,0x98,0x9d,0x52, }; const uint8_t tc_yb[] = { 0xe5,0x67,0x2f,0xc9,0xeb,0x4e,0x72,0x1f,0x41,0xd8,0x01,0x81, 0xec,0x4c,0x9f,0xd9,0x88,0x66,0x68,0xac,0xc4,0x80,0x24,0xd3, 0x3c,0x82,0xbb,0x10,0x2a,0xec,0xba,0x52, }; const uint8_t tc_ADb[] = { 0x41,0x44,0x62, }; const uint8_t tc_Yb[] = { 0x04,0xbb,0x27,0x83,0xa5,0x73,0x37,0xe7,0x46,0x71,0xf7,0x64, 0x52,0x87,0x6b,0x27,0x83,0x9c,0x0e,0xa9,0xe0,0x44,0xe3,0xaa, 0xda,0xad,0x2e,0x64,0x77,0x7e,0xd2,0x7a,0x90,0xe8,0x0a,0x99, 0x43,0x8e,0x2f,0x1c,0x07,0x24,0x62,0xf2,0x89,0x5c,0x6d,0xad, 0xf1,0xb4,0x38,0x67,0xb9,0x2f,0xfb,0x65,0x56,0x2b,0x78,0xc7, 0x93,0x94,0x7d,0xca,0xda, }; const uint8_t tc_K[] = { 0x8f,0xd1,0x2b,0x28,0x38,0x05,0x75,0x0a,0xee,0xe6,0x15,0x1b, 0xcd,0x42,0x11,0xa6,0xb7,0x10,0x19,0xe8,0xfc,0x41,0x62,0x93, 0xad,0xe2,0x4e,0xd2,0xbc,0xe1,0x2c,0x39, }; const uint8_t tc_ISK_IR[] = { 0x7a,0xe1,0xe9,0x16,0x60,0x6e,0x44,0x65,0x2e,0x3c,0x0d,0x72, 0x31,0x19,0x8a,0xf6,0x51,0x92,0x26,0x33,0x9c,0x24,0x1e,0x54, 0x6a,0xfd,0x0b,0xbf,0x48,0xe1,0xc9,0x6a, }; const uint8_t tc_ISK_SY[] = { 0xc5,0xb4,0xe6,0xd4,0x4f,0x5b,0xbb,0x76,0x37,0xa7,0x7e,0xc6, 0x7a,0xfd,0x76,0x8a,0x13,0x43,0xc4,0x10,0xf7,0xe1,0xf7,0x6f, 0x65,0x49,0xeb,0x00,0x26,0x23,0xc0,0xf1, }; B.5.8. Test case for scalar_mult_vfy with correct inputs Abdalla, et al. Expires 26 March 2024 [Page 58] Internet-Draft CPace September 2023 s: (length: 32 bytes) f012501c091ff9b99a123fffe571d8bc01e8077ee581362e1bd21399 0835643b X: (length: 65 bytes) 0424648eb986c2be0af636455cef0550671d6bcd8aa26e0d72ffa1b1 fd12ba4e0f78da2b6d2184f31af39e566aef127014b6936c9a37346d 10a4ab2514faef5831 G.scalar_mult(s,X) (full coordinates): (length: 65 bytes) 04f5a191f078c87c36633b78c701751159d56c59f3fe9105b5720673 470f303ab925b6a7fd1cdd8f649a21cf36b68d9e9c4a11919a951892 519786104b27033757 G.scalar_mult_vfy(s,X) (only X-coordinate): (length: 32 bytes) f5a191f078c87c36633b78c701751159d56c59f3fe9105b572067347 0f303ab9 B.5.9. Invalid inputs for scalar_mult_vfy For these test cases scalar_mult_vfy(y,.) MUST return the representation of the neutral element G.I. When including Y_i1 or Y_i2 in MSGa or MSGb the protocol MUST abort. s: (length: 32 bytes) f012501c091ff9b99a123fffe571d8bc01e8077ee581362e1bd21399 0835643b Y_i1: (length: 65 bytes) 0424648eb986c2be0af636455cef0550671d6bcd8aa26e0d72ffa1b1 fd12ba4e0f78da2b6d2184f31af39e566aef127014b6936c9a37346d 10a4ab2514faef5857 Y_i2: (length: 1 bytes) 00 G.scalar_mult_vfy(s,Y_i1) = G.scalar_mult_vfy(s,Y_i2) = G.I B.6. Test vector for CPace using group NIST P-384 and hash SHA-384 B.6.1. Test vectors for calculate_generator with group NIST P-384 Abdalla, et al. Expires 26 March 2024 [Page 59] Internet-Draft CPace September 2023 Inputs H = SHA-384 with input block size 128 bytes. PRS = b'Password' ; ZPAD length: 87 ; DSI = b'CPaceP384_XMD:SHA-384_SSWU_NU_' DST = b'CPaceP384_XMD:SHA-384_SSWU_NU__DST' CI = b'\nAinitiator\nBresponder' CI = 0a41696e69746961746f720a42726573706f6e646572 sid = 5b3773aa90e8f23c61563a4b645b276c Outputs generator_string(PRS,G.DSI,CI,sid,H.s_in_bytes): (length: 168 bytes) 1e4350616365503338345f584d443a5348412d3338345f535357555f 4e555f0850617373776f726457000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000160a41696e69746961746f72 0a42726573706f6e646572105b3773aa90e8f23c61563a4b645b276c generator g: (length: 97 bytes) 04f35a925fe82e54350e80b084a8013b1960cb3f73c49b0c2ae9b523 997846ddd14c66f24f62223112cf35b866065f91ad86674cce2a2876 84904e49f01287b54666bb518df2ea53cec627fa6e1283f14c6ed4bc d11b33fbb962da3e2e4ff1345c B.6.2. Test vector for MSGa Inputs ADa = b'ADa' ya (big endian): (length: 48 bytes) 7d5bc6a8959f9db2655b8b6642e393dc13d25150d69c6675fb3efd41 ae6255bf54202b960f9aacd97fd6d2841b461f18 Outputs Ya: (length: 97 bytes) 048b65b9ef4c5726664391ceeae241834b275960a6f9316799f5c974 eceb71dfb6d36e989addf2ae8c4e338f204b2cd754e1c43b43a12692 8d8d81ce2e6edbc22a99ed478ad3487b87e1052bce2d94b6464a2228 eab73c01f79d6b290af6b218cf Alternative correct value for Ya: g^(-ya): (length: 97 bytes) 048b65b9ef4c5726664391ceeae241834b275960a6f9316799f5c974 eceb71dfb6d36e989addf2ae8c4e338f204b2cd7541e3bc4bc5ed96d 72727e31d191243dd56612b8752cb784781efad431d26b49b8b5ddd7 1448c3fe086294d6f6094de730 MSGa = lv_cat(Ya,ADa): (length: 102 bytes) 61048b65b9ef4c5726664391ceeae241834b275960a6f9316799f5c9 74eceb71dfb6d36e989addf2ae8c4e338f204b2cd754e1c43b43a126 928d8d81ce2e6edbc22a99ed478ad3487b87e1052bce2d94b6464a22 28eab73c01f79d6b290af6b218cf03414461 Abdalla, et al. Expires 26 March 2024 [Page 60] Internet-Draft CPace September 2023 B.6.3. Test vector for MSGb Inputs ADb = b'ADb' yb (big endian): (length: 48 bytes) 5cc9465bdb3ae626b77521ea36218fc93a9693c36ff126899e3d8777 c126ef05483e34c05576c9e8c64b1a0b6f5b53d1 Outputs Yb: (length: 97 bytes) 04cb68451813699abda3dc0ed9d521baf9108bc2c4b2a1dbcd90a083 63f5e458938d6fe634ed6393bc8440ec9b9f8a30841ffcc7cd65d9cf 9617155e129bccded9888ac738f78e940f9887f9089ab6275d36c3ab 1bccbd048b088b1b80a0f56f27 Alternative correct value for Yb: g^(-yb): (length: 97 bytes) 04cb68451813699abda3dc0ed9d521baf9108bc2c4b2a1dbcd90a083 63f5e458938d6fe634ed6393bc8440ec9b9f8a3084e00338329a2630 69e8eaa1ed64332126777538c708716bf0677806f76549d8a1c93c54 e33342fb74f774e4805f0a90d8 MSGb = lv_cat(Yb,ADb): (length: 102 bytes) 6104cb68451813699abda3dc0ed9d521baf9108bc2c4b2a1dbcd90a0 8363f5e458938d6fe634ed6393bc8440ec9b9f8a30841ffcc7cd65d9 cf9617155e129bccded9888ac738f78e940f9887f9089ab6275d36c3 ab1bccbd048b088b1b80a0f56f2703414462 B.6.4. Test vector for secret points K scalar_mult_vfy(ya,Yb): (length: 48 bytes) c862709d6bfe7cc02f0c11dafdbf4ef8db1c5e4cb13a22985a83bef1 8631361ed7d8cd97b12931844b7ac61b2f31d332 scalar_mult_vfy(yb,Ya): (length: 48 bytes) c862709d6bfe7cc02f0c11dafdbf4ef8db1c5e4cb13a22985a83bef1 8631361ed7d8cd97b12931844b7ac61b2f31d332 B.6.5. Test vector for ISK calculation initiator/responder Abdalla, et al. Expires 26 March 2024 [Page 61] Internet-Draft CPace September 2023 unordered cat of transcript : (length: 204 bytes) 61048b65b9ef4c5726664391ceeae241834b275960a6f9316799f5c9 74eceb71dfb6d36e989addf2ae8c4e338f204b2cd754e1c43b43a126 928d8d81ce2e6edbc22a99ed478ad3487b87e1052bce2d94b6464a22 28eab73c01f79d6b290af6b218cf034144616104cb68451813699abd a3dc0ed9d521baf9108bc2c4b2a1dbcd90a08363f5e458938d6fe634 ed6393bc8440ec9b9f8a30841ffcc7cd65d9cf9617155e129bccded9 888ac738f78e940f9887f9089ab6275d36c3ab1bccbd048b088b1b80 a0f56f2703414462 DSI = G.DSI_ISK, b'CPaceP384_XMD:SHA-384_SSWU_NU__ISK': (length: 34 bytes) 4350616365503338345f584d443a5348412d3338345f535357555f4e 555f5f49534b lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 305 bytes) 224350616365503338345f584d443a5348412d3338345f535357555f 4e555f5f49534b105b3773aa90e8f23c61563a4b645b276c30c86270 9d6bfe7cc02f0c11dafdbf4ef8db1c5e4cb13a22985a83bef1863136 1ed7d8cd97b12931844b7ac61b2f31d33261048b65b9ef4c57266643 91ceeae241834b275960a6f9316799f5c974eceb71dfb6d36e989add f2ae8c4e338f204b2cd754e1c43b43a126928d8d81ce2e6edbc22a99 ed478ad3487b87e1052bce2d94b6464a2228eab73c01f79d6b290af6 b218cf034144616104cb68451813699abda3dc0ed9d521baf9108bc2 c4b2a1dbcd90a08363f5e458938d6fe634ed6393bc8440ec9b9f8a30 841ffcc7cd65d9cf9617155e129bccded9888ac738f78e940f9887f9 089ab6275d36c3ab1bccbd048b088b1b80a0f56f2703414462 ISK result: (length: 48 bytes) db1e8133be8359b9aa8cd563043ee784344f26580876862e28b3f98b 51b2f611a65362c1d77db66c879de466f5b6148a B.6.6. Test vector for ISK calculation parallel execution Abdalla, et al. Expires 26 March 2024 [Page 62] Internet-Draft CPace September 2023 ordered cat of transcript : (length: 204 bytes) 6104cb68451813699abda3dc0ed9d521baf9108bc2c4b2a1dbcd90a0 8363f5e458938d6fe634ed6393bc8440ec9b9f8a30841ffcc7cd65d9 cf9617155e129bccded9888ac738f78e940f9887f9089ab6275d36c3 ab1bccbd048b088b1b80a0f56f270341446261048b65b9ef4c572666 4391ceeae241834b275960a6f9316799f5c974eceb71dfb6d36e989a ddf2ae8c4e338f204b2cd754e1c43b43a126928d8d81ce2e6edbc22a 99ed478ad3487b87e1052bce2d94b6464a2228eab73c01f79d6b290a f6b218cf03414461 DSI = G.DSI_ISK, b'CPaceP384_XMD:SHA-384_SSWU_NU__ISK': (length: 34 bytes) 4350616365503338345f584d443a5348412d3338345f535357555f4e 555f5f49534b lv_cat(DSI,sid,K)||oCAT(MSGa,MSGb): (length: 305 bytes) 224350616365503338345f584d443a5348412d3338345f535357555f 4e555f5f49534b105b3773aa90e8f23c61563a4b645b276c30c86270 9d6bfe7cc02f0c11dafdbf4ef8db1c5e4cb13a22985a83bef1863136 1ed7d8cd97b12931844b7ac61b2f31d3326104cb68451813699abda3 dc0ed9d521baf9108bc2c4b2a1dbcd90a08363f5e458938d6fe634ed 6393bc8440ec9b9f8a30841ffcc7cd65d9cf9617155e129bccded988 8ac738f78e940f9887f9089ab6275d36c3ab1bccbd048b088b1b80a0 f56f270341446261048b65b9ef4c5726664391ceeae241834b275960 a6f9316799f5c974eceb71dfb6d36e989addf2ae8c4e338f204b2cd7 54e1c43b43a126928d8d81ce2e6edbc22a99ed478ad3487b87e1052b ce2d94b6464a2228eab73c01f79d6b290af6b218cf03414461 ISK result: (length: 48 bytes) 519bfbb1477652e8ed1b4ec5774e310c4f44da46f3c36be91b0dd6b4 e3a3245942cf4d9db8f79023dad6e1b57aed4891 B.6.7. Corresponding ANSI-C initializers const uint8_t tc_PRS[] = { 0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64, }; const uint8_t tc_CI[] = { 0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a, 0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72, }; const uint8_t tc_sid[] = { 0x5b,0x37,0x73,0xaa,0x90,0xe8,0xf2,0x3c,0x61,0x56,0x3a,0x4b, 0x64,0x5b,0x27,0x6c, }; const uint8_t tc_g[] = { 0x04,0xf3,0x5a,0x92,0x5f,0xe8,0x2e,0x54,0x35,0x0e,0x80,0xb0, 0x84,0xa8,0x01,0x3b,0x19,0x60,0xcb,0x3f,0x73,0xc4,0x9b,0x0c, 0x2a,0xe9,0xb5,0x23,0x99,0x78,0x46,0xdd,0xd1,0x4c,0x66,0xf2, 0x4f,0x62,0x22,0x31,0x12,0xcf,0x35,0xb8,0x66,0x06,0x5f,0x91, 0xad,0x86,0x67,0x4c,0xce,0x2a,0x28,0x76,0x84,0x90,0x4e,0x49, Abdalla, et al. Expires 26 March 2024 [Page 63] Internet-Draft CPace September 2023 0xf0,0x12,0x87,0xb5,0x46,0x66,0xbb,0x51,0x8d,0xf2,0xea,0x53, 0xce,0xc6,0x27,0xfa,0x6e,0x12,0x83,0xf1,0x4c,0x6e,0xd4,0xbc, 0xd1,0x1b,0x33,0xfb,0xb9,0x62,0xda,0x3e,0x2e,0x4f,0xf1,0x34, 0x5c, }; const uint8_t tc_ya[] = { 0x7d,0x5b,0xc6,0xa8,0x95,0x9f,0x9d,0xb2,0x65,0x5b,0x8b,0x66, 0x42,0xe3,0x93,0xdc,0x13,0xd2,0x51,0x50,0xd6,0x9c,0x66,0x75, 0xfb,0x3e,0xfd,0x41,0xae,0x62,0x55,0xbf,0x54,0x20,0x2b,0x96, 0x0f,0x9a,0xac,0xd9,0x7f,0xd6,0xd2,0x84,0x1b,0x46,0x1f,0x18, }; const uint8_t tc_ADa[] = { 0x41,0x44,0x61, }; const uint8_t tc_Ya[] = { 0x04,0x8b,0x65,0xb9,0xef,0x4c,0x57,0x26,0x66,0x43,0x91,0xce, 0xea,0xe2,0x41,0x83,0x4b,0x27,0x59,0x60,0xa6,0xf9,0x31,0x67, 0x99,0xf5,0xc9,0x74,0xec,0xeb,0x71,0xdf,0xb6,0xd3,0x6e,0x98, 0x9a,0xdd,0xf2,0xae,0x8c,0x4e,0x33,0x8f,0x20,0x4b,0x2c,0xd7, 0x54,0xe1,0xc4,0x3b,0x43,0xa1,0x26,0x92,0x8d,0x8d,0x81,0xce, 0x2e,0x6e,0xdb,0xc2,0x2a,0x99,0xed,0x47,0x8a,0xd3,0x48,0x7b, 0x87,0xe1,0x05,0x2b,0xce,0x2d,0x94,0xb6,0x46,0x4a,0x22,0x28, 0xea,0xb7,0x3c,0x01,0xf7,0x9d,0x6b,0x29,0x0a,0xf6,0xb2,0x18, 0xcf, }; const uint8_t tc_yb[] = { 0x5c,0xc9,0x46,0x5b,0xdb,0x3a,0xe6,0x26,0xb7,0x75,0x21,0xea, 0x36,0x21,0x8f,0xc9,0x3a,0x96,0x93,0xc3,0x6f,0xf1,0x26,0x89, 0x9e,0x3d,0x87,0x77,0xc1,0x26,0xef,0x05,0x48,0x3e,0x34,0xc0, 0x55,0x76,0xc9,0xe8,0xc6,0x4b,0x1a,0x0b,0x6f,0x5b,0x53,0xd1, }; const uint8_t tc_ADb[] = { 0x41,0x44,0x62, }; const uint8_t tc_Yb[] = { 0x04,0xcb,0x68,0x45,0x18,0x13,0x69,0x9a,0xbd,0xa3,0xdc,0x0e, 0xd9,0xd5,0x21,0xba,0xf9,0x10,0x8b,0xc2,0xc4,0xb2,0xa1,0xdb, 0xcd,0x90,0xa0,0x83,0x63,0xf5,0xe4,0x58,0x93,0x8d,0x6f,0xe6, 0x34,0xed,0x63,0x93,0xbc,0x84,0x40,0xec,0x9b,0x9f,0x8a,0x30, 0x84,0x1f,0xfc,0xc7,0xcd,0x65,0xd9,0xcf,0x96,0x17,0x15,0x5e, 0x12,0x9b,0xcc,0xde,0xd9,0x88,0x8a,0xc7,0x38,0xf7,0x8e,0x94, 0x0f,0x98,0x87,0xf9,0x08,0x9a,0xb6,0x27,0x5d,0x36,0xc3,0xab, 0x1b,0xcc,0xbd,0x04,0x8b,0x08,0x8b,0x1b,0x80,0xa0,0xf5,0x6f, 0x27, }; const uint8_t tc_K[] = { 0xc8,0x62,0x70,0x9d,0x6b,0xfe,0x7c,0xc0,0x2f,0x0c,0x11,0xda, 0xfd,0xbf,0x4e,0xf8,0xdb,0x1c,0x5e,0x4c,0xb1,0x3a,0x22,0x98, Abdalla, et al. Expires 26 March 2024 [Page 64] Internet-Draft CPace September 2023 0x5a,0x83,0xbe,0xf1,0x86,0x31,0x36,0x1e,0xd7,0xd8,0xcd,0x97, 0xb1,0x29,0x31,0x84,0x4b,0x7a,0xc6,0x1b,0x2f,0x31,0xd3,0x32, }; const uint8_t tc_ISK_IR[] = { 0xdb,0x1e,0x81,0x33,0xbe,0x83,0x59,0xb9,0xaa,0x8c,0xd5,0x63, 0x04,0x3e,0xe7,0x84,0x34,0x4f,0x26,0x58,0x08,0x76,0x86,0x2e, 0x28,0xb3,0xf9,0x8b,0x51,0xb2,0xf6,0x11,0xa6,0x53,0x62,0xc1, 0xd7,0x7d,0xb6,0x6c,0x87,0x9d,0xe4,0x66,0xf5,0xb6,0x14,0x8a, }; const uint8_t tc_ISK_SY[] = { 0x51,0x9b,0xfb,0xb1,0x47,0x76,0x52,0xe8,0xed,0x1b,0x4e,0xc5, 0x77,0x4e,0x31,0x0c,0x4f,0x44,0xda,0x46,0xf3,0xc3,0x6b,0xe9, 0x1b,0x0d,0xd6,0xb4,0xe3,0xa3,0x24,0x59,0x42,0xcf,0x4d,0x9d, 0xb8,0xf7,0x90,0x23,0xda,0xd6,0xe1,0xb5,0x7a,0xed,0x48,0x91, }; B.6.8. Test case for scalar_mult_vfy with correct inputs s: (length: 48 bytes) 6e8a99a5cdd408eae98e1b8aed286e7b12adbbdac7f2c628d9060ce9 2ae0d90bd57a564fd3500fbcce3425dc94ba0ade X: (length: 97 bytes) 045b4cd53c4506cc04ba4c44f2762d5d32c3e55df25b8baa5571b165 7ad9576efea8259f0684de065a470585b4be876748c7797054f3defe f21b77f83d53bac57c89d52aa4d6dd5872bd281989b138359698009f 8ac1f301538badcce9d9f4036e G.scalar_mult(s,X) (full coordinates): (length: 97 bytes) 0465c28db05fd9f9a93651c5cc31eae49c4e5246b46489b8f6105873 3173a033cda76c3e3ea5352b804e67fdbe2e334be8245dad5c8c993e 63bacf0456478f29b71b6c859f13676f84ff150d2741f028f560584a 0bdba19a63df62c08949c2fd6d G.scalar_mult_vfy(s,X) (only X-coordinate): (length: 48 bytes) 65c28db05fd9f9a93651c5cc31eae49c4e5246b46489b8f610587331 73a033cda76c3e3ea5352b804e67fdbe2e334be8 B.6.9. Invalid inputs for scalar_mult_vfy For these test cases scalar_mult_vfy(y,.) MUST return the representation of the neutral element G.I. When including Y_i1 or Y_i2 in MSGa or MSGb the protocol MUST abort. Abdalla, et al. Expires 26 March 2024 [Page 65] Internet-Draft CPace September 2023 s: (length: 48 bytes) 6e8a99a5cdd408eae98e1b8aed286e7b12adbbdac7f2c628d9060ce9 2ae0d90bd57a564fd3500fbcce3425dc94ba0ade Y_i1: (length: 97 bytes) 045b4cd53c4506cc04ba4c44f2762d5d32c3e55df25b8baa5571b165 7ad9576efea8259f0684de065a470585b4be876748c7797054f3defe f21b77f83d53bac57c89d52aa4d6dd5872bd281989b138359698009f 8ac1f301538badcce9d9f40302 Y_i2: (length: 1 bytes) 00 G.scalar_mult_vfy(s,Y_i1) = G.scalar_mult_vfy(s,Y_i2) = G.I B.7. Test vector for CPace using group NIST P-521 and hash SHA-512 B.7.1. Test vectors for calculate_generator with group NIST P-521 Inputs H = SHA-512 with input block size 128 bytes. PRS = b'Password' ; ZPAD length: 87 ; DSI = b'CPaceP521_XMD:SHA-512_SSWU_NU_' DST = b'CPaceP521_XMD:SHA-512_SSWU_NU__DST' CI = b'\nAinitiator\nBresponder' CI = 0a41696e69746961746f720a42726573706f6e646572 sid = 7e4b4791d6a8ef019b936c79fb7f2c57 Outputs generator_string(PRS,G.DSI,CI,sid,H.s_in_bytes): (length: 168 bytes) 1e4350616365503532315f584d443a5348412d3531325f535357555f 4e555f0850617373776f726457000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000160a41696e69746961746f72 0a42726573706f6e646572107e4b4791d6a8ef019b936c79fb7f2c57 generator g: (length: 133 bytes) 0400dc927958f0b69ccad8fb67ef008905354b58c7c9c92ad50060a9 e6afb10437d6ca8a26164e8573702b897275a25d05ed4407af2a3849 86dca7e243b92c5dd500d40057012121a9c8e34373fa619f918f7d47 9c23f85f0485379ef0f05284398de26653b49a155324c9d7b138be84 d0b49bb58e232b7bf697798de6ee8afd6b92b6fa2f B.7.2. Test vector for MSGa Abdalla, et al. Expires 26 March 2024 [Page 66] Internet-Draft CPace September 2023 Inputs ADa = b'ADa' ya (big endian): (length: 66 bytes) 006367e9c2aeff9f1db19af600cca73343d47cbe446cebbd1ccd783f 82755a872da86fd0707eb3767c6114f1803deb62d63bdd1e613f67e6 3e8c141ee5310e3ee819 Outputs Ya: (length: 133 bytes) 04003701ec35caafa3dd416cad29ba1774551f9d2ed89f7e1065706d ca230b86a11d02e4cee8b3fde64380d4a05983167d8a2414bc594ad5 286c068792ab7ca60ff6ea00919c41c00e789dabc2f42fd94178d7bf d8fbe1aff1c1854b3dafb3a0ea13f5a5fc1703860f022bd271740469 bb322b07c179c7c225499b31727c0ea3ee65578634 Alternative correct value for Ya: g^(-ya): (length: 133 bytes) 04003701ec35caafa3dd416cad29ba1774551f9d2ed89f7e1065706d ca230b86a11d02e4cee8b3fde64380d4a05983167d8a2414bc594ad5 286c068792ab7ca60ff6ea016e63be3ff18762543d0bd026be872840 27041e500e3e7ab4c2504c5f15ec0a5a03e8fc79f0fdd42d8e8bfb96 44cdd4f83e86383ddab664ce8d83f15c119aa879cb MSGa = lv_cat(Ya,ADa): (length: 139 bytes) 850104003701ec35caafa3dd416cad29ba1774551f9d2ed89f7e1065 706dca230b86a11d02e4cee8b3fde64380d4a05983167d8a2414bc59 4ad5286c068792ab7ca60ff6ea00919c41c00e789dabc2f42fd94178 d7bfd8fbe1aff1c1854b3dafb3a0ea13f5a5fc1703860f022bd27174 0469bb322b07c179c7c225499b31727c0ea3ee6557863403414461 B.7.3. Test vector for MSGb Abdalla, et al. Expires 26 March 2024 [Page 67] Internet-Draft CPace September 2023 Inputs ADb = b'ADb' yb (big endian): (length: 66 bytes) 009227bf8dc741dacc9422f8bf3c0e96fce9587bc562eaafe0dc5f6f 82f28594e4a6f98553560c62b75fa4abb198cecbbb86ebd41b0ea025 4cde78ac68d39a240ae7 Outputs Yb: (length: 133 bytes) 0400f5cb68bf0117bd1a65412a2bc800af92013f9969cf546e1ea6d3 bcf08643fdc482130aec1eecc33a2b5f33600be51295047fa3399fa2 82cc1a78de91f3a4e30b5d01a085b453f22bf3dc947386b042e5fc4e c691fee47fe3c3ec6408c22a17c26bc0ab73940910614d6fcee32daf bfd2d340d6e382d71b1fc763d7cec502fbcbcf93b4 Alternative correct value for Yb: g^(-yb): (length: 133 bytes) 0400f5cb68bf0117bd1a65412a2bc800af92013f9969cf546e1ea6d3 bcf08643fdc482130aec1eecc33a2b5f33600be51295047fa3399fa2 82cc1a78de91f3a4e30b5d005f7a4bac0dd40c236b8c794fbd1a03b1 396e011b801c3c139bf73dd5e83d943f548c6bf6ef9eb290311cd250 402d2cbf291c7d28e4e0389c28313afd0434306c4b MSGb = lv_cat(Yb,ADb): (length: 139 bytes) 85010400f5cb68bf0117bd1a65412a2bc800af92013f9969cf546e1e a6d3bcf08643fdc482130aec1eecc33a2b5f33600be51295047fa339 9fa282cc1a78de91f3a4e30b5d01a085b453f22bf3dc947386b042e5 fc4ec691fee47fe3c3ec6408c22a17c26bc0ab73940910614d6fcee3 2dafbfd2d340d6e382d71b1fc763d7cec502fbcbcf93b403414462 B.7.4. Test vector for secret points K scalar_mult_vfy(ya,Yb): (length: 66 bytes) 00503e75e38e012a6dc6f3561980e4cf540dbcff3de3a4a6f09d79c3 2cc45764d3a6605eb45df1dc63fb7937b7879f2820da1b3266b69fa0 99bf8720dd8f6a07e8ed scalar_mult_vfy(yb,Ya): (length: 66 bytes) 00503e75e38e012a6dc6f3561980e4cf540dbcff3de3a4a6f09d79c3 2cc45764d3a6605eb45df1dc63fb7937b7879f2820da1b3266b69fa0 99bf8720dd8f6a07e8ed B.7.5. Test vector for ISK calculation initiator/responder Abdalla, et al. Expires 26 March 2024 [Page 68] Internet-Draft CPace September 2023 unordered cat of transcript : (length: 278 bytes) 850104003701ec35caafa3dd416cad29ba1774551f9d2ed89f7e1065 706dca230b86a11d02e4cee8b3fde64380d4a05983167d8a2414bc59 4ad5286c068792ab7ca60ff6ea00919c41c00e789dabc2f42fd94178 d7bfd8fbe1aff1c1854b3dafb3a0ea13f5a5fc1703860f022bd27174 0469bb322b07c179c7c225499b31727c0ea3ee655786340341446185 010400f5cb68bf0117bd1a65412a2bc800af92013f9969cf546e1ea6 d3bcf08643fdc482130aec1eecc33a2b5f33600be51295047fa3399f a282cc1a78de91f3a4e30b5d01a085b453f22bf3dc947386b042e5fc 4ec691fee47fe3c3ec6408c22a17c26bc0ab73940910614d6fcee32d afbfd2d340d6e382d71b1fc763d7cec502fbcbcf93b403414462 DSI = G.DSI_ISK, b'CPaceP521_XMD:SHA-512_SSWU_NU__ISK': (length: 34 bytes) 4350616365503532315f584d443a5348412d3531325f535357555f4e 555f5f49534b lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 397 bytes) 224350616365503532315f584d443a5348412d3531325f535357555f 4e555f5f49534b107e4b4791d6a8ef019b936c79fb7f2c574200503e 75e38e012a6dc6f3561980e4cf540dbcff3de3a4a6f09d79c32cc457 64d3a6605eb45df1dc63fb7937b7879f2820da1b3266b69fa099bf87 20dd8f6a07e8ed850104003701ec35caafa3dd416cad29ba1774551f 9d2ed89f7e1065706dca230b86a11d02e4cee8b3fde64380d4a05983 167d8a2414bc594ad5286c068792ab7ca60ff6ea00919c41c00e789d abc2f42fd94178d7bfd8fbe1aff1c1854b3dafb3a0ea13f5a5fc1703 860f022bd271740469bb322b07c179c7c225499b31727c0ea3ee6557 86340341446185010400f5cb68bf0117bd1a65412a2bc800af92013f 9969cf546e1ea6d3bcf08643fdc482130aec1eecc33a2b5f33600be5 1295047fa3399fa282cc1a78de91f3a4e30b5d01a085b453f22bf3dc 947386b042e5fc4ec691fee47fe3c3ec6408c22a17c26bc0ab739409 10614d6fcee32dafbfd2d340d6e382d71b1fc763d7cec502fbcbcf93 b403414462 ISK result: (length: 64 bytes) ed208a15af3ef8a67a5cac4acb360d03154570e3b1b1c54867f53a72 53cb919d13aa47efc647375be2250cb39ad965afa4ddfcb6be47d586 d28c7eef6d654525 B.7.6. Test vector for ISK calculation parallel execution Abdalla, et al. Expires 26 March 2024 [Page 69] Internet-Draft CPace September 2023 ordered cat of transcript : (length: 278 bytes) 85010400f5cb68bf0117bd1a65412a2bc800af92013f9969cf546e1e a6d3bcf08643fdc482130aec1eecc33a2b5f33600be51295047fa339 9fa282cc1a78de91f3a4e30b5d01a085b453f22bf3dc947386b042e5 fc4ec691fee47fe3c3ec6408c22a17c26bc0ab73940910614d6fcee3 2dafbfd2d340d6e382d71b1fc763d7cec502fbcbcf93b40341446285 0104003701ec35caafa3dd416cad29ba1774551f9d2ed89f7e106570 6dca230b86a11d02e4cee8b3fde64380d4a05983167d8a2414bc594a d5286c068792ab7ca60ff6ea00919c41c00e789dabc2f42fd94178d7 bfd8fbe1aff1c1854b3dafb3a0ea13f5a5fc1703860f022bd2717404 69bb322b07c179c7c225499b31727c0ea3ee6557863403414461 DSI = G.DSI_ISK, b'CPaceP521_XMD:SHA-512_SSWU_NU__ISK': (length: 34 bytes) 4350616365503532315f584d443a5348412d3531325f535357555f4e 555f5f49534b lv_cat(DSI,sid,K)||oCAT(MSGa,MSGb): (length: 397 bytes) 224350616365503532315f584d443a5348412d3531325f535357555f 4e555f5f49534b107e4b4791d6a8ef019b936c79fb7f2c574200503e 75e38e012a6dc6f3561980e4cf540dbcff3de3a4a6f09d79c32cc457 64d3a6605eb45df1dc63fb7937b7879f2820da1b3266b69fa099bf87 20dd8f6a07e8ed85010400f5cb68bf0117bd1a65412a2bc800af9201 3f9969cf546e1ea6d3bcf08643fdc482130aec1eecc33a2b5f33600b e51295047fa3399fa282cc1a78de91f3a4e30b5d01a085b453f22bf3 dc947386b042e5fc4ec691fee47fe3c3ec6408c22a17c26bc0ab7394 0910614d6fcee32dafbfd2d340d6e382d71b1fc763d7cec502fbcbcf 93b403414462850104003701ec35caafa3dd416cad29ba1774551f9d 2ed89f7e1065706dca230b86a11d02e4cee8b3fde64380d4a0598316 7d8a2414bc594ad5286c068792ab7ca60ff6ea00919c41c00e789dab c2f42fd94178d7bfd8fbe1aff1c1854b3dafb3a0ea13f5a5fc170386 0f022bd271740469bb322b07c179c7c225499b31727c0ea3ee655786 3403414461 ISK result: (length: 64 bytes) aae7320b73ba2516f289f71088662d41c4314d00521c48ea3c9c85ea ca57112e55eb2b4094d4a0c7813ddd95c5d80c5596ad686d2eba876b a1cd92f90407aa3d B.7.7. Corresponding ANSI-C initializers const uint8_t tc_PRS[] = { 0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64, }; const uint8_t tc_CI[] = { 0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a, 0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72, }; const uint8_t tc_sid[] = { 0x7e,0x4b,0x47,0x91,0xd6,0xa8,0xef,0x01,0x9b,0x93,0x6c,0x79, 0xfb,0x7f,0x2c,0x57, Abdalla, et al. Expires 26 March 2024 [Page 70] Internet-Draft CPace September 2023 }; const uint8_t tc_g[] = { 0x04,0x00,0xdc,0x92,0x79,0x58,0xf0,0xb6,0x9c,0xca,0xd8,0xfb, 0x67,0xef,0x00,0x89,0x05,0x35,0x4b,0x58,0xc7,0xc9,0xc9,0x2a, 0xd5,0x00,0x60,0xa9,0xe6,0xaf,0xb1,0x04,0x37,0xd6,0xca,0x8a, 0x26,0x16,0x4e,0x85,0x73,0x70,0x2b,0x89,0x72,0x75,0xa2,0x5d, 0x05,0xed,0x44,0x07,0xaf,0x2a,0x38,0x49,0x86,0xdc,0xa7,0xe2, 0x43,0xb9,0x2c,0x5d,0xd5,0x00,0xd4,0x00,0x57,0x01,0x21,0x21, 0xa9,0xc8,0xe3,0x43,0x73,0xfa,0x61,0x9f,0x91,0x8f,0x7d,0x47, 0x9c,0x23,0xf8,0x5f,0x04,0x85,0x37,0x9e,0xf0,0xf0,0x52,0x84, 0x39,0x8d,0xe2,0x66,0x53,0xb4,0x9a,0x15,0x53,0x24,0xc9,0xd7, 0xb1,0x38,0xbe,0x84,0xd0,0xb4,0x9b,0xb5,0x8e,0x23,0x2b,0x7b, 0xf6,0x97,0x79,0x8d,0xe6,0xee,0x8a,0xfd,0x6b,0x92,0xb6,0xfa, 0x2f, }; const uint8_t tc_ya[] = { 0x00,0x63,0x67,0xe9,0xc2,0xae,0xff,0x9f,0x1d,0xb1,0x9a,0xf6, 0x00,0xcc,0xa7,0x33,0x43,0xd4,0x7c,0xbe,0x44,0x6c,0xeb,0xbd, 0x1c,0xcd,0x78,0x3f,0x82,0x75,0x5a,0x87,0x2d,0xa8,0x6f,0xd0, 0x70,0x7e,0xb3,0x76,0x7c,0x61,0x14,0xf1,0x80,0x3d,0xeb,0x62, 0xd6,0x3b,0xdd,0x1e,0x61,0x3f,0x67,0xe6,0x3e,0x8c,0x14,0x1e, 0xe5,0x31,0x0e,0x3e,0xe8,0x19, }; const uint8_t tc_ADa[] = { 0x41,0x44,0x61, }; const uint8_t tc_Ya[] = { 0x04,0x00,0x37,0x01,0xec,0x35,0xca,0xaf,0xa3,0xdd,0x41,0x6c, 0xad,0x29,0xba,0x17,0x74,0x55,0x1f,0x9d,0x2e,0xd8,0x9f,0x7e, 0x10,0x65,0x70,0x6d,0xca,0x23,0x0b,0x86,0xa1,0x1d,0x02,0xe4, 0xce,0xe8,0xb3,0xfd,0xe6,0x43,0x80,0xd4,0xa0,0x59,0x83,0x16, 0x7d,0x8a,0x24,0x14,0xbc,0x59,0x4a,0xd5,0x28,0x6c,0x06,0x87, 0x92,0xab,0x7c,0xa6,0x0f,0xf6,0xea,0x00,0x91,0x9c,0x41,0xc0, 0x0e,0x78,0x9d,0xab,0xc2,0xf4,0x2f,0xd9,0x41,0x78,0xd7,0xbf, 0xd8,0xfb,0xe1,0xaf,0xf1,0xc1,0x85,0x4b,0x3d,0xaf,0xb3,0xa0, 0xea,0x13,0xf5,0xa5,0xfc,0x17,0x03,0x86,0x0f,0x02,0x2b,0xd2, 0x71,0x74,0x04,0x69,0xbb,0x32,0x2b,0x07,0xc1,0x79,0xc7,0xc2, 0x25,0x49,0x9b,0x31,0x72,0x7c,0x0e,0xa3,0xee,0x65,0x57,0x86, 0x34, }; const uint8_t tc_yb[] = { 0x00,0x92,0x27,0xbf,0x8d,0xc7,0x41,0xda,0xcc,0x94,0x22,0xf8, 0xbf,0x3c,0x0e,0x96,0xfc,0xe9,0x58,0x7b,0xc5,0x62,0xea,0xaf, 0xe0,0xdc,0x5f,0x6f,0x82,0xf2,0x85,0x94,0xe4,0xa6,0xf9,0x85, 0x53,0x56,0x0c,0x62,0xb7,0x5f,0xa4,0xab,0xb1,0x98,0xce,0xcb, 0xbb,0x86,0xeb,0xd4,0x1b,0x0e,0xa0,0x25,0x4c,0xde,0x78,0xac, 0x68,0xd3,0x9a,0x24,0x0a,0xe7, }; Abdalla, et al. Expires 26 March 2024 [Page 71] Internet-Draft CPace September 2023 const uint8_t tc_ADb[] = { 0x41,0x44,0x62, }; const uint8_t tc_Yb[] = { 0x04,0x00,0xf5,0xcb,0x68,0xbf,0x01,0x17,0xbd,0x1a,0x65,0x41, 0x2a,0x2b,0xc8,0x00,0xaf,0x92,0x01,0x3f,0x99,0x69,0xcf,0x54, 0x6e,0x1e,0xa6,0xd3,0xbc,0xf0,0x86,0x43,0xfd,0xc4,0x82,0x13, 0x0a,0xec,0x1e,0xec,0xc3,0x3a,0x2b,0x5f,0x33,0x60,0x0b,0xe5, 0x12,0x95,0x04,0x7f,0xa3,0x39,0x9f,0xa2,0x82,0xcc,0x1a,0x78, 0xde,0x91,0xf3,0xa4,0xe3,0x0b,0x5d,0x01,0xa0,0x85,0xb4,0x53, 0xf2,0x2b,0xf3,0xdc,0x94,0x73,0x86,0xb0,0x42,0xe5,0xfc,0x4e, 0xc6,0x91,0xfe,0xe4,0x7f,0xe3,0xc3,0xec,0x64,0x08,0xc2,0x2a, 0x17,0xc2,0x6b,0xc0,0xab,0x73,0x94,0x09,0x10,0x61,0x4d,0x6f, 0xce,0xe3,0x2d,0xaf,0xbf,0xd2,0xd3,0x40,0xd6,0xe3,0x82,0xd7, 0x1b,0x1f,0xc7,0x63,0xd7,0xce,0xc5,0x02,0xfb,0xcb,0xcf,0x93, 0xb4, }; const uint8_t tc_K[] = { 0x00,0x50,0x3e,0x75,0xe3,0x8e,0x01,0x2a,0x6d,0xc6,0xf3,0x56, 0x19,0x80,0xe4,0xcf,0x54,0x0d,0xbc,0xff,0x3d,0xe3,0xa4,0xa6, 0xf0,0x9d,0x79,0xc3,0x2c,0xc4,0x57,0x64,0xd3,0xa6,0x60,0x5e, 0xb4,0x5d,0xf1,0xdc,0x63,0xfb,0x79,0x37,0xb7,0x87,0x9f,0x28, 0x20,0xda,0x1b,0x32,0x66,0xb6,0x9f,0xa0,0x99,0xbf,0x87,0x20, 0xdd,0x8f,0x6a,0x07,0xe8,0xed, }; const uint8_t tc_ISK_IR[] = { 0xed,0x20,0x8a,0x15,0xaf,0x3e,0xf8,0xa6,0x7a,0x5c,0xac,0x4a, 0xcb,0x36,0x0d,0x03,0x15,0x45,0x70,0xe3,0xb1,0xb1,0xc5,0x48, 0x67,0xf5,0x3a,0x72,0x53,0xcb,0x91,0x9d,0x13,0xaa,0x47,0xef, 0xc6,0x47,0x37,0x5b,0xe2,0x25,0x0c,0xb3,0x9a,0xd9,0x65,0xaf, 0xa4,0xdd,0xfc,0xb6,0xbe,0x47,0xd5,0x86,0xd2,0x8c,0x7e,0xef, 0x6d,0x65,0x45,0x25, }; const uint8_t tc_ISK_SY[] = { 0xaa,0xe7,0x32,0x0b,0x73,0xba,0x25,0x16,0xf2,0x89,0xf7,0x10, 0x88,0x66,0x2d,0x41,0xc4,0x31,0x4d,0x00,0x52,0x1c,0x48,0xea, 0x3c,0x9c,0x85,0xea,0xca,0x57,0x11,0x2e,0x55,0xeb,0x2b,0x40, 0x94,0xd4,0xa0,0xc7,0x81,0x3d,0xdd,0x95,0xc5,0xd8,0x0c,0x55, 0x96,0xad,0x68,0x6d,0x2e,0xba,0x87,0x6b,0xa1,0xcd,0x92,0xf9, 0x04,0x07,0xaa,0x3d, }; B.7.8. Test case for scalar_mult_vfy with correct inputs Abdalla, et al. Expires 26 March 2024 [Page 72] Internet-Draft CPace September 2023 s: (length: 66 bytes) 0182dd7925f1753419e4bf83429763acd37d64000cd5a175edf53a15 87dd986bc95acc1506991702b6ba1a9ee2458fee8efc00198cf0088c 480965ef65ff2048b856 X: (length: 133 bytes) 0400dc5078b24c4af1620cc10fbecc6cd8cf1cab0b011efb73c782f2 26dc21c7ca7eb406be74a69ecba5b4a87c07cfc6e687b4beca9a6eda c95940a3b4120573b26a80005e697833b0ba285fce7b3f1f25243008 860b8f1de710a0dcc05b0d20341efe90eb2bcca26797c2d85ae6ca74 c00696cb1b13e40bda15b27964d7670576647bfab9 G.scalar_mult(s,X) (full coordinates): (length: 133 bytes) 040122f88ce73ec5aa2d1c8c5d04148760c3d97ba87daa10d8cb8bb7 c73cf6e951fc922721bf1437995cfb13e132a78beb86389e60d3517c df6d99a8a2d6db19ef27bd0055af9e8ddcf337ce0a7c22a9c8099bc4 a44faeded1eb72effd26e4f322217b67d60b944b267b3df5046078fd 577f1785728f49b241fd5e8c83223a994a2d219281 G.scalar_mult_vfy(s,X) (only X-coordinate): (length: 66 bytes) 0122f88ce73ec5aa2d1c8c5d04148760c3d97ba87daa10d8cb8bb7c7 3cf6e951fc922721bf1437995cfb13e132a78beb86389e60d3517cdf 6d99a8a2d6db19ef27bd B.7.9. Invalid inputs for scalar_mult_vfy For these test cases scalar_mult_vfy(y,.) MUST return the representation of the neutral element G.I. When including Y_i1 or Y_i2 in MSGa or MSGb the protocol MUST abort. s: (length: 66 bytes) 0182dd7925f1753419e4bf83429763acd37d64000cd5a175edf53a15 87dd986bc95acc1506991702b6ba1a9ee2458fee8efc00198cf0088c 480965ef65ff2048b856 Y_i1: (length: 133 bytes) 0400dc5078b24c4af1620cc10fbecc6cd8cf1cab0b011efb73c782f2 26dc21c7ca7eb406be74a69ecba5b4a87c07cfc6e687b4beca9a6eda c95940a3b4120573b26a80005e697833b0ba285fce7b3f1f25243008 860b8f1de710a0dcc05b0d20341efe90eb2bcca26797c2d85ae6ca74 c00696cb1b13e40bda15b27964d7670576647bfaf9 Y_i2: (length: 1 bytes) 00 G.scalar_mult_vfy(s,Y_i1) = G.scalar_mult_vfy(s,Y_i2) = G.I Authors' Addresses Michel Abdalla DFINITY - Zurich Email: michel.abdalla@gmail.com Abdalla, et al. Expires 26 March 2024 [Page 73] Internet-Draft CPace September 2023 Bjoern Haase Endress + Hauser Liquid Analysis - Gerlingen Email: bjoern.m.haase@web.de Julia Hesse IBM Research Europe - Zurich Email: JHS@zurich.ibm.com Abdalla, et al. Expires 26 March 2024 [Page 74]