RFC 9021

Independent Submission                                         D. Atkins
Request for Comments: 9021                             Veridify Security
Category: Informational                                         May 2021
ISSN: 2070-1721

Use of the Walnut Digital Signature Algorithm with CBOR Object Signing
                         and Encryption (COSE)


   This document specifies the conventions for using the Walnut Digital
   Signature Algorithm (WalnutDSA) for digital signatures with the CBOR
   Object Signing and Encryption (COSE) syntax.  WalnutDSA is a
   lightweight, quantum-resistant signature scheme based on Group
   Theoretic Cryptography with implementation and computational
   efficiency of signature verification in constrained environments,
   even on 8- and 16-bit platforms.

   The goal of this publication is to document a way to use the
   lightweight, quantum-resistant WalnutDSA signature algorithm in COSE
   in a way that would allow multiple developers to build compatible
   implementations.  As of this publication, the security properties of
   WalnutDSA have not been evaluated by the IETF and its use has not
   been endorsed by the IETF.

   WalnutDSA and the Walnut Digital Signature Algorithm are trademarks
   of Veridify Security Inc.

Status of This Memo

   This document is not an Internet Standards Track specification; it is
   published for informational purposes.

   This is a contribution to the RFC Series, independently of any other
   RFC stream.  The RFC Editor has chosen to publish this document at
   its discretion and makes no statement about its value for
   implementation or deployment.  Documents approved for publication by
   the RFC Editor are not candidates for any level of Internet Standard;
   see Section 2 of RFC 7841.

   Information about the current status of this document, any errata,
   and how to provide feedback on it may be obtained at

Copyright Notice

   Copyright (c) 2021 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.

Table of Contents

   1.  Introduction
     1.1.  Motivation
     1.2.  Trademark Notice
   2.  Terminology
   3.  WalnutDSA Algorithm Overview
   4.  WalnutDSA Algorithm Identifiers
   5.  Security Considerations
     5.1.  Implementation Security Considerations
     5.2.  Method Security Considerations
   6.  IANA Considerations
     6.1.  COSE Algorithms Registry Entry
     6.2.  COSE Key Types Registry Entry
     6.3.  COSE Key Type Parameters Registry Entries
       6.3.1.  WalnutDSA Parameter: N
       6.3.2.  WalnutDSA Parameter: q
       6.3.3.  WalnutDSA Parameter: t-values
       6.3.4.  WalnutDSA Parameter: matrix 1
       6.3.5.  WalnutDSA Parameter: permutation 1
       6.3.6.  WalnutDSA Parameter: matrix 2
   7.  References
     7.1.  Normative References
     7.2.  Informative References

   Author's Address

1.  Introduction

   This document specifies the conventions for using the Walnut Digital
   Signature Algorithm (WalnutDSA) [WALNUTDSA] for digital signatures
   with the CBOR Object Signing and Encryption (COSE) syntax [RFC8152].
   WalnutDSA is a Group Theoretic signature scheme [GTC] where signature
   validation is both computationally and space efficient, even on very
   small processors.  Unlike many hash-based signatures, there is no
   state required and no limit on the number of signatures that can be
   made.  WalnutDSA private and public keys are relatively small;
   however, the signatures are larger than RSA and Elliptic Curve
   Cryptography (ECC), but still smaller than most all other quantum-
   resistant schemes (including all hash-based schemes).

   COSE provides a lightweight method to encode structured data.
   WalnutDSA is a lightweight, quantum-resistant digital signature
   algorithm.  The goal of this specification is to document a method to
   leverage WalnutDSA in COSE in a way that would allow multiple
   developers to build compatible implementations.

   As with all cryptosystems, the initial versions of WalnutDSA
   underwent significant cryptanalysis, and, in some cases, identified
   potential issues.  For more discussion on this topic, a summary of
   all published cryptanalysis can be found in Section 5.2.  Validated
   issues were addressed by reparameterization in updated versions of
   WalnutDSA.  Although the IETF has neither evaluated the security
   properties of WalnutDSA nor endorsed WalnutDSA as of this
   publication, this document provides a method to use WalnutDSA in
   conjunction with IETF protocols.  As always, users of any security
   algorithm are advised to research the security properties of the
   algorithm and make their own judgment about the risks involved.

1.1.  Motivation

   Recent advances in cryptanalysis [BH2013] and progress in the
   development of quantum computers [NAS2019] pose a threat to widely
   deployed digital signature algorithms.  As a result, there is a need
   to prepare for a day that cryptosystems such as RSA and DSA, which
   depend on discrete logarithm and factoring, cannot be depended upon.

   If large-scale quantum computers are ever built, these computers will
   be able to break many of the public key cryptosystems currently in
   use.  A post-quantum cryptosystem [PQC] is a system that is secure
   against quantum computers that have more than a trivial number of
   quantum bits (qubits).  It is open to conjecture when it will be
   feasible to build such computers; however, RSA, DSA, the Elliptic
   Curve Digital Signature Algorithm (ECDSA), and the Edwards-Curve
   Digital Signature Algorithm (EdDSA) are all vulnerable if large-scale
   quantum computers come to pass.

   WalnutDSA does not depend on the difficulty of discrete logarithms or
   factoring.  As a result, this algorithm is considered to be resistant
   to post-quantum attacks.

   Today, RSA and ECDSA are often used to digitally sign software
   updates.  Unfortunately, implementations of RSA and ECDSA can be
   relatively large, and verification can take a significant amount of
   time on some very small processors.  Therefore, we desire a digital
   signature scheme that verifies faster with less code.  Moreover, in
   preparation for a day when RSA, DSA, and ECDSA cannot be depended
   upon, a digital signature algorithm is needed that will remain secure
   even if there are significant cryptanalytic advances or a large-scale
   quantum computer is invented.  WalnutDSA, specified in [WALNUTSPEC],
   is a quantum-resistant algorithm that addresses these requirements.

1.2.  Trademark Notice

   WalnutDSA and the Walnut Digital Signature Algorithm are trademarks
   of Veridify Security Inc.

2.  Terminology

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "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.

3.  WalnutDSA Algorithm Overview

   This specification makes use of WalnutDSA signatures as described in
   [WALNUTDSA] and more concretely specified in [WALNUTSPEC].  WalnutDSA
   is a Group Theoretic cryptographic signature scheme that leverages
   infinite group theory as the basis of its security and maps that to a
   one-way evaluation of a series of matrices over small finite fields
   with permuted multiplicants based on the group input.  WalnutDSA
   leverages the SHA2-256 and SHA2-512 one-way hash algorithms [SHA2] in
   a hash-then-sign process.

   WalnutDSA is based on a one-way function, E-multiplication, which is
   an action on the infinite group.  A single E-multiplication step
   takes as input a matrix and permutation, a generator in the group,
   and a set of T-values (entries in the finite field) and outputs a new
   matrix and permutation.  To process a long string of generators (like
   a WalnutDSA signature), E-multiplication is iterated over each
   generator.  Due to its structure, E-multiplication is extremely easy
   to implement.

   In addition to being quantum resistant, the two main benefits of
   using WalnutDSA are that the verification implementation is very
   small and WalnutDSA signature verification is extremely fast, even on
   very small processors (including 16- and even 8-bit
   microcontrollers).  This lends it well to use in constrained and/or
   time-sensitive environments.

   WalnutDSA has several parameters required to process a signature.
   The main parameters are N and q.  The parameter N defines the size of
   the group by defining the number of strands in use and implies
   working in an NxN matrix.  The parameter q defines the number of
   elements in the finite field.  Signature verification also requires a
   set of T-values, which is an ordered list of N entries in the finite
   field F_q.

   A WalnutDSA signature is just a string of generators in the infinite
   group, packed into a byte string.

4.  WalnutDSA Algorithm Identifiers

   The CBOR Object Signing and Encryption (COSE) syntax [RFC8152]
   supports two signature algorithm schemes.  This specification makes
   use of the signature with appendix scheme for WalnutDSA signatures.

   The signature value is a large byte string.  The byte string is
   designed for easy parsing, and it includes a length (number of
   generators) and type codes that indirectly provide all of the
   information that is needed to parse the byte string during signature

   When using a COSE key for this algorithm, the following checks are

   *  The "kty" field MUST be present, and it MUST be "WalnutDSA".

   *  If the "alg" field is present, it MUST be "WalnutDSA".

   *  If the "key_ops" field is present, it MUST include "sign" when
      creating a WalnutDSA signature.

   *  If the "key_ops" field is present, it MUST include "verify" when
      verifying a WalnutDSA signature.

   *  If the "kid" field is present, it MAY be used to identify the
      WalnutDSA Key.

5.  Security Considerations

5.1.  Implementation Security Considerations

   Implementations MUST protect the private keys.  Use of a hardware
   security module (HSM) is one way to protect the private keys.
   Compromising the private keys may result in the ability to forge
   signatures.  As a result, when a private key is stored on non-
   volatile media or stored in a virtual machine environment, care must
   be taken to preserve confidentiality and integrity.

   The generation of private keys relies on random numbers.  The use of
   inadequate pseudorandom number generators (PRNGs) to generate these
   values can result in little or no security.  An attacker may find it
   much easier to reproduce the PRNG environment that produced the keys,
   searching the resulting small set of possibilities, rather than brute
   force searching the whole key space.  The generation of quality
   random numbers is difficult, and [RFC4086] offers important guidance
   in this area.

   The generation of WalnutDSA signatures also depends on random
   numbers.  While the consequences of an inadequate PRNG to generate
   these values are much less severe than the generation of private
   keys, the guidance in [RFC4086] remains important.

5.2.  Method Security Considerations

   The Walnut Digital Signature Algorithm has undergone significant
   cryptanalysis since it was first introduced, and several weaknesses
   were found in early versions of the method, resulting in the
   description of several attacks with exponential computational
   complexity.  A full writeup of all the analysis can be found in
   [WalnutDSAAnalysis].  In summary, the original suggested parameters
   (N=8, q=32) were too small, leading to many of these exponential-
   growth attacks being practical.  However, current parameters render
   these attacks impractical.  The following paragraphs summarize the
   analysis and how the current parameters defeat all the previous

   First, the team of Hart et al. found a universal forgery attack based
   on a group-factoring problem that runs in O(q^((N-1)/2)) with a
   memory complexity of log_2(q) N^2 q^((N-1)/2).  With parameters N=10
   and q=M31 (the Mersenne prime 2^31 - 1), the runtime is 2^139 and
   memory complexity is 2^151.  W. Beullens found a modification of this
   attack but its runtime is even longer.

   Next, Beullens and Blackburn found several issues with the original
   method and parameters.  First, they used a Pollard-Rho attack and
   discovered the original public key space was too small.
   Specifically, they require that q^(N(N-1)-1) > 2^(2*Security Level).
   One can clearly see that (N=10, q=M31) provides 128-bit security and
   (N=10, q=M61) provides 256-bit security.

   Beullens and Blackburn also found two issues with the original
   message encoder of WalnutDSA.  First, the original encoder was non-
   injective, which reduced the available signature space.  This was
   repaired in an update.  Second, they pointed out that the dimension
   of the vector space generated by the encoder was too small.
   Specifically, they require that q^dimension > 2^(2*Security Level).
   With N=10, the current encoder produces a dimension of 66, which
   clearly provides sufficient security with q=M31 or q=M61.

   The final issue discovered by Beullens and Blackburn was a process to
   theoretically "reverse" E-multiplication.  First, their process
   requires knowing the initial matrix and permutation (which are known
   for WalnutDSA).  But more importantly, their process runs at
   O(q^((N-1)/2)), which for (N=10, q=M31) is greater than 2^128.

   A team at Steven's Institute leveraged a length-shortening attack
   that enabled them to remove the cloaking elements and then solve a
   conjugacy search problem to derive the private keys.  Their attack
   requires both knowledge of the permutation being cloaked and also
   that the cloaking elements themselves are conjugates.  By adding
   additional concealed cloaking elements, the attack requires an N!
   search for each cloaking element.  By inserting k concealed cloaking
   elements, this requires the attacker to perform (N!)^k work.  This
   allows k to be set to meet the desired security level.

   Finally, Merz and Petit discovered that using a Garside Normal Form
   of a WalnutDSA signature enabled them to find commonalities with the
   Garside Normal Form of the encoded message.  Using those
   commonalities, they were able to splice into a signature and create
   forgeries.  Increasing the number of cloaking elements, specifically
   within the encoded message, sufficiently obscures the commonalities
   and blocks this attack.

   In summary, most of these attacks are exponential in runtime and it
   can be shown that current parameters put the runtime beyond the
   desired security level.  The final two attacks are also sufficiently
   blocked to the desired security level.

6.  IANA Considerations

   IANA has added entries for WalnutDSA signatures in the "COSE
   Algorithms" registry and WalnutDSA public keys in the "COSE Key
   Types" and "COSE Key Type Parameters" registries.

6.1.  COSE Algorithms Registry Entry

   The following new entry has been registered in the "COSE Algorithms"

   Name:  WalnutDSA

   Value:  -260

   Description:  WalnutDSA signature

   Reference:  RFC 9021

   Recommended:  No

6.2.  COSE Key Types Registry Entry

   The following new entry has been registered in the "COSE Key Types"

   Name:  WalnutDSA

   Value:  6

   Description:  WalnutDSA public key

   Reference:  RFC 9021

6.3.  COSE Key Type Parameters Registry Entries

   The following sections detail the additions to the "COSE Key Type
   Parameters" registry.

6.3.1.  WalnutDSA Parameter: N

   The new entry, N, has been registered in the "COSE Key Type
   Parameters" registry as follows:

   Key Type:  6

   Name:  N

   Label:  -1

   CBOR Type:  uint

   Description:  Group and Matrix (NxN) size

   Reference:  RFC 9021

6.3.2.  WalnutDSA Parameter: q

   The new entry, q, has been registered in the "COSE Key Type
   Parameters" registry as follows:

   Key Type:  6

   Name:  q

   Label:  -2

   CBOR Type:  uint

   Description:  Finite field F_q

   Reference:  RFC 9021

6.3.3.  WalnutDSA Parameter: t-values

   The new entry, t-values, has been registered in the "COSE Key Type
   Parameters" registry as follows:

   Key Type:  6

   Name:  t-values

   Label:  -3

   CBOR Type:  array (of uint)

   Description:  List of T-values, entries in F_q

   Reference:  RFC 9021

6.3.4.  WalnutDSA Parameter: matrix 1

   The new entry, matrix 1, has been registered in the "COSE Key Type
   Parameters" registry as follows:

   Key Type:  6

   Name:  matrix 1

   Label:  -4

   CBOR Type:  array (of array of uint)

   Description:  NxN Matrix of entries in F_q in column-major form

   Reference:  RFC 9021

6.3.5.  WalnutDSA Parameter: permutation 1

   The new entry, permutation 1, has been registered in the "COSE Key
   Type Parameters" registry as follows:

   Key Type:  6

   Name:  permutation 1

   Label:  -5

   CBOR Type:  array (of uint)

   Description:  Permutation associated with matrix 1

   Reference:  RFC 9021

6.3.6.  WalnutDSA Parameter: matrix 2

   The new entry, matrix 2, has been registered in the "COSE Key Type
   Parameters" registry as follows:

   Key Type:  6

   Name:  matrix 2

   Label:  -6

   CBOR Type:  array (of array of uint)

   Description:  NxN Matrix of entries in F_q in column-major form

   Reference:  RFC 9021

7.  References

7.1.  Normative References

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997,

   [RFC8152]  Schaad, J., "CBOR Object Signing and Encryption (COSE)",
              RFC 8152, DOI 10.17487/RFC8152, July 2017,

   [RFC8174]  Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
              2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
              May 2017, <https://www.rfc-editor.org/info/rfc8174>.

   [SHA2]     National Institute of Standards and Technology (NIST),
              "Secure Hash Standard (SHS)", DOI 10.6028/NIST.FIPS.180-4,
              August 2015, <https://doi.org/10.6028/NIST.FIPS.180-4>.

              Anshel, I., Atkins, D., Goldfeld, D., and P. Gunnells,
              "WalnutDSA(TM): A group theoretic digital signature
              algorithm", DOI 10.1080/23799927.2020.1831613, November
              2020, <https://doi.org/10.1080/23799927.2020.1831613>.

7.2.  Informative References

   [BH2013]   Ptacek, T., Ritter, J., Samuel, J., and A. Stamos, "The
              Factoring Dead: Preparing for the Cryptopocalypse", August
              2013, <https://www.slideshare.net/astamos/bh-slides>.

   [GTC]      Vasco, M. and R. Steinwandt, "Group Theoretic
              Cryptography", ISBN 9781584888369, April 2015,

   [NAS2019]  National Academies of Sciences, Engineering, and Medicine,
              "Quantum Computing: Progress and Prospects",
              DOI 10.17226/25196, 2019,

   [PQC]      Bernstein, D., "Introduction to post-quantum
              cryptography", DOI 10.1007/978-3-540-88702-7, 2009,

   [RFC4086]  Eastlake 3rd, D., Schiller, J., and S. Crocker,
              "Randomness Requirements for Security", BCP 106, RFC 4086,
              DOI 10.17487/RFC4086, June 2005,

              Anshel, I., Atkins, D., Goldfeld, D., and P. Gunnells,
              "Defeating the Hart et al, Beullens-Blackburn, Kotov-
              Menshov-Ushakov, and Merz-Petit Attacks on WalnutDSA(TM)",
              May 2019, <https://eprint.iacr.org/2019/472>.

              Anshel, I., Atkins, D., Goldfeld, D., and P. Gunnells,
              "The Walnut Digital Signature Algorithm Specification",
              Post-Quantum Cryptography, November 2018,


   A big thank you to Russ Housley for his input on the concepts and
   text of this document.

Author's Address

   Derek Atkins
   Veridify Security
   100 Beard Sawmill Rd, Suite 350
   Shelton, CT 06484
   United States of America

   Phone: +1 617 623 3745
   Email: datkins@veridify.com