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Use of the Walnut Digital Signature Algorithm with CBOR Object Signing and Encryption (COSE)
SecureRF Corporation
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Security
Internet Engineering Task Force
COSE
WalnutDSA
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 (see
and ) with implementation and
computational efficiency of signature verification in constrained
environments, even on 8- and 16-bit platforms.
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 Group-Theoretic signature scheme
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 ECC, but still smaller than most all
other quantum-resistant schemes (including all hash-based
schemes).
There have been recent advances in cryptanalysis and advances in the
development of quantum computers. Each of these advances pose a
threat to widely deployed digital signature algorithms.
At Black Hat USA 2013, some researchers gave a presentation on the
current state of public key cryptography. They said: "Current
cryptosystems depend on discrete logarithm and factoring which has
seen some major new developments in the past 6 months"
. Due to advances in cryptanalysis,
they encouraged preparation for a day when RSA and DSA cannot be
depended upon.
Peter Shor showed that a large-scale quantum computer could be used
to factor a number in polynomial time , effectively breaking
RSA. 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
is a system that is secure against
quantum computers that have more than a trivial number of
quantum bits (qu-bits). It is open to conjecture when it will
be feasible to build such a machine; however, RSA, DSA, ECDSA,
and EdDSA are all vulnerable if large-scale uantum computers
come to pass.
WalnutDSA does not depend on the difficulty of discrete logarithm or factoring.
As a result this algorithm is considered to be post-quantum secure.
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 cryptoanalytic advances or a large-scale
quantum computer is invented. WalnutDSA, specified in , is one such algorithm.
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 RFC 2119
RFC 8174 when, and only when,
they appear in all capitals, as shown here.
This specification makes use of WalnutDSA signatures as
described in and more concretely
specified in . 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 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 MCUs). 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 and implies working in an NxN
matrix. The parameter q defines the size of the finite field
(in q elements). 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.
The CBOR Object Signing and Encryption (COSE) 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 validation.
When using a COSE key for this algorithm, the following checks are
made:
The 'kty' field MUST be present, and it MUST be 'WalnutDSA'.
If the 'alg' field is present, and 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.

Implementations must protect the private keys. Use of a hardware
security module (HSM) is one way to protect the private keys.
Compromise of 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 pseudo-random 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.
offers important guidance in this area.
The generation of WalnutDSA signatures also depends on random
numbers. While the consequences of an inadequate pseudo-random
number generator (PRNGs) to generate these values is much less severe
than the generation of private keys, the guidance in
remains important.
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 exponential attacks.
A full writeup of all the analysis can be found in
. In summary,
the original suggested parameters were too small, leading to
many of these exponential 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 attacks.
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 (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.
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 is 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 run
time and 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.
IANA is requested to add entries for WalnutDSA signatures in the
"COSE Algorithms" registry and WalnutDSA public keys in the "COSE
Key Types" and "COSE Key Type Parameters" registries.
The new entry in the "COSE Algorithms" registry has the following
columns:
Name: WalnutDSA
Value: TBD1 (Value to be assigned by IANA)
Description: WalnutDSA signature
Reference: This document (Number to be assigned by RFC Editor)
Recommended: Yes

The new entry in the "COSE Key Types" registry has the following
columns:
Name: WalnutDSA
Value: TBD2 (Value to be assigned by IANA)
Description: WalnutDSA public key
Reference: This document (Number to be assigned by RFC Editor)

The following sections detail the additions to the "COSE Key Type Parameters" registry.
The new entry N in the "COSE Key Type Parameters" registry
has the following columns:
Key Type: TBD2 (Value assigned by IANA above)
Name: N
Label: TBD (Value to be assigned by IANA)
CBOR Type: uint
Description: Group and Matrix (NxN) size
Reference: This document (Number to be assigned by RFC Editor)

The new entry q in the "COSE Key Type Parameters" registry
has the following columns:
Key Type: TBD2 (Value assigned by IANA above)
Name: q
Label: TBD (Value to be assigned by IANA)
CBOR Type: uint
Description: Finite field F_q
Reference: This document (Number to be assigned by RFC Editor)

The new entry t-values in the "COSE Key Type Parameters" registry
has the following columns:
Key Type: TBD2 (Value assigned by IANA above)
Name: t-values
Label: TBD (Value to be assigned by IANA)
CBOR Type: array (of uint)
Description: List of T-values, enties in F_q
Reference: This document (Number to be assigned by RFC Editor)

The new entry matrix 1 in the "COSE Key Type Parameters" registry
has the following columns:
Key Type: TBD2 (Value assigned by IANA above)
Name: matrix 1
Label: TBD (Value to be assigned by IANA)
CBOR Type: array (of array of uint)
Description: NxN Matrix of enties in F_q
Reference: This document (Number to be assigned by RFC Editor)

The new entry permutation 1 in the "COSE Key Type Parameters" registry
has the following columns:
Key Type: TBD2 (Value assigned by IANA above)
Name: permutation 1
Label: TBD (Value to be assigned by IANA)
CBOR Type: array (of uint)
Description: Permutation associated with matrix 1
Reference: This document (Number to be assigned by RFC Editor)

The new entry matrix 2 in the "COSE Key Type Parameters" registry
has the following columns:
Key Type: TBD2 (Value assigned by IANA above)
Name: matrix 2
Label: TBD (Value to be assigned by IANA)
CBOR Type: array (of array of uint)
Description: NxN Matrix of enties in F_q
Reference: This document (Number to be assigned by RFC Editor)

&RFC2119;
&RFC8174;
&RFC8152;
The Walnut Digital Signature Algorithm Specification
FIPS Publication 180-3: Secure Hash Standard
National Institute of Standards and Technology (NIST)
Group Theoretic Cryptography
WalnutDSA(TM): A Quantum-Resistant Digital Signature Algorithm
Defeating the Hart et al, Beullens-Blackburn, Kotov-Menshov-Ushakov, and Merz-Petit Attacks on WalnutDSA(TM)
&RFC4086;
The Factoring Dead: Preparing for the Cryptopocalypse
Introduction to post-quantum cryptography
Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer
A big thank you to Russ Housley for his input on the concepts and text of this document.