# Round 1 (Additional Signatures) OFFICIAL COMMENT: FuLeeca

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### Wessel van Woerden

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Jul 20, 2023, 7:33:58 AM7/20/23
to pqc-forum, Felicitas Hörmann
Dear FuLeeca Team,

The FuLeeca scheme is a hash-and-sign scheme based on quasi-cyclic codes with the Lee metric. The secret key consists of a quasi-cyclic generator matrix G of a length n and dimension n/2 linear code over F_p with p=65521.

Signatures are codewords of small Lee weight, and messages are bound to this by some sort of parity-count condition. Following the specification we work with row vectors here.
Looking at the signing procedure each signature (low-weight codeword) v is in fact a small linear combination of the rows of G, i.e., v = x*G mod p for some small integer vector x.

We observe however experimentally that with the current parameters the coefficients of x*G (without modulo) are already well within the range [-(p-1)/2,(p-1)/2]. E.g., experimentally the coefficients of v are in the range [-5000,5000] for all the three variants. This implies that we have a real equality v = x*G, and therefore, signatures leak the ZZ-span of (the rows of) G, which thus reveals the rank n/2 lattice L(G) in Z^n. In fact, it is enough to only consider the first n/2 coefficients of each vector of v, and therefore recover the lattice L(G') where G=(G'|G'').

We thus ignore the Lee metric and move to lattices with the standard Euclidean metric. The n/2 shortest vectors of the lattice L(G') are precisely the rows of the secret key G'. The gap between the norm of these shortest vectors, and the Gaussian Heuristic of the lattice L(G') is of order O(sqrt(n)), i.e., on a concrete example generated by the FuLeeca1 parameters with n/2=659 we obtain gh/lambda1=4.722.

So the recovery of G' is an unusual SVP instance in an n/2 dimensional lattice with gap O(sqrt(n)). Asymptotically this can be broken by BKZ with blocksize beta=n/4+o(n). Concretely, for the FuLeeca1 parameters a quick estimate gives that BKZ recovers G' with blocksize less than 310, which is significantly below the security level 1. We expect a similar reduction in security for the other variants.

Some additional comments:
• The signature vectors themselves are quite short (shorter than what one would get by BKZ-310), which could reduce the blocksize even further.
• The lattice L(G') is in fact a circulant lattice. Firstly, due to this structure 2 signatures can be enough to recover the full lattice span. Secondly, this extra structure might make the attack classically sub-exponential exp(O(sqrt(n))) or quantum-polynomial-time by known ideal-lattice attacks.
• Given the leakage of the signature procedure, and the similarity of the Lee metric to the Euclidean one, we expect a more advanced learning attack à la [Nguyen-Regev,2006] on GGH to also apply to this scheme.

Best regards,

Felicitas Hörmann and Wessel van Woerden

### Patrick Karl

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Jul 21, 2023, 1:27:49 PM7/21/23
to wesselva...@gmail.com, pqc-...@list.nist.gov, felicita...@gmail.com
Dear Wessel and Felicitas,

Thank you for the comment on FuLeeca and your helpful insights to BKZ also prior to submission.
We acknowledge that FuLeeca in its current form is susceptible to the attack.

Best,
The FuLeeca Team

> Dear FuLeeca Team,
>
> The FuLeeca scheme is a hash-and-sign scheme based on quasi-cyclic codes with the Lee metric. The secret key consists of a quasi-cyclic generator
> matrix G of a length n and dimension n/2 linear code over F_p with p=65521.
>
> Signatures are codewords of small Lee weight, and messages are bound to this by some sort of parity-count condition. Following the specification we
> work with row vectors here.
> Looking at the signing procedure each signature (low-weight codeword) v is in fact a small linear combination of the rows of G, i.e., v = x*G mod p
> for some small integer vector x.
>
> We observe however experimentally that with the current parameters the coefficients of x*G (without modulo) are already well within the range [-(p-
> 1)/2,(p-1)/2]. E.g., experimentally the coefficients of v are in the range [-5000,5000] for all the three variants. This implies that we have a real
> equality v = x*G, and therefore, signatures leak the ZZ-span of (the rows of) G, which thus reveals the rank n/2 lattice L(G) in Z^n. In fact, it is
> enough to only consider the first n/2 coefficients of each vector of v, and therefore recover the lattice L(G') where G=(G'|G'').
>
> We thus ignore the Lee metric and move to lattices with the standard Euclidean metric. The n/2 shortest vectors of the lattice L(G') are precisely
> the rows of the secret key G'. The gap between the norm of these shortest vectors, and the Gaussian Heuristic of the lattice L(G') is of order
> O(sqrt(n)), i.e., on a concrete example generated by the FuLeeca1 parameters with n/2=659 we obtain gh/lambda1=4.722.
>
> So the recovery of G' is an unusual SVP instance in an n/2 dimensional lattice with gap O(sqrt(n)). Asymptotically this can be broken by BKZ with
> blocksize beta=n/4+o(n). Concretely, for the FuLeeca1 parameters a quick estimate gives that BKZ recovers G' with blocksize less than 310, which is
> significantly below the security level 1. We expect a similar reduction in security for the other variants.
>
> Some additional comments:
>  * The signature vectors themselves are quite short (shorter than what one would get by BKZ-310), which could reduce the blocksize even further.
>  * The lattice L(G') is in fact a circulant lattice. Firstly, due to this structure 2 signatures can be enough to recover the full lattice span.
> Secondly, this extra structure might make the attack classically sub-exponential exp(O(sqrt(n))) or quantum-polynomial-time by known ideal-lattice
> attacks.
>  * Given the leakage of the signature procedure, and the similarity of the Lee metric to the Euclidean one, we expect a more advanced learning
> attack à la [Nguyen-Regev,2006] on GGH to also apply to this scheme.
>
> Best regards,
>
> Felicitas Hörmann and Wessel van Woerden

--
Patrick Karl, M.Sc.

Technical University of Munich, Germany
TUM School of Computation, Information and Technology
Chair of Security in Information Technology
Head: Prof. Dr.-Ing. Georg Sigl

Arcisstr. 21
80333 Munich
Germany

Email: patric...@tum.de
PGP: F045 90FD F999 1112 E888 6926 322D 2EC5 2E81 A510
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