International Association for Cryptologic Research

International Association
for Cryptologic Research


Ward Beullens


Sigma protocols for MQ, PKP and SIS, and fishy signature schemes 📺
Ward Beullens
This work presents sigma protocols to prove knowledge of: - a solution to a system of quadratic polynomials, - a solution to an instance of the Permuted Kernel Problem and - a witness for a variety of lattice statements (including SIS). Our sigma protocols have soundness error 1/q', where q' is any number bounded by the size of the underlying finite field. This is much better than existing proofs, which have soundness error 2/3 or (q'+1)/2q'. The prover and verifier time our proofs are O(q'). We achieve this by first constructing so-called sigma protocols with helper, which are sigma protocols where the prover and the verifier are assisted by a trusted third party, and then eliminating the helper from the proof with a "cut-and-choose" protocol. We apply the Fiat-Shamir transform to obtain signature schemes with security proof in the QROM. We show that the resulting signature schemes, which we call the "MUltivariate quaDratic FIat-SHamir" scheme (MUDFISH) and the "ShUffled Solution to Homogeneous linear SYstem FIat-SHamir" scheme (SUSHSYFISH), are more efficient than existing signatures based on the MQ problem and the Permuted Kernel Problem. Our proof system can be used to improve the efficiency of applications relying on (generalizations of) Stern's protocol. We show that the proof size of our SIS proof is smaller than that of Stern's protocol by an order of magnitude and that our proof is more efficient than existing post-quantum secure SIS proofs.
Cryptanalysis of the Legendre PRF and Generalizations
The Legendre PRF relies on the conjectured pseudorandomness properties of the Legendre symbol with a hidden shift. Originally proposed as a PRG by Damgård at CRYPTO 1988, it was recently suggested as an efficient PRF for multiparty computation purposes by Grassi et al. at CCS 2016. Moreover, the Legendre PRF is being considered for usage in the Ethereum 2.0 blockchain.This paper improves previous attacks on the Legendre PRF and its higher-degree variant due to Khovratovich by reducing the time complexity from O(< (p log p/M) to O(p log2 p/M2) Legendre symbol evaluations when M ≤ 4√ p log2 p queries are available. The practical relevance of our improved attack is demonstrated by breaking three concrete instances of the PRF proposed by the Ethereum foundation. Furthermore, we generalize our attack in a nontrivial way to the higher-degree variant of the Legendre PRF and we point out a large class of weak keys for this construction. Lastly, we provide the first security analysis of two additional generalizations of the Legendre PRF originally proposed by Damgård in the PRG setting, namely the Jacobi PRF and the power residue PRF.
Obfuscating Simple Functionalities from Knowledge Assumptions
Ward Beullens Hoeteck Wee
This paper shows how to obfuscate several simple functionalities from a new Knowledge of OrthogonALity Assumption (KOALA) in cyclic groups which is shown to hold in the Generic Group Model. Specifically, we give simpler and stronger security proofs for obfuscation schemes for point functions, general-output point functions and pattern matching with wildcards. We also revisit the work of Bishop et al. (CRYPTO 2018) on obfuscating the pattern matching with wildcards functionality. We improve upon the construction and the analysis in several ways:attacks and stronger guarantees: We show that the construction achieves virtual black-box security for a simulator that runs in time roughly $$2^{n/2}$$, as well as distributional security for larger classes of distributions. We give attacks that show that our results are tight.weaker assumptions: We prove security under KOALA.better efficiency: We also provide a construction that outputs $$n+1$$ instead of 2n group elements. We obtain our results by first obfuscating a simpler “big subset functionality”, for which we establish full virtual black-box security; this yields a simpler and more modular analysis for pattern matching. Finally, we extend our distinguishing attacks to a large class of simple linear-in-the-exponent schemes.
CSI-FiSh: Efficient Isogeny Based Signatures Through Class Group Computations
In this paper we report on a new record class group computation of an imaginary quadratic field having 154-digit discriminant, surpassing the previous record of 130 digits. This class group is central to the CSIDH-512 isogeny based cryptosystem, and knowing the class group structure and relation lattice implies efficient uniform sampling and a canonical representation of its elements. Both operations were impossible before and allow us to instantiate an isogeny based signature scheme first sketched by Stolbunov. We further optimize the scheme using multiple public keys and Merkle trees, following an idea by De Feo and Galbraith. We also show that including quadratic twists allows to cut the public key size in half for free. Optimizing for signature size, our implementation takes 390 ms to sign/verify and results in signatures of 263 bytes, at the expense of a large public key. This is 300 times faster and over 3 times smaller than an optimized version of SeaSign for the same parameter set. Optimizing for public key and signature size combined, results in a total size of 1468 bytes, which is smaller than any other post-quantum signature scheme at the 128-bit security level.
Practical Attacks Against the Walnut Digital Signature Scheme
Ward Beullens Simon R. Blackburn
Recently, NIST started the process of standardizing quantum-resistant public-key cryptographic algorithms. WalnutDSA, the subject of this paper, is one of the 20 proposed signature schemes that are being considered for standardization. Walnut relies on a one-way function called E-Multiplication, which has a rich algebraic structure. This paper shows that this structure can be exploited to launch several practical attacks against the Walnut cryptosystem. The attacks work very well in practice; it is possible to forge signatures and compute equivalent secret keys for the 128-bit and 256-bit security parameters submitted to NIST in less than a second and in less than a minute respectively.