A Simple Deterministic Algorithm for Systems of Quadratic Polynomials over F2\mathbb{F}_2

This article discusses a simple deterministic algorithm for solving quadratic Boolean systems which is essentially a special case of more sophisticated methods. The main idea fits in a single sentence: guess enough variables so that the remaining quadratic equations can be solved by linearization (i.e. by considering each remaining monomial as an independent variable and solving the resulting linear system) and restart until the solution is found. Under strong heuristic assumptions, this finds all the solutions of $m$ quadratic polynomials in $n$ variables with $\mathcal{\tilde O}({2^{n-\sqrt{2m}}})$ operations. Although the best known algorithms require exponentially less time, the present technique has the advantage of being simpler to describe and easy to implement. In strong contrast with the state-of-the-art, it is also quite efficient in practice

Faster Modular Arithmetic For Isogeny Based Crypto on Embedded Devices

We show how to implement the Montgomery reduction algorithm for isogeny based cryptography such that it can utilize the unsigned multiply accumulate accumulate long instruction present on modern ARM architectures. This results in a practical speed-up of a factor 1.34 compared to the approach used by SIKE: the supersingular isogeny based submission to the ongoing post-quantum standardization effort. Moreover, motivated by the recent work of Costello and Hisil (ASIACRYPT 2017), which shows that there is only a moderate degradation in performance when evaluating large odd degree isogenies, we search for more general supersingular isogeny friendly moduli. Using graphics processing units to accelerate this search we find many such moduli which allow for faster implementations on embedded devices. By combining these two approaches we manage to make the modular reduction 1.5 times as fast on a 32-bit ARM platform

Another Look at the Cost of Cryptographic Attacks

This paper makes the case for considering the cost of cryptographic attacks as the main measure of their efficiency, instead of their time complexity. This allows, in our opinion, a more realistic assessment of the "risk" these attacks represent. This is half-and-half a position and a technical paper. Cryptographic attacks described in the literature are rarely implemented. Most exist only "on paper", and their main characteristic is that their estimated time complexity is small enough to break a given security property. However, when a cryptanalyst actually considers implementing an attack, she soon realizes that there is more to the story than time complexity. For instance, Wiener has shown that breaking the double-DES costs 2 6n/5 , asymptotically more than exhaustive search on n bits. We put forward the asymptotic cost of cryptographic attacks as a measure of their practicality. We discuss the shortcomings of the usual computational model and propose a simple abstract cryptographic machine on which it is easy to estimate the cost. We then study the asymptotic cost of several relevant algorithm: collision search, the three-list birthday problem (3XOR) and solving multivariate quadratic polynomial equations. We find that some smart algorithms cost much more than what their time complexity suggest, while naive and simple algorithms may cost less. Some algorithms can be tuned to reduce their cost (this increases their time complexity). Foreword A celebrated High Performance Computing paper entitled "Hitting the Memory Wall: Implications of the Obvious" [47] opens with these words: This brief note points out something obvious-something the authors "knew" without really understanding. With apologies to those who did understand, we offer it to those others who, like us, missed the point. We would like to do the same-but this note is not so short

Variable Elimination - a Tool for Algebraic Cryptanalysis

Techniques for eliminating variables from a system of nonlinear equations are used to find solutions of the system. We discuss how these methods can be used to attack certain types of symmetric block ciphers, by solving sets of equations arising from known plain text attacks. The systems of equations corresponding to these block ciphers have the characteristics that the solution is determined by a small subset of the variables (i.e., the secret key), and also that it is known that there always exists at least one solution (again corresponding to the key which is actually used in the encryption). It turns out that some toy ciphers can be solved simpler than anticipated by this method, and that the method can take advantage of overdetermined systems

Shorter VOLEitH Signature from Multivariate Quadratic

The VOLE-in-the-Head paradigm, recently introduced by Baum et al. (Crypto 2023), is a compiler that uses SoftspokenOT (Crypto 2022) to transfer any VOLE-based designated verifier zero-knowledge protocol into a publicly verifiable zero-knowledge protocol. Together with the Fiat-Shamir transformation, a new digital signature scheme FAEST (faest.info) is proposed, and it outperforms all MPC-in-the-Head signatures. We propose a new candidate post-quantum signature scheme from the Multivariate Quadratic (MQ) problem in the VOLE-in-the-Head framework, which significantly reduces the signature size compared to previous works. We achieve a signature size ranging from 3.5KB to 6KB for the 128-bit security level. Compared to the state-of-the-art MQ-based signature schemes and existing VOLE-in-the-Head signatures, our scheme achieves the smallest signature size (1.5 to 2 times compared to MQ-based schemes) while keeping the computational efficiency competitive

A crossbred algorithm for solving Boolean polynomial systems

International audienceWe consider the problem of solving multivariate systems of Boolean polynomial equations: starting from a system of m polynomials of degree at most d in n variables, we want to find its solutions over F2. Except for d = 1, the problem is known to be NP-hard, and its hardness has been used to create public cryptosystems; this motivates the search for faster algorithms to solve this problem. After reviewing the state of the art, we describe a new algorithm and show that it outperforms previously known methods in a wide range of relevant parameters. In particular, the first named author has been able to solve all the Fukuoka Type I MQ challenges, culminating with the resolution of a system of 148 quadratic equations in 74 variables in less than a day (and with a lot of luck)