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# A Simple Deterministic Algorithm for Systems of Quadratic Polynomials over

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... 2. By utilizing a simple version of the crossbred algorithm [JV17] proposed in [BDT22] to solve an overdefined quadratic equation system, our attacks on 3-round LowMC require negligible memory and can achieve better time-memory tradeoffs than Dinur's algorithm [Din21]. ...
... In our attacks, we will adopt a simple version of the crossbred algorithm [JV17] to solve an overdefined system of quadratic equations, which is described in [BDT22]. This algorithm fits very well with our attacks on LowMC for its simplicity to bound the time complexity and to implement in practice. ...
... The first attack is a new and simple guess-and-determine (GnD) attack on 3-round LowMC by using Banik et al.'s strategy [BBDV20] to linearize the 3-bit S-box, where we solve a system of quadratic equations with the standard linearization technique. The second attack is a much simpler yet more efficient GnD attack on 3-round LowMC by using a naive guess strategy to linearize the 3-bit S-box, where we solve quadratic equations with the simplified version of the crossbred algorithm [BDT22]. The third attack is for full-round (4-round) LowMC, where we still adopt the naive guess strategy but use Dinur's algorithm [Din21] to solve equations of degree 4. ...
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Andreas Björklund, Petteri Kaski, and Ryan Williams; licensed under Creative Commons License CC-BY We consider the problem of finding solutions to systems of polynomial equations over a finite field. Lokshtanov et al. [SODA'17] recently obtained the first worst-case algorithms that beat exhaustive search for this problem. In particular for degree-d equations modulo two in n variables, they gave an O∗2(1−1/(5d))n time algorithm, and for the special case d = 2 they gave an O∗20.876n time algorithm. We modify their approach in a way that improves these running times to O∗2(1−1/(27d))n and O∗20.804n, respectively. In particular, our latter bound - that holds for all systems of quadratic equations modulo 2 - comes close to the O∗20.792n expected time bound of an algorithm empirically found to hold for random equation systems in Bardet et al. [J. Complexity, 2013]. Our improvement involves three observations: 1. The Valiant-Vazirani lemma can be used to reduce the solution-finding problem to that of counting solutions modulo 2. 2. The monomials in the probabilistic polynomials used in this solution-counting modulo 2 have a special form that we exploit to obtain better bounds on their number than in Lokshtanov et al. [SODA'17]. 3. The problem of solution-counting modulo 2 can be “embedded” in a smaller instance of the original problem, which enables us to apply the algorithm as a subroutine to itself.
Chapter
At SODA 2017 Lokshtanov et al. presented the first worst-case algorithms with exponential speedup over exhaustive search for solving polynomial equation systems of degree d in n variables over finite fields. These algorithms were based on the polynomial method in circuit complexity which is a technique for proving circuit lower bounds that has recently been applied in algorithm design. Subsequent works further improved the asymptotic complexity of polynomial method-based algorithms for solving equations over the field F2. However, the asymptotic complexity formulas of these algorithms hide significant low-order terms, and hence they outperform exhaustive search only for very large values of n. In this paper, we devise a concretely efficient polynomial method-based algorithm for solving multivariate equation systems over F2. We analyze our algorithm’s performance for solving random equation systems, and bound its complexity by about n2·20.815n bit operations for d=2 and n2·21-1/2.7dn for any d≥2. We apply our algorithm in cryptanalysis of recently proposed instances of the Picnic signature scheme (an alternate third-round candidate in NIST’s post-quantum standardization project) that are based on the security of the LowMC block cipher. Consequently, we show that 2 out of 3 new instances do not achieve their claimed security level. As a secondary application, we also improve the best-known preimage attacks on several round-reduced variants of the Keccak hash function. Our algorithm combines various techniques used in previous polynomial method-based algorithms with new optimizations, some of which exploit randomness assumptions about the system of equations. In its cryptanalytic application to Picnic, we demonstrate how to further optimize the algorithm for solving structured equation systems that are constructed from specific cryptosystems.
Chapter
After initial publication of the book, various errors were identified that needed correction. The following corrections have been updated within the current version, along with all known typographical errors.
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GeMSS: A Great Multivariate Short Signature
• Antoine Casanova
• Jean-Charles Faugère
• Gilles Macario-Rat
• Jacques Patarin
• Ludovic Perret
• Jocelyn Ryckeghem
Antoine Casanova, Jean-Charles Faugère, Gilles Macario-Rat, Jacques Patarin, Ludovic Perret, and Jocelyn Ryckeghem. GeMSS: A Great Multivariate Short Signature. Research report, UPMC -Paris 6 Sorbonne Universités ; INRIA Paris Research Centre, MAMBA Team, F-75012, Paris, France ; LIP6 -Laboratoire d'Informatique de Paris 6, December 2017. URL: https://hal.inria.fr/hal-01662158.
A Crossbred Algorithm for Solving Boolean Polynomial Systems
• Antoine Joux
• Vanessa Vitse
Antoine Joux and Vanessa Vitse. A Crossbred Algorithm for Solving Boolean Polynomial Systems. In NuTMiC, volume 10737 of Lecture Notes in Computer Science, pages 3-21. Springer, 2017. https://eprint.iacr.org/2017/ 372.pdf.