MRHS Equation Systems that can be Solved in Polynomial Time

Zajac, Pavol

doi:10.1515/tmmp-2016-0040

Cited by 3 publications

(1 citation statement)

References 12 publications

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“…Solving MRHS systems is believed to be difficult (in general). We have already shown that it is an NP-complete problem to decide, whether an MRHS system has a solution [12]. There are multiple existing exponential time algorithms that can solve an MRHS system [9,10,11].…”

Section: Preliminariesmentioning

confidence: 99%

On solving sparse MRHS equations with bit-flipping

Zajac¹

2022

Publ. Math. Debrecen

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We present a new class of probabilistic algorithms that can be used to solve (sparse) MRHS equation systems. The algorithms are based on the idea of bit-flipping: start from a random vector as a potential solution, and try to improve individual bits according to some selected heuristic strategy. We have evaluated a group of algorithms experimentally on a model of sparse MRHS systems based on AND-XOR circuits with low gate count. We compare bit-flipping algorithms with a more complex hill-climbing algorithm and show that bit-flipping algorithms can achieve better success probability for the same number of MRHS evaluations.

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Section: Preliminariesmentioning

confidence: 99%

On solving sparse MRHS equations with bit-flipping

Zajac¹

2022

Publ. Math. Debrecen

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Connecting the Complexity of MQ- and Code-Based Cryptosystems

Zajac

2017

Tatra Mountains Mathematical Publications

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We study the connection between the MQ problem and the decoding problem, through the intermediate MRHS representation. The main goal of this study is to explicitly bound the complexity of solving MQ systems with decoding tools. The main observation is that although the MQ problem over GF(2) can be efficiently transformed to syndrome decoding, the existing general decoding methods are not suitable to solve the system as efficiently as expected from the MQ representation.

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Algebraic Cryptanalysis of Ascon Using MRHS Equations

Smičík,

Zajac

2024

Tatra Mountains Mathematical Publications

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Ascon is a family of lightweight authenticated encryption and hashing algorithms, which is a finalist in the NIST Lightweight Cryptography competition. We study the Ascon algorithm from the perspective of algebraic cryptanalysis based on the MRHS representation of the cipher. We call such an approach an MRHS cryptanalysis. We represent the system on the gate level (focusing on individual AND-gates) and the S-box level (basing MRHS equations on 5-bit S-boxes). We compare the results from the application of two custom MRHS solvers. The RZ solver is based on linear algebra and exhaustive search. The HC solver is based on adaptive bit-flipping with restarts. We show that both the choice of the solver and the choice of the system representation influence the total complexity of the attack. On the other hand, these choices do not change the fundamental properties of the attack, such as scaling with the amount of information the attacker possesses. A similar assessment holds for using a scaled-down version of Ascon for the experiments. Our method can be used for the experimental evaluation of cipher designs against algebraic attacks.

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MRHS Equation Systems that can be Solved in Polynomial Time

Cited by 3 publications

References 12 publications

On solving sparse MRHS equations with bit-flipping

On solving sparse MRHS equations with bit-flipping

Connecting the Complexity of MQ- and Code-Based Cryptosystems

Algebraic Cryptanalysis of Ascon Using MRHS Equations

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