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Title:
The complexity of solving a random polynomial system
Authors: Giulia Gaggero, Elisa Gorla
Abstract:In this paper, we discuss what it means for a polynomial system to be random and how hard it is to solve a random polynomial system. We propose an algebraic definition of randomness, that we call algebraic randomness. Using a conjecture from commutative algebra, we produce a sharp upper bound for the degree of regularity, hence the complexity of solving an algebraically random polynomial system by Groebner bases methods. As a proof of concept, we apply our result to Rainbow and GeMSS and show that these systems are far from being algebraically random.
ePrint: https://eprint.iacr.org/2024/1924
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