Ni, J.; Zhang, J.; Wang, G.; Li, R.; Shen, Y. Algebraic Attacks against Grendel: An Arithmetization-Oriented Primitive with the Legendre Symbol. Symmetry2023, 15, 1563.
Ni, J.; Zhang, J.; Wang, G.; Li, R.; Shen, Y. Algebraic Attacks against Grendel: An Arithmetization-Oriented Primitive with the Legendre Symbol. Symmetry 2023, 15, 1563.
Ni, J.; Zhang, J.; Wang, G.; Li, R.; Shen, Y. Algebraic Attacks against Grendel: An Arithmetization-Oriented Primitive with the Legendre Symbol. Symmetry2023, 15, 1563.
Ni, J.; Zhang, J.; Wang, G.; Li, R.; Shen, Y. Algebraic Attacks against Grendel: An Arithmetization-Oriented Primitive with the Legendre Symbol. Symmetry 2023, 15, 1563.
Abstract
Modern cryptographic protocols such as zero-knowledge proofs and secure multi-party computation have increased the demand for a novel category of symmetric primitives. These primitives are not optimized for traditional platforms such as servers, microcontrollers, and desktop computers but rather for their ability to be implemented in arithmetic circuits. To enable efficient arithmetic operations, they define operations over larger finite fields and use low-degree invertible functions to construct their non-linear layers. Grendel is an arithmetization-oriented permutation that leverages the Legendre Symbol to enhance the growth of algebraic degrees in its non-linear layer. In this paper, we present a preimage attack on the sponge hash function instantiated with the full rounds of the Grendel permutation using algebraic methods. We introduce a technique that allows us to eliminate two full rounds of substitution permutation networks (SPN) in the sponge hash function with minimal or no additional cost. This method can be combined with univariate root-finding techniques and Gröbner basis attacks to break the number of rounds claimed by the designers. By utilizing this strategy, our attack achieves an improvement of two additional rounds compared to the previous state-of-the-art attack. While not breaking its security margin, it allows us to further understand the design and analysis of such cryptographic primitives.
Copyright:
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