### An Algebraic Attack on Ciphers with Low-Degree Round Functions: Application to Full MiMC

Maria Eichlseder, Lorenzo Grassi, Reinhard Lüftenegger, Morten Øygarden, Christian Rechberger, Markus Schofnegger, and Qingju Wang

##### Abstract

Algebraically simple PRFs, ciphers, or cryptographic hash functions are becoming increasingly popular, for example due to their attractive properties for MPC and new proof systems (SNARKs, STARKs, among many others). In this paper, we focus on the algebraically simple construction MiMC, which became an attractive cryptanalytic target due to its simplicity, but also due to its use as a baseline in a competition for more recent algorithms exploring this design space. For the first time, we are able to describe key-recovery attacks on all full-round versions of MiMC over GF(2^n), requiring half the code book. In the chosen-ciphertext scenario, recovering the key from this data for the n-bit full version of MiMC takes the equivalent of less than 2^(n - log_2(n) + 1) calls to MiMC and negligible amounts of memory. The attack procedure is a generalization of higher-order differential cryptanalysis, and it is based on two main ingredients. First, we present a higher-order distinguisher which exploits the fact that the algebraic degree of MiMC grows significantly slower than originally believed. Secondly, we describe an approach to turn this distinguisher into a key-recovery attack without guessing the full subkey. Finally, we show that approximately ceil(log_3(2 * R)) more rounds (where R = ceil(n * log_3(2)) is the current number of rounds of MiMC-n/n) can be necessary and sufficient to restore the security against the key-recovery attack presented here. The attack has been practically verified on toy versions of MiMC. Note that our attack does not affect the security of MiMC over prime fields.

Note: Updated acknowledgements.

Available format(s)
Category
Secret-key cryptography
Publication info
A major revision of an IACR publication in ASIACRYPT 2020
DOI
10.1007/978-3-030-64837-4_16
Keywords
Algebraic attackMiMCHigher-order differential
Contact author(s)
markus schofnegger @ iaik tugraz at
History
2020-12-16: last of 3 revisions
See all versions
Short URL
https://ia.cr/2020/182

CC BY

BibTeX

@misc{cryptoeprint:2020/182,
author = {Maria Eichlseder and Lorenzo Grassi and Reinhard Lüftenegger and Morten Øygarden and Christian Rechberger and Markus Schofnegger and Qingju Wang},
title = {An Algebraic Attack on Ciphers with Low-Degree Round Functions: Application to Full MiMC},
howpublished = {Cryptology ePrint Archive, Paper 2020/182},
year = {2020},
doi = {10.1007/978-3-030-64837-4_16},
note = {\url{https://eprint.iacr.org/2020/182}},
url = {https://eprint.iacr.org/2020/182}
}

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