Abstract
Isomorphism of polynomials with two secrets (IP2S) problem was proposed by Patarin et al. at Eurocrypt 1996 and the problem is to find two secret linear maps filling in the gap between two polynomial maps over a finite field. At PQC 2020, Santoso proposed a problem originated from IP2S, which is called block isomorphism of polynomials with circulant matrices (BIPC) problem. The BIPC problem is obtained by linearizing IP2S and restricting secret linear maps to linear maps represented by circulant matrices. Using the commutativity of products of circulant matrices, Santoso also proposed an ElGamal-like encryption scheme based on the BIPC problem. In this paper, we give a new security analysis on the ElGamal-like encryption scheme. In particular, we introduce a new attack (called linear stack attack) which finds an equivalent key of the ElGamal-like encryption scheme by using the linearity of the BIPC problem. We see that the attack is a polynomial-time algorithm and can break some 128-bit proposed parameters of the ElGamal-like encryption scheme within 10 h on a standard PC.
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Notes
- 1.
The earlier draft of our attack is published as a preprint in IACR Cryptology ePrint Archive [11].
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Acknowledgements
This work was supported by JST CREST Grant Number JPMJCR14D6, JSPS KAKENHI Grant Number JP19K20266, JP20K19802, JP20K03741, JP18H01438, and JP18K11292
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Ikematsu, Y., Nakamura, S., Santoso, B., Yasuda, T. (2021). Security Analysis on an ElGamal-Like Multivariate Encryption Scheme Based on Isomorphism of Polynomials. In: Yu, Y., Yung, M. (eds) Information Security and Cryptology. Inscrypt 2021. Lecture Notes in Computer Science(), vol 13007. Springer, Cham. https://doi.org/10.1007/978-3-030-88323-2_12
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