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Title:
Proof of Mirror Theory for \xi_{\max}=2
Authors: Avijit Dutta, Mridul Nandi, Abishanka Saha
Abstract:In ICISC-05, and in the ePrint 2010/287, Patarin claimed a lower bound on the number of 2q tuples of n-bit strings (P_1, \ldots, P_{2q}) \in (\{0,1\}^{n})^{2q} satisfying P_{2i - 1} \oplus P_{2i} = \lambda_i for 1 \leq i \leq q such that P_1, P_2, \ldots, P_{2q} are distinct and \lambda_i \in \{0,1\}^n \setminus \{0^n\}. This result is known as {\em Mirror theory} and widely used in cryptography. It stands as a powerful tool to provide a high-security guarantee for many block cipher-(or even ideal permutation-) based designs. In particular, Mirror theory has a direct application in the security of XOR of block ciphers. Unfortunately, the proof of Mirror theory contains some unverifiable gaps and several mistakes. This paper provides a simple and verifiable proof of Mirror theory.
ePrint: https://eprint.iacr.org/2020/669
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