Proof of Mirror Theory for ξmax= 2

Article Type

Research Article

Publication Title

IEEE Transactions on Information Theory

Abstract

In ICISC-05, and in the ePrint 2010/287, Patarin claimed a lower bound on the number of $2 q$ 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 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.

First Page

6218

Last Page

6232

DOI

10.1109/TIT.2022.3171178

Publication Date

9-1-2022

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