Introduction to Mirror Theory: Analysis of Systems of Linear Equalities and Linear Non Equalities for Cryptography

Jacques Patarin

Paper 2010/287

Introduction to Mirror Theory: Analysis of Systems of Linear Equalities and Linear Non Equalities for Cryptography

Jacques Patarin

Abstract

\begin{abstract} In this paper we will first study two closely related problems:\ 1. The problem of distinguishing $f (x ‖ 0) \oplus f (x ‖ 1)$ where $f$ is a random permutation on $n$ bits. This problem was first studied by Bellare and Implagliazzo in~\cite{BI}.\ 2. The so-called ``Theorem $P_{i} \oplus P_{j}$ '' of Patarin (cf~\cite{P05}). Then, we will see many variants and generalizations of this ``Theorem $P_{i} \oplus P_{j}$ '' useful in Cryptography. In fact all these results can be seen as part of the theory that analyzes the number of solutions of systems of linear equalities and linear non equalities in finite groups. We have nicknamed these analysis ``Mirror Theory'' due to the multiples induction properties that we have in it. \end{abstract}

Note: rectification of typos

Metadata

Available format(s): PDF
Category: Secret-key cryptography
Publication info: Preprint. MINOR revision.
Keywords: Xor of random permutations Security proofs beyond the Birthday Bound
Contact author(s): valerie nachef @ u-cergy fr
History: 2017-02-19: last of 2 revisions; 2010-05-17: received; See all versions
Short URL: https://ia.cr/2010/287
License: CC BY

BibTeX

@misc{cryptoeprint:2010/287,
      author = {Jacques Patarin},
      title = {Introduction to Mirror Theory: Analysis of Systems of Linear Equalities and Linear Non Equalities for Cryptography},
      howpublished = {Cryptology {ePrint} Archive, Paper 2010/287},
      year = {2010},
      url = {https://eprint.iacr.org/2010/287}
}