**Mirror Theory: A simple proof of the Pi+Pj Theorem with xi_max=2**

*Benoît Cogliati and Jacques Patarin*

**Abstract: **We provide a simple and complete proof of the famous Pi⊕Pj Theorem in the particular case where ξmax=2. This Theorem gives a lower bound for the number of solutions of simple linear systems of equations in the case where all the variables have to be pairwise distinct. Such systems often occur in cryptographic proofs of security, and this particular Theorem can be used to prove that the function x↦P(0||x)⊕P(1||x) is an optimally secure pseudorandom function when P is a uniformly random permutation.

**Category / Keywords: **secret-key cryptography / Mirror Theory, security proofs, optimal security

**Date: **received 17 Jun 2020, last revised 17 Jun 2020

**Contact author: **benoitcogliati at hotmail fr

**Available format(s): **PDF | BibTeX Citation

**Version: **20200618:154737 (All versions of this report)

**Short URL: **ia.cr/2020/734

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