## Cryptology ePrint Archive: Report 2011/686

**Analysis of some natural variants of the PKP Algorithm**

*Rodolphe LAMPE and Jacques PATARIN*

**Abstract: **In 1989, Adi Shamir proposed a new zero-knowledge identification scheme based
on a NP-complete problem called PKP for Permuted Kernel Problem. For a given
prime p, a given matrix A and a given vector V , the problem is to find a
permutation \pi such that the permuted vector V_\pi verifies A.V_\pi = 0
mod p. This scheme is still in 2011 known as one of the most efficient
identification scheme based on a combinatorial problem. However, we will see
in this paper that it is possible to improve this scheme significantly by
combining new ideas in order to reduce the total number of computations to be
performed and to improve very efficiently the security against side channel
attacks using precomputations. We will obtain like this a new scheme that we
have called SPKP. Moreover, if we use precomputed values in the scheme SPKP,
then the prover will need to perform no computations (i.e. only selection and
transmission of precomputed values). This is very interesting for security
against side channel attacks because our scheme is zero-knowledge and we don't
perform any computations using the key during the identification so we prove
that any attacker (even using side channel attacks) being successfully
identified implies that he has a solution to the NP-complete problem PKP.

**Category / Keywords: **public-key cryptography / identification scheme, zero knowledge, permuted kernel problem

**Date: **received 16 Dec 2011, last revised 4 Jul 2012

**Contact author: **rodolphe lampe at gmail com

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

**Version: **20120704:182542 (All versions of this report)

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