# Analysis of Some Natural Variants of the PKP Algorithm

### Rodolphe Lampe, Jacques Patarin

#### Abstract

In 1989, (Shamir, 1989) 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 p such that the permuted vector Vp verifies A·Vp = 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.

#### References

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#### Paper Citation

#### in Harvard Style

Lampe R. and Patarin J. (2012). **Analysis of Some Natural Variants of the PKP Algorithm** . In *Proceedings of the International Conference on Security and Cryptography - Volume 1: SECRYPT, (ICETE 2012)* ISBN 978-989-8565-24-2, pages 209-214. DOI: 10.5220/0004012202090214

#### in Bibtex Style

@conference{secrypt12,

author={Rodolphe Lampe and Jacques Patarin},

title={Analysis of Some Natural Variants of the PKP Algorithm},

booktitle={Proceedings of the International Conference on Security and Cryptography - Volume 1: SECRYPT, (ICETE 2012)},

year={2012},

pages={209-214},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0004012202090214},

isbn={978-989-8565-24-2},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the International Conference on Security and Cryptography - Volume 1: SECRYPT, (ICETE 2012)

TI - Analysis of Some Natural Variants of the PKP Algorithm

SN - 978-989-8565-24-2

AU - Lampe R.

AU - Patarin J.

PY - 2012

SP - 209

EP - 214

DO - 10.5220/0004012202090214