Cryptography from tensor problems

We describe a new proposal for a trap-door one-way function. The new proposal belongs to the "multivariate quadratic" family but the trap-door is different from existing methods, and is simpler

Fast Quantum Algorithm for Solving Multivariate Quadratic Equations

In August 2015 the cryptographic world was shaken by a sudden and surprising announcement by the US National Security Agency NSA concerning plans to transition to post-quantum algorithms. Since this announcement post-quantum cryptography has become a topic of primary interest for several standardization bodies. The transition from the currently deployed public-key algorithms to post-quantum algorithms has been found to be challenging in many aspects. In particular the problem of evaluating the quantum-bit security of such post-quantum cryptosystems remains vastly open. Of course this question is of primarily concern in the process of standardizing the post-quantum cryptosystems. In this paper we consider the quantum security of the problem of solving a system of {\it $m$ Boolean multivariate quadratic equations in $n$ variables} (\MQb); a central problem in post-quantum cryptography. When $n=m$, under a natural algebraic assumption, we present a Las-Vegas quantum algorithm solving \MQb{} that requires the evaluation of, on average, $O(2^{0.462n})$ quantum gates. To our knowledge this is the fastest algorithm for solving \MQb{}

Improved Key Recovery of the HFEv- Signature Scheme

The HFEv- signature scheme is a twenty year old multivariate public key signature scheme. It uses the Minus and the Vinegar modifier on the original HFE scheme. An instance of the HFEv- signature scheme called GeMSS is one of the alternative candidates for signature schemes in the third round of the NIST Post Quantum Crypto (PQC) Standardization Project. In this paper, we propose a new key recovery attack on the HFEv- signature scheme. We show that the Minus modification does not enhance the security of cryptosystems of the HFE family, while the Vinegar modification increases the complexity of our attack only by a polynomial factor. By doing so, we show that the proposed parameters of the GeMSS scheme are not as secure as claimed. Our attack shows that it is very difficult to build a secure and efficient signature scheme on the basis of HFEv-

Group theory in cryptography

This paper is a guide for the pure mathematician who would like to know more about cryptography based on group theory. The paper gives a brief overview of the subject, and provides pointers to good textbooks, key research papers and recent survey papers in the area.Comment: 25 pages References updated, and a few extra references added. Minor typographical changes. To appear in Proceedings of Groups St Andrews 2009 in Bath, U