Cryptanalysis of Cryptosystems Based on Non-commutative Skew Polynomials

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Abstract

Ten years ago, Ko et al. described a Diffie-Hellman like protocol based on decomposition with respect to a non-commutative semigroup law. Instantiation with braid groups had first been considered, however intense research on braid groups revealed vulnerabilities of the group structure and of the braid based DH problem itself.

Recently, Boucher et al. proposed a similar scheme based on a particular non-commutative multiplication of polynomials over a finite field. These so called skew polynomials have a well-studied theory and have many applications in mathematics and coding theory, however they are quite unknown in a cryptographic application.

In this paper, we show that the Diffie-Hellman problem based on skew polynomials is susceptible to a very efficient attack. This attack is in fact general in nature, it uses the availability of a one-sided notion of gcd and exact division. Given such tools, one can shift the Diffie-Hellman problem to a linear algebra type problem.