Signature scheme based on the root extraction problem over braid groups

Abstract

Several public key cryptosystems and authentication schemes based on the conjugator search and root extraction problems over braid groups have been proposed. However, security analysis showed that it is not necessary to solve the underlying conjugator search problem or the root extraction problem in order to break these public key cryptographic algorithms. Hence, these cryptographic primitives suffer from some security drawbacks. A digital signature scheme based on the root extraction problem over braid groups is proposed. It is proven that the only way for the attacker to forge a signature is to extract the eth root for a given braid in the braid group. It is also shown that given sufficiently many message-signature pairs, the attacker needs to solve an intractable problem, the group factorisation problem, in order to forge a signature. Furthermore, it is pointed out that the attacker cannot learn much useful information by reconstructing braid equations with respect to the public and secret keys. Performance analysis shows that the proposed signature scheme is efficient and practical, and the key sizes are acceptable. The computational overheads to sign a message and to verify a signature are only equivalent to several 1024-RSA modular multiplications.