Modification of Anshel-Anshel-Goldfeld Postquantum Algorithm / Protocol Based on Algebraic Braid Groups, in Order to the Span–Cyberattack Neutralization

This article deals with relatively young field of cryptography, namely cryptography based on the theory of infinite abstract groups. The research identifies the main problems on which this type of cryptography is based, and the cryptoanalysis conducts of one of the algorithms grounded on the problem of mating (conjugate) element finding, on the base of which the protocol for common key generating is developed. As the algorithm under study, the protocol for generating the common key based on Anshel-Anshel-Goldfeld algorithm, built on the algebraic braid groups, is investigated. During the study of this protocol, one of possible cyberattacks was identified, which allows to get hold of the secret keys of subscribers. To eliminate and to neutralize this span–cyberattack, the new modification of Anshel–Anshel– Goldfeld (AAG) algorithm was developed, which significantly reduces the probability of this cyberattack successful implementation. Analysis of this modification operating time was carried out also.

Linear time algorithm for the conjugacy problem in the first Grigorchuk group

We prove that the conjugacy problem in the first Grigorchuk group [Formula: see text] can be solved in linear time. Furthermore, the problem to decide if a list of elements [Formula: see text] contains a pair of conjugate elements can be solved in linear time. We also show that a conjugator for a pair of conjugate element [Formula: see text] can be found in polynomial time.