Braided Ribbon Group C_n-based Asymmetric Cryptography

Abstract

Braided ribbons are two-dimensional analogues of classical braids. These are basically intertwining, non-intersecting braids of strips in , instead of strings or curves which geometrically composed the usual braids. The set of isotopy classes of braided ribbons with the same number of strips is essentially an infinite nonabelian group endowed with a natural product structure analogous to that of the braid groups. We call this the braided ribbon group  of  strips, and denote it by . In this article, we introduce braided ribbons and give explicitly some of its group-theoretical properties as support to the construction of the -based cryptosystem. These properties include the group presentation, center, finitely-presentedness, and linearity. This also shows that  is the semidirect product  of an abelian group by the usual braid group, and exhibits its relation to other known groups. Moreover, this paper looks into the potential of braided ribbon groups as platform groups in noncommutative algebraic public-key cryptosystems. In particular, we explicitly define a trapdoor function, key agreement scheme and public key cryptography using  that are based on the version of the generalized conjugacy search problem for .