ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

13. Combinatorics

1991 MS Classification: 5, 20, 94 Keedwell, Anthony Donald, University of Surrey, England Crossed-inverse quasigroups and their use in cryptography. A finite quasigroup (Q,_{0} ) of order n consists of a set Q of symbols on which a binary operation (_{0}) is defned such that (i) for all pairs of elements a, b ∈ Q, a_{0}b ∈ Q; and (ii) for all pairs a, b ∈ Q, there exists unique elements x, y ∈ Q such that x_{0}b = b and a_{0}y = b. If, for each a ∈ Q, there ecists an element a′ ∈ Q such that (a_{0}b)_{0}a′ = b for all b ∈ Q, the quasigroup is said to have the right crossed-inverse property. Such a quasigroup also has the left crossed-inverse property and is called a crossed-inverse quasigroup. We show how crossed-inverse quasigroup with long inverse cycles may be constructed and suggest several applications to cryptography. Reference: R. Artzy, On loops with a special property, Proc. Amer. Math. Soc. 6(1955), 448 - 452.