A Note on Elliptic Curve Based Transformations in Symmetric-Key Cryptography
by
Michele Elia
Politecnico di Torino - Italy
Coauthors: Guglielmo Morgari (Telsy Elettronica - Torino - Italy)

The addition group structure of super-singular elliptic curves E(F) over a Galois field F=GF(2^m) is used to define a function Q which may be profitably used in symmetric-key cryptography. The value u of Q(x) is defined as the abscissa of a point-sum Q(u, v)=P(X(x), Y(x))+kK(a, b) on E(F) for all x Î F, where K(a, b) is a fixed point on E(F), k is an integer, and P(X(x), Y(x)) is a point on E(F) which can be inexpensively computed as a function of x.
A thorough analysis of Q(x) as a function of k, a, and b shows that many properties of these functions have cryptographic appeal. Some examples of applications to symmetric-key encryption are thus presented, and their cryptographic characteristics investigated.

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Date received: October 5, 2005