The number of isomorphism elliptic curve during the transformation of the canonical form of the equation

Abstract

The results of analyze of analytic expressions for transformation of nonsupersingular elliptic curves in the canonical form for cryptographic purposes are shown. New results of the estimate of an upper bound estimate of the number of isomorphic transformations of elliptic curve of the curves in the canonical form over finite Galois field were obtained. For a field with characteristic p from canonical to normal form, an upper bound of the number of isomorphisms increases proportionally to p⁴. The using of full set of base elliptic curve transformations gives possibility to increase cardinality of set parameters of cryptosystems on elliptic curves and also use it as additional entropy source. Implementation of these results in cryptographic random bit generators can allow to cut size of Galois field module.