Sur les courbes elliptiques et application à la cryptographie
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Date
2022
Authors
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Journal ISSN
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Publisher
Université de M'sila
Abstract
This thesis deals with the study of the elliptic curves over finite rings and their cryptographic
applications. Firstly, we defined the elliptic curves Ea,b(Fq["]) and Ea,b(F3d ["]) over the rings
Fq["] and F3d ["] respectively, with "4 = "3 by its projective equations, then we studied the
classification of elements in these elliptic curves. Moreover, using the elliptic curve Ea,b(Fq["]),
we introduced new non-commutative cryptography schemes on a special rings so that these
cryptosystems are based on the two hard problems, the conjugal classical problem and the
discrete logarithm problem, and they have strong security and very difficult to solve for
decryption. At the end of the thesis we have given a numerical example of cryptography
(encryption and decryption) on the elliptic curve E4
a,b over the ring F3d ["] where "4 = 0 by
using two methods (with a secret key and a password).
Description
Keywords
Elliptic curve; Finite ring; Local ring; Elliptic curve cryptography; Noncommutative cryptography; Fully homomorphic encryption.