Inferring sequences produced by elliptic curve generators using Coppersmith's methods

Thierry Mefenza (1, 2) , Damien Vergnaud (3, 4)

1 UY1 - Université de Yaoundé I
2 Département de Mathématiques [Yaoundé I] = Department of Mathematics [Yaoundé, Cameroon]
3 IUF - Institut universitaire de France
4 ALMASTY - ALgorithms for coMmunicAtion SecuriTY

We analyze the security of two number-theoretic pseudo-random generators based on elliptic curves: the elliptic curve linear congruential generator and the elliptic curve power generator. We show that these recursive generators are insecure if sufficiently many bits are output at each iteration (improving notably the prior cryptanalysis of Gutierrez and Ibeas from 2007). We present several theoretical attacks based on Coppersmith's techniques for finding small roots on polynomial equations. Our results confirm that these generators are not appropriate for cryptographic purposes.

Mots clés: Elliptic curve power generator; Elliptic curve linear congruential generator; Pseudo-random generator; Elliptic curves; Coppersmith's methods

Domaines: Cryptographie et sécurité [cs.CR]

Thierry Mefenza, Damien Vergnaud. Inferring sequences produced by elliptic curve generators using Coppersmith's methods. Theoretical Computer Science, 2020, 830-831, pp.20-42. ⟨10.1016/j.tcs.2020.04.025⟩. ⟨hal-02568170⟩