### Inferring sequences produced by a linear congruential generator on elliptic curves missing high--order bits

Jaime Gutierrez and Alvar Ibeas

##### Abstract

Let $p$ be a prime and let $E(\F_p)$ be an elliptic curve defined over the finite field $\F_p$ of $p$ elements. For a given point $G \in E(\F_p)$ the linear congruential genarator on elliptic curves (EC-LCG) is a sequence $(U_n)$ of pseudorandom numbers defined by the relation $$U_n=U_{n-1}\oplus G=nG\oplus U_0,\quad n=1,2,\ldots,$$ where $\oplus$ denote the group operation in $E(\F_p)$ and $U_0 \in E(\F_p)$ is the initial value or seed. We show that if $G$ and sufficiently many of the most significants bits of two consecutive values $U_n, U_{n+1}$ of the EC-LCG are given, one can recover the seed $U_0$ (even in the case where the elliptic curve is private) provided that the former value $U_n$ does not lie in a certain small subset of exceptional values. We also estimate limits of a heuristic approach for the case where $G$ is also unknown. This suggests that for cryptographic applications EC-LCG should be used with great care. Our results are somewhat similar to those known for the linear and non-linear pseudorandom number congruential generator.

Available format(s)
Category
Foundations
Publication info
Published elsewhere. pseudorandom numbers, elliptic curves
Contact author(s)
jaime gutierrez @ unican es
History
Short URL
https://ia.cr/2007/099

CC BY

BibTeX

@misc{cryptoeprint:2007/099,
author = {Jaime Gutierrez and Alvar Ibeas},
title = {Inferring sequences produced by a linear congruential generator on elliptic curves missing high--order bits},
howpublished = {Cryptology ePrint Archive, Paper 2007/099},
year = {2007},
note = {\url{https://eprint.iacr.org/2007/099}},
url = {https://eprint.iacr.org/2007/099}
}

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