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Inferring sequences produced by a linear congruential generator on elliptic curves missing high--order bits
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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. |
BibTeX
@misc{eprint-2007-13381, title={Inferring sequences produced by a linear congruential generator on elliptic curves missing high--order bits}, booktitle={IACR Eprint archive}, keywords={foundations /}, url={http://eprint.iacr.org/2007/099}, note={pseudorandom numbers, elliptic curves jaime.gutierrez@unican.es 13591 received 19 Mar 2007}, author={Jaime Gutierrez and Alvar Ibeas}, year=2007 }