Abstract
Lattice tests are quality measures for assessing the intrinsic structure of pseudorandom number generators. Recently a new lattice test has been introduced by Niederreiter and Winterhof. In this paper, we present a general inequality that is satisfied by any periodic sequence. Then, we analyze the behavior of the linear congruential generators on elliptic curves (EC-LCG) under this new lattice test and prove that the EC-LCG passes it up to very high dimensions. We also use a result of Brandstatter and Winterhof on the linear complexity profile related to the correlation measure of order \(k\) to present lower bounds on the linear complexity profile of some binary sequences derived from the EC-LCG.
Highlights
1.1 Lattice testsLattice tests are quality measures for assessing the intrinsic structure of pseudorandom generators
Recent developments point towards an interest in the elliptic curve analogues of pseudo-random number generators, which are reasonably new sources of pseudorandom numbers based on the group structure of elliptic curves over finite fields
From Proposition 1 and the result in [14] we get a lower bound for S(ηn, T ), in the Theorem, we prove a stronger lower bound
Summary
Lattice tests are quality measures for assessing the intrinsic structure of pseudorandom generators. For given integers s ≥ 1 and N ≥ 2, (ηn) passes the s-dimensional N -lattice test if the vectors {ηn − η0 : 1 ≤ n ≤ N − 1} span Fsq, where ηn = Let (ηn) be a T -periodic sequence over the finite field Fq. For given integers s ≥ 1, 0 < d1 < d2 < · · · < ds−1 < T , and N ≥ 2, we say that (ηn) passes the s-dimensional N -lattice test with lags d1, . Let (ηn) be any T -periodic sequence with T prime, if s satisfies the following inequality log T + log s + 2 s≤. We assume that the sequence (ηn) does not pass the s-dimensional T lattice test for some lags 0 < d1 < · · · < ds−1 < T.
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