Pseudorandomness and Dynamics of Fermat Quotients

Ostafe, Alina; Shparlinski, Igor E.

doi:10.1137/100798466

Cited by 69 publications

(74 citation statements)

References 34 publications

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“…There are several results which involve the distribution and structure of Fermat quotients q p (u) modulo p and it has numerous applications in computational and algebraic number theory, see e.g. [8,10,11,21] and references therein.…”

Section: Introductionmentioning

confidence: 99%

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Structure of Pseudorandom Numbers Derived from Fermat Quotients

Chen

Ostafe

Winterhof

2010

Arithmetic of Finite Fields

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Abstract.We study the distribution of s-dimensional points of Fermat quotients modulo p with arbitrary lags. If no lags coincide modulo p the same technique as in [21] works. However, there are some interesting twists in the other case. We prove a discrepancy bound which is unconditional for s = 2 and needs restrictions on the lags for s > 2. We apply this bound to derive results on the pseudorandomness of the binary threshold sequence derived from Fermat quotients in terms of bounds on the well-distribution measure and the correlation measure of order 2, both introduced by Mauduit and Sárközy. We also prove a lower bound on its linear complexity profile. The proofs are based on bounds on exponential sums and earlier relations between discrepancy and both measures above shown by Mauduit, Niederreiter and Sárközy. Moreover, we analyze the lattice structure of Fermat quotients modulo p with arbitrary lags.

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Section: Introductionmentioning

confidence: 99%

“…If no lags coincide modulo p the same technique as in [21] works. However, there are some interesting twists in the other case.…”

mentioning

confidence: 99%

Structure of Pseudorandom Numbers Derived from Fermat Quotients

Chen

Ostafe

Winterhof

2010

Arithmetic of Finite Fields

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“…• discrepancy of Fermat quotients from [6], Theorems 18-19, • new bound for the size of the image of q(n), see [11], Theorem 1, • estimates for Ihara sum, [12], • better bounds for the sums…”

Section: Theorem 3 One Hasmentioning

confidence: 99%

“…Then |S(a)| ≪ p Heilbronn's exponential sum is connected (see e.g. [1], [2], [5], [6], [11], [12]) with so-called Fermat quotients defined as q(n) = n p−1 − 1 p , n = 0 (mod p) .…”

Section: Introductionmentioning

confidence: 99%