Character Sums with Exponential Functions and their Applications

Konyagin, Sergeĭ; Shparlinski, Igor E.

doi:10.1017/cbo9780511542930

Cited by 185 publications

(217 citation statements)

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“…The proof is analogous to the proof of Lemma 2.1 using in this case Theorem 5.5 of [7] instead of (1) and (2). For each prime p ≡ 1 (mod T ) we fix an element g p,T of multiplicative order T .…”

Section: Distribution Of G X Modulo Pmentioning

confidence: 93%

“…For each prime p ≡ 1 (mod T ) we fix an element g p,T of multiplicative order T . Then Theorem 5.5 of [7] claims that for any U > 1 and any integer ν ≥ 2, for all primes p ≡ 1 (mod T ) except at most O(U/ log U ) of them, the bound max gcd(c,p)=1…”

Section: Distribution Of G X Modulo Pmentioning

confidence: 99%

“…It would be very interesting to replace the condition T ≥ p ε for the smallest size of the multiplicative order of g in Lemma 2.2 by a weaker condition of the form T ≥ (log p) c with some constant c. Although a more careful analysis of the proof of Theorem 5.5 of [7] should allow to replace p ε with a slower growing function, it seems unlikely that the present method can be applied to T as small as a power of log p.…”

Section: Remarksmentioning

confidence: 99%

“…Here we use new bounds of exponential sums from [7] to extend some results of [2] to the case of elements g of arbitrary multiplicative order T , provided that T ≥ p 1/3+ε . This allows us to prove that the statement of Theorem 2 of [2] holds for all pairs (a, b).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the Security of Diffie-Hellman Bits

Vasco

Shparlinski

2001

Cryptography and Computational Number Theory

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Abstract. Boneh and Venkatesan have recently proposed a polynomial time algorithm for recovering a "hidden" element α of a finite field IFp of p elements from rather short strings of the most significant bits of the remainder modulo p of αt for several values of t selected uniformly at random from IF * p . We use some recent bounds of exponential sums to generalize this algorithm to the case when t is selected from a quite small subgroup of IF * p . Namely, our results apply to subgroups of size at least p 1/3+ε for all primes p and to subgroups of size at least p ε for almost all primes p, for any fixed ε > 0. We also use this generalization to improve (and correct) one of the statements of the aforementioned work about the computational security of the most significant bits of the Diffie-Hellman key.

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Section: Distribution Of G X Modulo Pmentioning

confidence: 93%

Section: Distribution Of G X Modulo Pmentioning

confidence: 99%

Section: Remarksmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the Security of Diffie-Hellman Bits

Vasco

Shparlinski

2001

Cryptography and Computational Number Theory

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“…. , M +N , where N ≤ t, in finite fields and residue rings: see [6,7,17,[21][22][23][26][27][28] for several results in this direction together with their numerous applications.Here we consider a presumably harder question about exponential sums with the roots ϑ 1/n , for n = M + 1, . .…”

mentioning

confidence: 99%

Distribution of consecutive modular roots of an integer

Bourgain

Shparlinski

2008

Acta Arith.

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1. Introduction. For a prime p we denote by F p the finite field of p elements, which we assume to be represented by the set {0, 1, . . . , p − 1}. For an integer t we denote by Z t the residue ring modulo t and by Z * t the group of units of Z t .Let ϑ ∈ F * p be of multiplicative order t ≥ 1. There is a rather long history of studying the exponential sums with the powers ϑ n , n = M +1, . . . , M +N , where N ≤ t, in finite fields and residue rings: see [6,7,17,[21][22][23][26][27][28] for several results in this direction together with their numerous applications.Here we consider a presumably harder question about exponential sums with the roots ϑ 1/n , for n = M + 1, . . . , M + N with gcd(n, t) = 1 instead of the powers. More precisely, for an integer m > 0, we put

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One‐bit sigma‐delta quantization with exponential accuracy

Güntürk

2003

Comm Pure Appl Math

166

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One-bit quantization is a method of representing band-limited signals by ±1 sequences that are computed from regularly spaced samples of these signals; as the sampling density λ → ∞, convolving these one-bit sequences with appropriately chosen filters produces increasingly close approximations of the original signals. This method is widely used for analog-to-digital and digital-to-analog conversion, because it is less expensive and simpler to implement than the more familiar critical sampling followed by fine-resolution quantization. However, unlike fine-resolution quantization, the accuracy of one-bit quantization is not well understood. A natural error lower bound that decreases like 2 −λ can easily be given using information-theoretic arguments. Yet, no one-bit quantization algorithm was known with an error decay estimate even close to exponential decay. In this paper we construct an infinite family of one-bit sigma-delta quantization schemes that achieves this goal. In particular, using this family, we prove that the error signal for π -band-limited signals is at most O(2 −.07λ ).

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Character Sums with Exponential Functions and their Applications

Cited by 185 publications

References 0 publications

On the Security of Diffie-Hellman Bits

On the Security of Diffie-Hellman Bits

Distribution of consecutive modular roots of an integer

One‐bit sigma‐delta quantization with exponential accuracy

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