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Cryptography and Communications, 2019
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V A Zinoviev, Zinoviev V A
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V A Zinoviev, Zinoviev V A
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Doklady Mathematics, 2022
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Designs, Codes and Cryptography, 2011
Let \(p\) be a prime number, \(m\) a positive integer, \(\mathbb F_q\) the finite field of order \(q=p^m\) and \(\mathbb F^*= \mathbb F_q \setminus \{0 \}\). The Kloosterman sum on \(\mathbb F\) is the map \(K_q: \mathbb F_q \to \mathbb R \) defined by \[ K_q(a): = 1+ \sum_{x \in \mathbb F_{q}^{*}} e^{2\pi i \text{tr}(x^{-1}+ax)/p}, \] where \(\text{tr}
Petr Lisonek, Marko J. Moisio
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Let \(p\) be a prime number, \(m\) a positive integer, \(\mathbb F_q\) the finite field of order \(q=p^m\) and \(\mathbb F^*= \mathbb F_q \setminus \{0 \}\). The Kloosterman sum on \(\mathbb F\) is the map \(K_q: \mathbb F_q \to \mathbb R \) defined by \[ K_q(a): = 1+ \sum_{x \in \mathbb F_{q}^{*}} e^{2\pi i \text{tr}(x^{-1}+ax)/p}, \] where \(\text{tr}
Petr Lisonek, Marko J. Moisio
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Russian Academy of Sciences. Izvestiya Mathematics, 1993
In the paper under review the classical Kloosterman sum \[ K(d_ 1, d_ 2;Q) = \sum_{{x_ 1, x_ 2 \text{mod} Q \atop x_ 1x_ 2 \equiv 1 \pmod Q}} \exp \left( 2\pi i {d_ 1x_ 2 + d_ 2 x_ 2 \over Q} \right) \] is expressed in terms of numbers connected with the arithmetic of \(\mathbb{Z}/m \mathbb{Z}\) where ...
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In the paper under review the classical Kloosterman sum \[ K(d_ 1, d_ 2;Q) = \sum_{{x_ 1, x_ 2 \text{mod} Q \atop x_ 1x_ 2 \equiv 1 \pmod Q}} \exp \left( 2\pi i {d_ 1x_ 2 + d_ 2 x_ 2 \over Q} \right) \] is expressed in terms of numbers connected with the arithmetic of \(\mathbb{Z}/m \mathbb{Z}\) where ...
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Journal of Mathematical Sciences, 2005
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Izvestiya: Mathematics, 1995
Kloosterman sums are of the type \[ S(a,b;m)=\sum _{1\leq n\leq m, (m,n)=1} e((an^*+bn)/m), \] where \(nn^*\equiv 1\pmod{m}\). The author restricts the sum here to integers \(n=xy\) with \((xy,m)=1\) and \(x\), \(y\) lying in certain intervals. A complicated but perfectly explicit bound is given for such a modified sum. The bilinear shape of the sum is
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Kloosterman sums are of the type \[ S(a,b;m)=\sum _{1\leq n\leq m, (m,n)=1} e((an^*+bn)/m), \] where \(nn^*\equiv 1\pmod{m}\). The author restricts the sum here to integers \(n=xy\) with \((xy,m)=1\) and \(x\), \(y\) lying in certain intervals. A complicated but perfectly explicit bound is given for such a modified sum. The bilinear shape of the sum is
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Canadian Mathematical Bulletin, 1973
Let p be an odd prime, n an integer not divisible by p and α a positive integer. For any integer h with (h,pα)=l, is defined as any solution of the congruence (mod,pα). The Kloosterman sum Ap α(n) (see for example [4]) is defined by(1.1)where the dash (') indicates that the letter of summation runs only through a reduced residue system with respect to
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Let p be an odd prime, n an integer not divisible by p and α a positive integer. For any integer h with (h,pα)=l, is defined as any solution of the congruence (mod,pα). The Kloosterman sum Ap α(n) (see for example [4]) is defined by(1.1)where the dash (') indicates that the letter of summation runs only through a reduced residue system with respect to
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On the General Kloosterman Sums
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On the Values of Kloosterman Sums
IEEE Transactions on Information Theory, 2009Given a prime p and a positive integer n, we show that the shifted Kloosterman sums SigmaxisinF p nPsi(x + alphaxpn-2)=SigmaxisinF* p nPsi(x+alphax-1)+1, alphaisinF*pn where Psi is a nontrivial additive character of a finite field Fpn of pn elements, do not vanish if alpha belongs to a small subfield Fpm sube Fpn.
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