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**Authors:**
Shparlinski, Igor E
**Date:** 2013
**Subject Keyword:**
080200 Computation Theory and Mathematics
**Language:** eng
**Resource Type:** book chapter
**Identifier:** http://hdl.handle.net/1959.14/224276
**Description:**
We present a collection of several natural questions about elliptic curves, mostly over finite fields, that have led to some interesting number theoretic questions and whose solutions require rather i
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**Authors:**
Shparlinski, Igor E
**Date:** 2013
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.14/271249
**Description:**
We show that for any integers a and m with m ≥ 1 and gcd(a,m) = 1, there is a solution to the congruence pr ≡ a (modm) where p is prime, r is a product of at most k = 17 prime factors and p, r ≤ m. Th
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**Reviewed:**
**Authors:**
Shparlinski, Igor E
**Date:** 2013
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.14/276005
**Description:**
We obtain an asymptotic formula for the number of square-free integers in N consecutive values of polynomials on average over integral polynomials of degree at most k and of height at most H, where H
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**Reviewed:**
**Date:** 2013
**Subject Keyword:**
010100 Pure Mathematics
| Discrepancy
| Distribution of values
| Distribution of zeros
| Polynomials over finite fields
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.14/221895
**Description:**
11 page(s)
**Reviewed:**
**Date:** 2013
**Subject Keyword:**
010100 Pure Mathematics
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.14/221888
**Description:**
8 page(s)
**Reviewed:**
**Date:** 2013
**Subject Keyword:**
080200 Computation Theory and Mathematics
| Arithmetic functions
| Eisenstein criterion
| Irreducible polynomial
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.14/231936
**Description:**
We obtain a more precise version of an asymptotic formula of A. Dubickas for the number of monic Eisenstein polynomials of fixed degree d and of height at most H, as H → ∞. In particular, we give an e
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**Reviewed:**
**Authors:**
Baker, Roger C
| Banks, William D
| Brudern, Jörg
| Shparlinski, Igor E
| Weingartner, Andreas J
**Date:** 2013
**Subject Keyword:**
010100 Pure Mathematics
| Carmichael numbers
| Exponential sums with monomials
| Piatetski-Shapiro primes
| Piatetski-Shapiro sequences
| Smooth numbers
| Squarefree numbers
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.14/221582
**Description:**
32 page(s)
**Reviewed:**
**Date:** 2013
**Subject Keyword:**
080200 Computation Theory and Mathematics
| Cryptography
| Finite fields
| Pseudorandom numbers
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.14/221905
**Description:**
We study the security of the linear generator over a finite field. It is shown that the seed of a linear generator can be deduced from partial information of a short sequence of consecutive outputs of
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**Reviewed:**
**Date:** 2013
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.14/272100
**Description:**
We present some theoretical bounds and algorithms concerning the statistics of different reduction types in the family of Fermat curves Yp = Xs(1 - X ), where p is prime and s = 1,..., p - 2.
**Reviewed:**
**Authors:**
Shparlinski, Igor E
**Date:** 2012
**Subject Keyword:**
010100 Pure Mathematics
| character sums
| congruences
| exponential sums
| modular hyperbola
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.14/202794
**Description:**
We give a survey of a variety of recent results about the distribution and some geometric properties of points (x, y) on modular hyperbolas xy ≡ a (mod m). We also outline a very diverse range of appl
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**Reviewed:**
**Date:** 2012
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.14/173573
**Description:**
We estimate multiplicative character sums over the integers with a fixed sum of binary digits and apply these results to study the distribution of products of such integers in residues modulo a prime
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**Reviewed:**
**Authors:**
Shparlinski, Igor E
**Date:** 2012
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.14/172365
**Description:**
We give an upper bound on the number of vertices of the convex hull of the set of solutions to multivariate polynomial congruences modulo a prime p. The result is based on a combination of an estimate
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**Reviewed:**
**Authors:**
Shparlinski, Igor E
**Date:** 2012
**Subject Keyword:**
010100 Pure Mathematics
| Distribution of points
| Multivariate congruences
| Polynomials
| Well-shaped domains
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.14/198238
**Description:**
We use a result of É. Fouvry about the distribution of solutions to systems of congruences with multivariate polynomials in small cubic boxes and some ideas of W. Schmidt to derive an asymptotic formu
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**Reviewed:**
**Date:** 2012
**Subject Keyword:**
080200 Computation Theory and Mathematics
| Circulant graphs
| random graphs
| random walks
**Language:** eng
**Resource Type:** conference paper
**Identifier:** http://hdl.handle.net/1959.14/183072
**Description:**
14 page(s)
**Reviewed:**
**Date:** 2012
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.14/218019
**Description:**
We obtain several versions of the sum-product theorem with k-fold sums and product sets. We also give several applications of these estimates.
**Reviewed:**

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