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**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:**
**Date:** 2012
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.14/195239
**Description:**
We obtain explicit formulas for the number of nonisomorphic elliptic curves with a given group structure (considered as an abstract abelian group) and the number of distinct group structures of all el
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**Reviewed:**
**Date:** 2012
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.14/195249
**Description:**
We study the collection of group structures that can be realized as a group of rational points on an elliptic curve over a finite field (such groups are well known to be of rank at most two). We also
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**Reviewed:**
**Date:** 2012
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.14/195607
**Description:**
We give upper and lower bounds on the count of positive integers n <= x dividing the nth term of a non-degenerate linearly recurrent sequence with simple roots.
**Full Text:**
**Reviewed:**
**Date:** 2012
**Subject Keyword:**
010100 Pure Mathematics
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.14/176439
**Description:**
We recall that a polynomial f(X)= K[X] over a field K is called stable if all its iterates are irreducible over K. We show that almost all monic quadratic polynomials f(X) = ℤ[X] are stable over ℚ. We
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**Reviewed:**
**Date:** 2012
**Subject Keyword:**
010100 Pure Mathematics
| Additive combinatorics
| Orbits
| Polynomial congruences
| Vinogradov mean value theorem
| Visible points
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.14/205033
**Description:**
For a polynomial f∈Fp[X] , we obtain upper bounds on the number of points (x, f (x)) modulo a prime p which belong to an arbitrary square with the side length H. Our results in particular are based on
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**Reviewed:**
**Date:** 2012
**Subject Keyword:**
010100 Pure Mathematics
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.14/172451
**Description:**
10 page(s)
**Full Text:**
**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:**
010100 Pure Mathematics
| 080200 Computation Theory and Mathematics
| Black-box interpolation
| Character sums
| Hidden shifted power problem
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.14/202843
**Description:**
We consider the problem of recovering a hidden element s of a finite field Fq of q elements from queries to an oracle that for a given x ∈ Fq returns (x+s)e for a given divisor e | q - 1. We use some
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**Full Text:**
**Reviewed:**
**Date:** 2012
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.14/173394
**Description:**
We give a rigorous deterministic polynomial time algorithm for the modular inversion hidden number problem introduced by D.Boneh, S. Halevi and N.A. Howgrave-Graham in 2001. For our algorithm, we need
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**Reviewed:**
**Date:** 2012
**Subject Keyword:**
080200 Computation Theory and Mathematics
| Elliptic curves
| Embedding degree
| MNT curves
| Pairing based cryptography
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.14/172505
**Description:**
We give an upper bound on the number of finite fields over which elliptic curves of cryptographic interest with a given embedding degree and small complex multiplication discriminant may exist, and pr
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**Full Text:**
**Reviewed:**
**Date:** 2012
**Subject Keyword:**
080200 Computation Theory and Mathematics
| Binomial exponential sums
| Discrepancy
| Power generator
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.14/177143
**Description:**
We obtain a new estimate on the discrepancy of the power generator over a part of the period that improves several previous results. We also introduce a multidimensional analogue and show that the cor
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**Reviewed:**
**Authors:**
Shparlinski, Igor E
**Date:** 2012
**Subject Keyword:**
010100 Pure Mathematics
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.14/175289
**Description:**
8 page(s)
**Reviewed:**

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