arXiv subjects

Igor E. Shparlinski

Publications and source records attributed to Igor E. Shparlinski.

At least 217 records · Page 12Linked to original sources

arXiv2010

Pseudorandom Bits From Points on Elliptic Curves

Let $\E$ be an elliptic curve over a finite field $\F_{q}$ of $q$ elements, with $\gcd(q,6)=1$, given by an affine Weierstraß equation. We also use $x(P)$ to denote the $x$-component of a point $P = (x(P),y(P))\in \E$. We estimate character sums of the form $$ \sum_{n=1}^N χ$x(nP)x(nQ)$ \quad \text{and}\quad \sum_{n_1, \ldots, n_k=1}^N ψ$\sum_{j=1}^k c_j x\(\(\prod_{i =1}^j n_i$ R\)\) $$ on average over all $\F_q$ rational points $P$, $Q$ and $R$ on $\E$, where $χ$ is a quadratic character, $ψ$ is a nontrivial additive character in $\F_q$ and $(c_1, \ldots, c_k)\in \F_q^k$ is a non-zero vector. These bounds confirm several recent conjectures of D. Jao, D. Jetchev and R. Venkatesan, related to extracting random bits from various sequences of points on elliptic curves.

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arXiv2010

On Pseudopoints of Algebraic Curves

Following Kraitchik and Lehmer, we say that a positive integer $n\equiv1\pmod 8$ is an $x$-pseudosquare if it is a quadratic residue for each odd prime $p\le x$, yet is not a square. We extend this defintion to algebraic curves and say that $n$ is an $x$-pseudopoint of a curve $f(u,v) = 0$ (where $f \in \Z[U,V]$) if for all sufficiently large primes $p \le x$ the congruence $f(n,m)\equiv 0 \pmod p$ is satisfied for some $m$. We use the Bombieri bound of exponential sums along a curve to estimate the smallest $x$-pseudopoint, which shows the limitations of the modular approach to searching for points on curves.

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arXiv2010

On group structures realized by elliptic curves over a fixed finite field

We obtain explicit formulas for the number of non-isomorphic elliptic curves with a given group structure (considered as an abstract abelian group). Moreover, we give explicit formulas for the number of distinct group structures of all elliptic curves over a finite field. We use these formulas to derive some asymptotic estimates and tight upper and lower bounds for various counting functions related to classification of elliptic curves accordingly to their group structure. Finally, we present results of some numerical tests which exhibit several interesting phenomena in the distribution of group structures. We pose getting an explanation to these as an open problem.

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arXiv2010

On group structures realized by elliptic curves over arbitrary finite fields

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 study various subsets of this collection which correspond to curves over prime fields or to curves with a prescribed torsion. Some of our results are rigorous and are based on recent advances in analytic number theory, some are conditional under certain widely believed conjectures, and others are purely heuristic in nature.

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arXiv2010

On the Distribution of the Number of Points on Algebraic Curves in Extensions of Finite Fields

Let $\cC$ be a smooth absolutely irreducible curve of genus $g \ge 1$ defined over $\F_q$, the finite field of $q$ elements. Let $# \cC(\F_{q^n})$ be the number of $\F_{q^n}$-rational points on $\cC$. Under a certain multiplicative independence condition on the roots of the zeta-function of $\cC$, we derive an asymptotic formula for the number of $n =1, ..., N$ such that $(# \cC(\F_{q^n}) - q^n -1)/2gq^{n/2}$ belongs to a given interval $\cI \subseteq [-1,1]$. This can be considered as an analogue of the Sato-Tate distribution which covers the case when the curve $\E$ is defined over $\Q$ and considered modulo consecutive primes $p$, although in our scenario the distribution function is different. The above multiplicative independence condition has, recently, been considered by E. Kowalski in statistical settings. It is trivially satisfied for ordinary elliptic curves and we also establish it for a natural family of curves of genus $g=2$.

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arXiv2010

On the Number of Solutions of Exponential Congruences

For a prime $p$ and an integer $a \in \Z$ we obtain nontrivial upper bounds on the number of solutions to the congruence $x^x \equiv a \pmod p$, $1 \le x \le p-1$. We use these estimates to estimate the number of solutions to the congruence $x^x \equiv y^y \pmod p$, $1 \le x,y \le p-1$, which is of cryptographic relevance.

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arXiv2010

Pseudorandomness and Dynamics of Fermat Quotients

We obtain some theoretic and experimental results concerning various properties (the number of fixed points, image distribution, cycle lengths) of the dynamical system naturally associated with Fermat quotients acting on the set $\{0, ..., p-1\}$. We also consider pseudorandom properties of Fermat quotients such as joint distribution and linear complexity.

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arXiv2010

Pseudorandom Numbers and Hash Functions from Iterations of Multivariate Polynomials

Dynamical systems generated by iterations of multivariate polynomials with slow degree growth have proved to admit good estimates of exponential sums along their orbits which in turn lead to rather stronger bounds on the discrepancy for pseudorandom vectors generated by these iterations. Here we add new arguments to our original approach and also extend some of our recent constructions and results to more general orbits of polynomial iterations which may involve distinct polynomials as well. Using this construction we design a new class of hash functions from iterations of polynomials and use our estimates to motivate their "mixing" properties.

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arXiv2009

On the Length of Critical Orbits of Stable Quadratic Polynomials

We use the Weil bound of multiplicative character sums together with some recent results of N. Boston and R. Jones, to show that the critical orbit of quadratic polynomials over a finite field of $q$ elements is of length $O(q^{3/4})$, improving upon the trivial bound $q$.

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arXiv2009

Short Cycles in Repeated Exponentiation Modulo a Prime

Given a prime $p$, we consider the dynamical system generated by repeated exponentiations modulo $p$, that is, by the map $u \mapsto f_g(u)$, where $f_g(u) \equiv g^u \pmod p$ and $0 \le f_g(u) \le p-1$. This map is in particular used in a number of constructions of cryptographically secure pseudorandom generators. We obtain nontrivial upper bounds on the number of fixed points and short cycles in the above dynamical system.

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arXiv2009

Rank statistics for a family of elliptic curves over a function field

We show that the average and typical ranks in a certain parametric family of elliptic curves described by D. Ulmer tend to infinity as the parameter $d \to\infty$. This is perhaps unexpected since by a result of A. Brumer, the average rank for all elliptic curves over a function field of positive characteristic is asymptotically bounded above by 2.3.

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arXiv2009

On Point Sets in Vector Spaces over Finite Fields That Determine Only Acute Angle Triangles

For three points $\vec{u}$,$\vec{v}$ and $\vec{w}$ in the $n$-dimensional space $\F_q^n$ over the finite field $\F_q$ of $q$ elements we give a natural interpretation of an acute angle triangle defined by this points. We obtain an upper bound on the size of a set $\cZ$ such that all triples of distinct points $\vec{u}, \vec{v}, \vec{w} \in \cZ$ define acute angle triangles. A similar question in the real space $\cR^n$ dates back to P. Erd{\H o}s and has been studied by several authors.

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arXiv2009

Divisibility, Smoothness and Cryptographic Applications

This paper deals with products of moderate-size primes, familiarly known as smooth numbers. Smooth numbers play a crucial role in information theory, signal processing and cryptography. We present various properties of smooth numbers relating to their enumeration, distribution and occurrence in various integer sequences. We then turn our attention to cryptographic applications in which smooth numbers play a pivotal role.

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arXiv2008

On the Distribution of the Euler Function of Shifted Smooth Numbers

We give asymptotic formulas for some average values of the Euler function on shifted smooth numbers. The result is based on various estimates on the distribution of smooth numbers in arithmetic progressions which are due to A. Granville and É. Fouvry & G. Tenenbaum.

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arXiv2008

Classical and Quantum Algorithms for Exponential Congruences

We discuss classical and quantum algorithms for solvability testing and finding integer solutions x,y of equations of the form af^x + bg^y = c over finite fields GF(q). A quantum algorithm with time complexity q^(3/8) (log q)^O(1) is presented. While still superpolynomial in log q, this quantum algorithm is significantly faster than the best known classical algorithm, which has time complexity q^(9/8) (log q)^O(1). Thus it gives an example of a natural problem where quantum algorithms provide about a cubic speed-up over classical ones.

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arXiv2007

On Pseudosquares and Pseudopowers

Introduced by Kraitchik and Lehmer, an $x$-pseudosquare is a positive integer $n\equiv1\pmod 8$ that is a quadratic residue for each odd prime $p\le x$, yet is not a square. We use bounds of character sums to prove that pseudosquares are equidistributed in fairly short intervals. An $x$-pseudopower to base $g$ is a positive integer which is not a power of $g$ yet is so modulo $p$ for all primes $p\le x$. It is conjectured by Bach, Lukes, Shallit, and Williams that the least such number is at most $\exp(a_g x/\log x)$ for a suitable constant $a_g$. A bound of $\exp(a_g x\log\log x/\log x)$ is proved conditionally on the Riemann Hypothesis for Dedekind zeta functions, thus improving on a recent conditional exponential bound of Konyagin and the present authors. We also give a GRH-conditional equidistribution result for pseudopowers that is analogous to our unconditional result for pseudosquares.

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arXiv2007

On the Distribution of Pseudopowers

An $x$-pseudopower to base $g$ is a positive integer which is not a power of $g$ yet is so modulo $p$ for all primes $p\le x$. We improve an upper bound for the least such number due to E. Bach, R. Lukes, J. Shallit, and H. C. Williams. The method is based on a combination of some bounds of exponential sums with new results about the average behaviour of the multiplicative order of $g$ modulo prime numbers.

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arXiv2007

Tate-Shafarevich Groups and Frobenius Fields of Reductions of Elliptic Curves

Let $\E/\Q$ be a fixed elliptic curve over $\Q$ which does not have complex multiplication. Assuming the Generalized Riemann Hypothesis, A. C. Cojocaru and W. Duke have obtained an asymptotic formula for the number of primes $p\le x$ such that the reduction of $\E$ modulo p has a trivial Tate-Shafarevich group. Recent results of A. C. Cojocaru and C. David lead to a better error term. We introduce a new argument in the scheme of the proof which gives further improvement.

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