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Mathematics of Cryptography

Title
Mathematics of Cryptography
Funding
ARC | Discovery Projects
Contract (GA) number
DP1092835
Start Date
2010/01/01
End Date
2012/12/31
Open Access mandate
no
Organizations
-
More information
http://purl.org/au-research/grants/arc/DP1092835

 

  • Fermat quotients: Exponential sums, value set and primitive roots

    Shparlinski, Igor E. (2011)
    Projects: ARC | Mathematics of Cryptography (DP1092835)
    For a prime $p$ and an integer $u$ with $\gcd(u,p)=1$, we define Fermat quotients by the conditions $$ q_p(u) \equiv \frac{u^{p-1} -1}{p} \pmod p, \qquad 0 \le q_p(u) \le p-1. $$ D. R. Heath-Brown has given a bound of exponential sums with $N$ consecutive Fermat quotients that is nontrivial for $N\ge p^{1/2+\epsilon}$ for any fixed $\epsilon>0$. We use a recent idea of M. Z. Garaev together with a form of the large sieve inequality due to S. Baier and L. Zhao, to show that on average over $p$...

    On the Product of Small Elkies Primes

    Shparlinski, Igor (2012)
    Projects: ARC | Mathematics of Cryptography (DP1092835)
    Given an elliptic curve $E$ over a finite field $\F_q$ of $q$ elements, we say that an odd prime $\ell \nmid q$ is an Elkies prime for $E$ if $t_E^2 - 4q$ is a quadratic residue modulo $\ell$, where $t_E = q+1 - #E(\F_q)$ and $#E(\F_q)$ is the number of $\F_q$-rational points on $E$. These primes are used in the presently most efficient algorithm to compute $#E(\F_q)$. In particular, the bound $L_q(E)$ such that the product of all Elkies primes for $E$ up to $L_q(E)$ exceeds $4q^{1/2}$ is a c...

    On small solutions to quadratic congruences

    Shparlinski, Igor E. (2011)
    Projects: ARC | Mathematics of Cryptography (DP1092835)
    We estimate the deviation of the number of solutions of the congruence $$ m^2-n^2 \equiv c \pmod q, \qquad 1 \le m \le M, \ 1\le n \le N, $$ from its expected value on average over $c=1, ..., q$. This estimate is motivated by the recently established by D. R. Heath-Brown connection between the distibution of solution to this congruence and the pair correlation problem for the fractional parts of the quadratic function $\alpha k^2$, $k=1,2,...$ with a real $\alpha$.

    On digit patterns in expansions of rational numbers with prime denominator

    Shparlinski , Igor ,; Steiner , Wolfgang (2013)
    Projects: ARC | Mathematics of Cryptography (DP1092835)
    International audience; We show that, for any fixed $\varepsilon > 0$ and almost all primes $p$, the $g$-ary expansion of any fraction $m/p$ with $\gcd(m,p) = 1$ contains almost all $g$-ary strings of length $k < (5/24 - \varepsilon) \log_g p$. This complements a result of J. Bourgain, S. V. Konyagin, and I. E. Shparlinski that asserts that, for almost all primes, all $g$-ary strings of length $k < (41/504 -\varepsilon) \log_g p$ occur in the $g$-ary expansion of $m/p$.

    On the Counting Function of Elliptic Carmichael Numbers

    Luca, Florian; Shparlinski, Igor E. (2012)
    Projects: ARC | Mathematics of Cryptography (DP1092835)
    We give an upper bound for the number elliptic Carmichael numbers $n \le x$ that have recently been introduced by J. H. Silverman. We also discuss several possible ways for further improvements.

    On the Fixed Points of the Map x -> x^x Modulo a Prime

    Kurlberg, Pär; Luca, Florian; Shparlinski, Igor (2014)
    Projects: ARC | Mathematics of Cryptography (DP1092835)
    In this paper, we show that for almost all primes p there is an integer solution x in [2,p-1] to the congruence x^x == x mod p. The solutions can be interpretated as fixed points of the map x -> x^x mod p, and we study numerically and discuss some unexpected properties of the dynamical system associated with this map.

    On vanishing Fermat quotients and a bound of the Ihara sum

    Shparlinski, Igor E. (2013)
    Projects: ARC | Mathematics of Cryptography (DP1092835)
    We improve an estimate of A.Granville (1987) on the number of vanishing Fermat quotients $q_p(\ell)$ modulo a prime $p$ when $\ell$ runs through primes $\ell \le N$. We use this bound to obtain an unconditional improvement of the conditional (under the Generalised Riemann Hypothesis) estimate of Y. Ihara (2006) on a certain sum, related to vanishing Fermat quotients. In turn this sum appears in the study of the index of certain subfields of of cyclotomic fields $\Q(\exp(2 \pi i/p^2))$.

    On the Convex Hull of the Points on Modular Hyperbolas

    Konyagin, Sergei V.; Shparlinski, Igor E. (2010)
    Projects: ARC | Mathematics of Cryptography (DP1092835)
    Given integers $a$ and $m\ge 2$, let $\Hm$ be the following set of integral points $$ \Hm= \{(x,y) \ : \ xy \equiv a \pmod m,\ 1\le x,y \le m-1\} $$ We improve several previously known upper bounds on $v_a(m)$, the number of vertices of the convex closure of $\Hm$, and show that uniformly over all $a$ with $\gcd(a,m)=1$ we have $v_a(m) \le m^{1/2 + o(1)}$ and furthermore, we have $v_a(m) \le m^{5/12 + o(1)}$ for $m$ which are almost squarefree.

    On Solutions to Some Polynomial Congruences in Small Boxes

    Shparlinski, Igor E. (2013)
    Projects: ARC | Mathematics of Cryptography (DP1092835)
    We use bounds of mixed character sum to study the distribution of solutions to certain polynomial systems of congruences modulo a prime $p$. In particular, we obtain nontrivial results about the number of solution in boxes with the side length below $p^{1/2}$, which seems to be the limit of more general methods based on the bounds of exponential sums along varieties.

    Distribution on elements of cosets of small subgroups and applications

    Bourgain, Jean; Konyagin, Sergei; Shparlinski, Igor (2011)
    Projects: ARC | Mathematics of Cryptography (DP1092835)
    We obtain a series of estimates on the number of small integers and small order Farey fractions which belong to a given coset of a subgroup of order $t$ of the group of units of the residue ring modulo a prime $p$, in the case when $t$ is small compared to $p$. We give two applications of these results: to the simultaneous distribution of two high degree monomials $x^{k_1}$ and $x^{k_2}$ modulo $p$ and to a question of J.Holden and P.Moree on fixed points of the discrete logarithm.
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