Igor E. Shparlinski

Andrew M. Childs, Wim van Dam, Shih-Han Hung et al.

We consider the number of quantum queries required to determine the coefficients of a degree-d polynomial over GF(q). A lower bound shown independently by Kane and Kutin and by Meyer and Pommersheim shows that d/2+1/2 quantum queries are needed to solve this problem with bounded error, whereas an algorithm of Boneh and Zhandry shows that d quantum queries are sufficient. We show that the lower bound is achievable: d/2+1/2 quantum queries suffice to determine the polynomial with bounded error. Furthermore, we show that d/2+1 queries suffice to achieve probability approaching 1 for large q. These upper bounds improve results of Boneh and Zhandry on the insecurity of cryptographic protocols against quantum attacks. We also show that our algorithm's success probability as a function of the number of queries is precisely optimal. Furthermore, the algorithm can be implemented with gate complexity poly(log q) with negligible decrease in the success probability. We end with a conjecture about the quantum query complexity of multivariate polynomial interpolation.

Oscar Garcia-Morchon, Ronald Rietman, Igor E. Shparlinski et al.

Motivated by a recently introduced HIMMO key distribution scheme, we consider a modification of the noisy polynomial interpolation problem of recovering an unknown polynomial $f(X) \in Z[X]$ from approximate values of the residues of $f(t)$ modulo a prime $p$ at polynomially many points $t$ taken from a short interval.

Jonas Kaszian, Pieter Moree, Igor E. Shparlinski

We investigate the periodic structure of the exponential pseudorandom number generator obtained from the map $x\mapsto g^x\pmod p$ that acts on the set $\{1, \ldots, p-1\}$.

Igor Shparlinski

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 crucial parameter of this algorithm. We show that there are infinitely many pairs $(p, E)$ of primes $p$ and curves $E$ over $\F_p$ with $L_p(E) \ge c \log p \log \log \log p$ for some absolute constant $c>0$, while a naive heuristic estimate suggests that $L_p(E) \sim \log p$. This complements recent results of Galbraith and Satoh (2002), conditional under the Generalised Riemann Hypothesis, and of Shparlinski and Sutherland (2012), unconditional for almost all pairs $(p,E)$.