Igor E. Shparlinski

Character sums with exponential functions and their applications

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The Insecurity of the Elliptic Curve Digital Signature Algorithm with Partially Known Nonces

Nguyen and Shparlinski have recently presented a polynomial-time algorithm that provably recovers the signers secret DSA key when a few consecutive bits of the random nonces k (used at each signature generation) are known for a number of DSA signatures at most linear in log q (q denoting as usual the small prime of DSA), under a reasonable assumption on the hash function used in DSA. The number of required bits is about log1/2q, but can be decreased to log log q with a running time qO(1/log log q) subexponential in log q, and even further to two in polynomial time if one assumes access to ideal lattice basis reduction, namely an oracle for the lattice closest vector problem for the infinity norm. All previously known results were only heuristic, including those of Howgrave-Graham and Smart who introduced the topic. Here, we obtain similar results for the elliptic curve variant of DSA (ECDSA).

PARAMETERS OF INTEGRAL CIRCULANT GRAPHS AND PERIODIC QUANTUM DYNAMICS

The means for simultaneous scouring of metal surfaces contains a waste product in manufacture of fodder yeast, citric acid, ammonium citrate, aqueous solution of sodium gluconate, sulphonated ricinic oil and an inorganic acid, f.e. sulphuric acid respectively in the following weight ratios: 60 to 95%; 2 to 6%; 0.1 to 10%; 0.0 to 4.0%; 0.0 to 20% and 0.0 to 15%. The waste product from fodder yeast manufacture contains itself 0.06 to 0.1% reducing agents; 0.01 to 0.4% phosphates; 0.2 to 0.4% ammonium sulphate; 0.0 to 0.06% furfurol and 0.0 to 0.1% yeast. The means for simultaneous scouring of metal surface from corrosion products, scale and scoria is used in metallurgy, machine construction, agriculture, energetics and all fields where there are conditions for metal corrosion.

On Polynomial Approximation of the Discrete Logarithm and the Diffie--Hellman Mapping

Abstract. We obtain several lower bounds, exponential in terms of lg p , on the degrees of polynomials and algebraic functions coinciding with values of the discrete logarithm modulo a prime p at sufficiently many points; the number of points can be as little as p1/2 + ɛ . We also obtain improved lower bounds on the degree and sensitivity of Boolean functions on bits of x deciding whether x is a quadratic residue. Similar bounds are also proved for the Diffie—Hellman mapping gx→ gx2 , where g is a primitive root of a finite field of q elements Fq . These results can be used to obtain lower bounds on the parallel arithmetic and Boolean complexity of computing the discrete logarithm and breaking the Diffie—Hellman cryptosystem. The method is based on bounds of character sums and numbers of solutions of some polynomial equations.

On Exponential Sums and Group Generators for Elliptic Curves over Finite Fields

In the paper an upper bound is established for certain exponential sums, analogous to Gaussian sums, defined on the points of an elliptic curve over a prime finite field. The bound is applied to prove the existence of group generators for the set of points on an elliptic curve over \(\mathbb{F}_{q}\) among certain sets of bounded size. We apply this estimate to obtain a deterministic O(q 1/2 + e) algorithm for finding generators of the group in echelon form, and in particular to determine its group structure.

Recent Advances in the Theory of Nonlinear Pseudorandom Number Generators

We present a survey of recent developments in the theory of nonlinear generators for uniform pseudorandom numbers. The emphasis is on discrepancybased tests for inversive generators where most of the progress has taken place.

Distribution of Modular Sums and the Security of the Server Aided Exponentiation

We obtain some uniformity of distribution results for the values of modular sums of the form

On the distribution of inversive congruential pseudorandom numbers in parts of the period

\sum\limits_{j = 1}^n {a_j x_j } \left( {\bmod M} \right)\left( {x_1 , \ldots ,x_n } \right) \in \beta