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Two-sources Randomness Extractors for Elliptic Curves
This paper studies the task of two-sources randomness extractors for elliptic
curves defined over finite fields , where can be a prime or a binary
field. In fact, we introduce new constructions of functions over elliptic
curves which take in input two random points from two differents subgroups. In
other words, for a ginven elliptic curve defined over a finite field
and two random points and , where and are two subgroups of
, our function extracts the least significant bits of the
abscissa of the point when is a large prime, and the -first
coefficients of the asbcissa of the point when , where is a prime greater than . We show that the extracted bits
are close to uniform.
Our construction extends some interesting randomness extractors for elliptic
curves, namely those defined in \cite{op} and \cite{ciss1,ciss2}, when
. The proposed constructions can be used in any
cryptographic schemes which require extraction of random bits from two sources
over elliptic curves, namely in key exchange protole, design of strong
pseudo-random number generators, etc
Character sums with division polynomials
We obtain nontrivial estimates of quadratic character sums of division
polynomials , , evaluated at a given point on an
elliptic curve over a finite field of elements. Our bounds are nontrivial
if the order of is at least for some fixed . This work is motivated by an open question about statistical
indistinguishability of some cryptographically relevant sequences which has
recently been brought up by K. Lauter and the second author
Kloosterman sums, elliptic curves, and irreducible polynomials with prescribed trace and norm
Let \F_q () be a finite field. In this paper the number of
irreducible polynomials of degree in \F_q[x] with prescribed trace and
norm coefficients is calculated in certain special cases and a general bound
for that number is obtained improving the bound by Wan if is small compared
to . As a corollary, sharp bounds are obtained for the number of elements in
\F_{q^3} with prescribed trace and norm over \F_q improving the estimates
by Katz in this special case. Moreover, a characterization of Kloosterman sums
over \F_{2^r} divisible by three is given generalizing the earlier result by
Charpin, Helleseth, and Zinoviev obtained only in the case odd. Finally, a
new simple proof for the value distribution of a Kloosterman sum over the field
\F_{3^r}, first proved by Katz and Livne, is given.Comment: 21 pages; revised version with somewhat more clearer proofs; to
appear in Acta Arithmetic
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