We present an algorithm to invert the Euler function $\varphi(m)$. The algorithm, for a given integer $n\ge 1$, in polynomial time ``on average'', finds theset $\Psi(n)$ of all solutions $m$ to the equation $\varphi(m) =n$. In fact, in the worst case the set $\Psi(n)$ is exponentially large and cannot be ... more >>>

We show that for several natural classes of ``structured'' matrices, including symmetric, circulant, Hankel and Toeplitz matrices, approximating the permanent modulo a prime $p$ is as hard as computing the exact value. Results of this kind are well known for the class of arbitrary matrices; however the techniques used do ... more >>>

Boneh and Venkatesan have recently proposed a polynomial time
algorithm for recovering a ``hidden'' element $\alpha$ of a
finite field $\F_p$ of $p$ elements from rather short
strings of the most significant bits of the remainder
mo\-du\-lo $p$ of $\alpha t$ for several values of $t$ selected
uniformly at ... more >>>

D. Boneh and R. Venkatesan have recently proposed an approach to proving
that a reasonably small portions of most significant bits of the
Diffie--Hellman key modulo a prime are as secure the the whole key. Some
further improvements and generalizations have been obtained by
I. M. Gonzales Vasco ... more >>>

Boneh and Venkatesan have recently proposed a polynomial time
algorithm for recovering a ``hidden'' element $\alpha$ of a
finite field $\F_p$ of $p$ elements from rather short
strings of the most significant bits of the remainder
mo\-du\-lo $p$ of $\alpha t$ for several values of $t$ selected uniformly
at random ... more >>>

We show that deciding square-freeness of a sparse univariate
polynomial over the integer and over the algebraic closure of a
finite field is NP-hard. We also discuss some related open
problems about sparse polynomials.

We show that a pseudo-random number generator,
introduced recently by M. Naor and O. Reingold,
possess one more attractive and useful property.
Namely, it is proved that for almost all values of parameters it
produces a uniformly distributed sequence.
The proof is based on some recent bounds of exponential
more >>>

Recent work by Bernasconi, Damm and Shparlinski
proved lower bounds on the circuit complexity of the square-free
numbers, and raised as an open question if similar (or stronger)
lower bounds could be proved for the set of prime numbers. In
this short note, we answer this question ... more >>>

In this paper we extend the area of applications of
the Abstract Harmonic Analysis to the field of Boolean function complexity.
In particular, we extend the class of functions to which
a spectral technique developed in a series of works of the first author
can be applied.
This extension ... more >>>

Lower bounds are obtained on the degree and the number of monomials of
Boolean functions, considered as a polynomial over $GF(2)$,
which decide if a given $r$-bit integer is square-free.
Similar lower bounds are also obtained for polynomials
over the reals which provide a threshold representation
more >>>