Attila Bérczes

A one-way function based on norm form equations

In this paper we present a new one-way function with collision resistance. The security of this function is based on the difficulty of solving a norm form equation. We prove that this function is collision resistant, so it can be used as a one-way hash function. We show that this construction probably provides a family of one-way functions.

On generalized Lebesgue-Ramanujan-Nagell equations

Abstract We give a brief survey on some classical and recent results concerning the generalized Lebesgue-Ramanujan-Nagell equation. Moreover, we solve completely the equation x2 + 11a 17b = yn in nonnegative integer unknowns with n ≧ 3 and gcd(x, y) = 1.

Parameterized Norm Form Equations with Arithmetic Progressions

Let be a zero of the Thomas polynomial X 3 (a 1)X 2 (a+2)X 1. We find all algebraic numbers µ = x0+x1 +x2 2 2 Z[ ], such that x0,x1,x2 2 Z forms an arithmetic progression and the norm of µ is less than |2a + 1|. In order to find all progressions we reduce our problem to solve a family of Thue equations and solve this family completely.

On the number of equivalence classes of binary forms of given degree and given discriminant

We give an explicit upper bound for the number of equivalence classes of binary forms with rational integral coefficients of given degree and given discriminant, and with given splitting field. Further, we give an explicit upper bound for the number of irreducible binary forms with rational integral coefficients with given invariant order. Our bounds depend on as few parameters as possible. For instance, we show that the number of equivalence classes of irreducible binary forms with rational integral coefficients of degree r with given invariant order has an upper bound depending only on r. We have proved more general results for binary forms with coefficients in the ring of S-integers of a number field.

On the Diophantine equation (x + 1)k + (x + 2)k + ... + (2x)k = yn

Abstract In this work, we give upper bounds for n on the title equation. Our results depend on assertions describing the precise exponents of 2 and 3 appearing in the prime factorization of T k ( x ) = ( x + 1 ) k + ( x + 2 ) k + . . . + ( 2 x ) k . Further, on combining Bakers method with the explicit solution of polynomial exponential congruences (see e.g. [6] ), we show that for 2 ≤ x ≤ 13 , k ≥ 1 , y ≥ 2 and n ≥ 3 the title equation has no solutions.

Effective results for hyper- and superelliptic equations over number fields

We consider hyper- and superelliptic equations

ON A FAMILY OF PREIMAGE-RESISTANT FUNCTIONS

f(x)=by^m

An application of index forms in cryptography

with unknowns x,y from the ring of S-integers of a given number field K. Here, f is a polynomial with S-integral coefficients of degree n with non-zero discriminant and b is a non-zero S-integer. Assuming that n>2 if m=2 or n>1 if m>2, we give completely explicit upper bounds for the heights of the solutions x,y in terms of the heights of f and b, the discriminant of K, and the norms of the prime ideals in S. Further, we give a completely explicit bound C such that

Effective results for points on certain subvarieties of tori

f(x)=by^m

On the number of solutions of norm form equations

has no solutions in S-integers x,y if m>C, except if y is 0 or a root of unity. We will apply these results in another paper where we consider hyper- and superelliptic equations with unknowns taken from an arbitrary finitely generated integral domain of characteristic 0.