Elliptic curve analogues of a pseudorandom generator

Using the discrete logarithm in [7] and [9] a large family of pseudorandom binary sequences was constructed. Here we extend this construction. An interesting feature of this extension is that in certain special cases we get sequences involving points on elliptic curves.     

On divisibility properties of integers of the form ab + 1

We give estimates for the cardinality of the sets A , B ⊆ {1, 2,...,N} with the property that ab + 1 is squarefree for all a ∈ A, b ∈ B .     

On a family of pseudorandom binary sequences

Recently, numerous constructions have been given for finite pseudorandom binary sequences. However, in many applications, e.g., in cryptography one needs large families of good pseudorandom sequences. Very Recently L.~Goubin, C.~Mauduit, A.~Sárkzy succeeded in constructing large families of pseudorandom binary sequences based on the Legendre symbol. In this paper we will generate another type of large family of pseudorandom sequences by using the notion of index (discrete logarithm).     

On the shape of the alpha-potential in direct alpha-transfer

The variation in the shape of the potential generating the α form factor leaves the DWBA cross section of α-transfer essentially unaffected.     

Equations in finite fields with restricted solution sets. I (Character sums)

In earlier papers, for “large” (but otherwise unspecified) subsets A, B of Z _p and for h(x) ∈ Z _p [x], Gyarmati studied the solvability of the equations a + b = h(x), resp. ab = h(x) with a ∈ A, b ∈ B, x ∈ Z _p , and for large subsets A, B, C, D of Z _p Sárközy showed the solvability of the equations a + b = cd, resp. ab + 1 = cd with a ∈ A, b ∈ B, c ∈ C, d ∈ D. In this series of papers equations of this type will be studied in finite fields. In particular, in Part I of the series we will prove the necessary character sum estimates of independent interest some of which generalize earlier results.     

Equations in finite fields with restricted solution sets. II (Algebraic equations)

Generalizing earlier results, it is shown that if are “large” subsets of a finite field F _q , then the equations a + b = cd, resp. ab + 1 = cd can be solved with . Other algebraic equations with solutions restricted to “large” subsets of F _q are also studied. The proofs are based on character sum estimates proved in Part I of the paper. Research partially supported by the Hungarian National Foundation for Scientific Research, Grants No. T 043623, T 043631 and T 049693.     

On the linear complexity of binary lattices

Linear complexity is an important and frequently used measure of unpredictability and pseudorandomness of binary sequences. In this paper our goal is to extend this notion to two dimensions. We will define and study the linear complexity of binary lattices. The linear complexity of a truly random binary lattice will be estimated. Finally, we will analyze the connection between linear complexity and correlation measures, and we will utilize the inequalities obtained in this way for estimating the linear complexity of an important special binary lattice. Finally, we will study the connection between the linear complexity of binary lattices and of the associated binary sequences.     

A rigorous foundation of an easy-to-apply approximation method for bound state problems

Recently the application of a non-expensive scattering theory method to bound state problems has been proposed on the grounds of intuitive arguments. Subsequent realistic calculations have manifested advantageous features of the method especially in problems in which the asymptotic behaviour of the wave function is important. Here it is rigorously proven that the method is convergent, and mathematically correct procedures are described that speed up convergence and make the approximate energies approach the exact value from one side only. The general considerations are visualized by numerical results of a model problem.