Algorithm Of The Electronic Digital Subscript On The Basis Of The Composition Of Computing Complexities

In article the new algorithm of a digital signature in composition of the existing difficulties is developed: discrete logarithming in a final field, decomposition of rather large natural number on simple multipliers, additions of points with rational coordinates of an elliptic curve. On the basis of a combination of difficulties of discrete logarithming on a final field with the characteristic of large number, decomposition of rather large odd number on simple multipliers and additions of points of an elliptic curve develops algorithm of a digital signature for formation. The conventional scheme (model) of a digital signature covers three processes: generation of keys of EDS; formation of EDS; check (authenticity confirmation) of EDS. The idea of a design of the offered algorithm allows modifying and increasing crypto stability with addition to other computing difficulties. It is intended for use in systems of information processing of different function during forming and confirmation of authenticity of digital signature.

The asymptotic formula in Waring’s problem: Higher order expansions

When {k>1} and s is sufficiently large in terms of k, we derive an explicit multi-term asymptotic expansion for the number of representations of a large natural number as the sum of s positive integral k-th powers.

The use in additive number theory of numbers without large prime factors

In the past few years considerable progress has been made with regard to the known upper bounds for G ( k ) in Waring’s problem, that is, the smallest s such that every sufficiently large natural number is the sum of at most 8 k th powers of natural numbers. This has come about through the development of techniques using properties of numbers having only relatively small prime factors. In this article an account of these developments is given, and they are illustrated initially in a historical perspective through the special case of cubes. In particular the connection with the classical work of Davenport on smaller values of k is demonstrated. It is apparent that the fundamental ideas and the underlying mean value theorems and estimates for exponential sums have numerous applications and a brief account is given of some of them.

On a theorem of Birch concerning sums of distinct integers taken from certain sequences

In [1] B. J. Birch, solving in the affirmative a conjecture of Erdὅs, proved the following result:Theorem 1. Let p and q be coprime integers greater than 1. Then every large natural number may be written as a sum of distinct terms of type paqb.In fact Birch pointed out that, with similar arguments, one could obtain a stronger version where the exponent b of q can be bounded in terms of p and q. The proofs were entirely elementary.

Note on the representation of integers as sums of certain powers

In the paper entitled ‘Asymptotic formula in a generalized Waring's problem’, I established an asymptotic formula for the number of representations of a large natural number N in the formwhere x1, x2, …, x7 and k are natural numbers with k ≥ 4 (see (2) Theorem 2).