Jerzy Urbanowicz

On elements of given order in K2F

We generalize Steinberg symbols in K2 of a ring, and we use them to investigate torsion elements in K2. The results are based on papers of Rosset-Tate and Suslin.

Remarks on the Classical Threshold Secret Sharing Schemes

We survey some results related to classical secret sharing schemes defined in Shamir [10] and Blakley [1], and developed in Brickell [2] and Lai and Ding [4]. Using elementary symmetric polynomials, we describe in a unified way which allocations of identities to participants define Shamirs threshold scheme, or its generalization by Lai and Ding, with a secret placed as a fixed coefficient of the scheme polynomial. This characterization enabled proving in Schinzel et al. [8], [9] and Spiez et al. [13] some new and non-trivial properties of such schemes. Also a characterization of matrices corresponding to the threshold secret sharing schemes of Blakley and Brickells type is given. Using Gaussian elimination we provide an algorithm to construct all such matrices which is efficient in the case of relatively small matrices. The algorithm may be useful in constructing systems where dynamics is important (one may generate new identities using it). It can also be used to construct all possible MDS codes. MSC: primary 94A62; secondary 11T71; 11C20

Remarks on the equation 1k + 2k + + (x − 1)k= xk

Summary For every t there is an explicitly given number k 0 such that the equation 1 k + 2 k + + (x − 1) k = x k has no integer solutions x ≥2 for all k 0 for which the denominator of the k th Bernoulli number B k has at most t distinct prime factors.

On constructing privileged coalitions in Shamirʼs type scheme

Abstract We consider Shamirʼs type secret sharing scheme with the secret placed as a coefficient a i of the scheme polynomial f ( x ) = ∑ i = 0 k − 1 a i x i over F q . A coalition of shareholders equipped with pairwise different public identities t j belonging to F q is called ( k , i ) -authorized if the holders using their secret shares y j = f ( t j ) ( 1 ⩽ j ⩽ n ) are able to reconstruct the secret by themselves. A ( k , i ) -authorized coalition of k − 1 or less shareholders is called a ( k , i ) -privileged coalition. We construct all ( k , i ) -privileged coalitions of k − 1 shareholders if i ≠ 0 , k − 1 . This construction allows us to deduce a new estimate for the number of such coalitions, which implies that they exist if q > 2 k − 1 . We also give a method of extending a given minimal ( k , i ) -privileged coalition of r shareholders to a coalition of n shareholders ( n ⩾ r ) each of whose ( k , i ) -privileged subcoalitions contains the given one provided q ⩾ n + r ( n − 2 k − 2 ) .

On linear congruence relations for Kubota–Leopoldt 2-adic L-functions

Abstract Our purpose in the paper is to find the most general linear congruence relation of the Hardy–Williams type for linear combinations of special values of Kubota–Leopoldt 2-adic L-functions L2(k,χω1−k) with k running over any finite subset of Z not necessarily consisting of consecutive integers (see Acta Arith. 47 (1986) 263; Publ. Math. Fac. Sci. Besancon, Theorie des Nombres, 1995/1996; Publ. Math. Debrecen 56 (2000) 677 and cf. Mathematics and Its Applications, Vol. 511, Kluwer Academic Publishers, Dordrecht, Boston, London, 2000). If k runs over finite subsets of Z consisting of consecutive integers see Compositio Math. 111 (1998) 289; Publ. Math. Debrecen 56 (2000) 677; Hardy and Williams, 1986; Compositio Math. 75 (1990) 271; Acta Arith. 71 (1995) 273; 52 (1989) 147; J. Number Theory 34 (1990) 362. In order to obtain the most general congruences of this type we make use of divisibility properties of the generalized Vandermonde determinants obtained in Spiez et al. (Divisibility properties of generalized Vandermonde and Cauchy determinants, Preprint 627, Institute of Mathematics, Polish Academy of Sciences, Warsaw, 2002). This allows us to simplify our main Theorem 2 and obtain Theorem 3 where the most general form of the linear congruence relation is given.

Divisibility properties of generalized Vandermonde determinants

Given n ≥ 2 let a denote an increasing n-tuple of non-negative integers ai and let x denote an n-tuple of indeterminates xi. Denote by Va(x) the generalized Vandermonde determinant, the polynomial obtained by computing the determinant of the matrix with (i, j) entry equal to x aj i . Let s be the standard n-tuple of consecutive integers from the interval [0, n−1] and given c ≥ 1 assume that x is an n-tuple of distinct 2-integral odd rational numbers xi such that xi ≡ xj ( mod 2 ). Several years ago one of the authors, investigating some properties of KubotaLeopoldt 2-adic L-functions, asked whether for any n-tuples a and x with c = 1 the identity ord2Va(x) = ord2Vs(x) + ord2Vs(a)− ord2Vs(s) (1.1)

Congruences Between the Orders of K 2 -Groups

This chapter will focus on the results which appear in [Urbanowicz, 1990a, b] and [Urbanowicz, 1990/1991a, b]. In section 7.2 of Chapter I we observed that Browkin and Schinzel’s theorem shows that

Applications of Zagier’s Formula (II)

{k_2}\left( d \right) \equiv \left\{ {\begin{array}{*{20}{c}} 0&{\left( {\bmod {2^{v + s}}} \right),}&{ifd \succ 0isodd,} \\ 0&{\left( {{{\bmod }^{2v + s - 1}}} \right),}&{ifd \succ 0iseven,ord \prec 0isodd,} \\ 0&{\left( {\bmod {2^{v + s - 2}}} \right),}&{ifd \prec 0iseven,} \end{array}} \right.