Latin squares and Hurwitz theorem

Designs, Codes and Cryptography, 2000

It turns out that Latin squares which are hard to approximate by a polynomial are suitable to be used as a part of block cipher algorithms (BCA). In this paper we state basic properties of those Latin squares and provide their construction.

International Journal of Number Theory, 2005

We give a short proof of Milne's formulas for sums of 4n2 and 4n2 + 4n integer squares using the theory of modular forms. Other identities of Milne are also discussed.

Journal of the Australian Mathematical Society, 1982

Jacobi's four-square theorem, which gives the number of representations of a positive integer as a sum of four squares, is shown to follow simply from the triple-product identity.

Proceedings of the American Mathematical Society, 2012

We investigate here the representability of integers as sums of triangular numbers, where the n-th triangular number T n = n(n + 1)/2. In particular, we show that f (x 1 , x 2 ,. .. , x k) = b 1 T x1 + • • • + b k T x k , for fixed positive integers b 1 , b 2 ,. .. , b k , represents every nonnegative integer if and only if it represents 1, 2, 4, 5, and 8. Moreover, if 'cross-terms' are allowed in f , we show that no finite set of positive integers can play an analogous role, in turn showing that there is no overarching finiteness theorem which generalizes from positive definite quadratic forms to totally positive quadratic polynomials.

Discrete Mathematics, 2004

Periodica Mathematica Hungarica, 2012

Let n ≥ 3 be a positive integer. We show that the number of equivalence classes of generalized Latin squares of order n with n 2 − 1 distinct elements is 4 if n = 3 and 5 if n ≥ 4. It is also shown that all these squares are embeddable in groups. As an application, we obtain a lower bound for the number of isomorphism classes of certain Eulerian graphs with n 2 + 2n − 1 vertices.

In this paper, we present eighteen interesting infinite products and their Lambert series expansions. From these, we deduce formulae for the number of representations of an integer n by eighteen quadratic forms in terms of divisor sums. -Dedicated to the memory of my grandmother Yuet Kwai Mah.

2004

Let λ = (λ1, • • • , λm) be a partition of k. Let r λ (n) denote the number of solutions in integers of λ1x 2 1 + • • • + λmx 2 m = n, and let t λ (n) denote the number of solutions in non negative integers of λ1x1(x1 + 1)/2 + • • • + λmxm(xm + 1)/2 = n. We prove that if 1 ≤ k ≤ 7, then there is a constant c λ , depending only on λ, such that r λ (8n + k) = c λ t λ (n), for all integers n.

Calcolo, 2012

In this paper, we conjecture a formula for the value of the Pythagoras number for real multivariate sum of squares polynomials as a function of the (total or coordinate) degree and the number of variables. The conjecture is based on the comparison between the number of parameters and the number of conditions for a corresponding low-rank representation. This is then numerically verified for a number of examples. Additionally, we discuss the Pythagoras number of (complex) multivariate Laurent polynomials that are sum of square magnitudes of polynomials on the n-torus. For both types of polynomials, we also propose an algorithm to numerically compute the Pythagoras number and give some numerical illustrations.

We investigate the permutation behaviour of a class of quadratic polynomials over a finite field of characteristic 2 of the form P (X) = L 1 (X)(L 2 (X) + L 1 (X)L 3 (X)) where L 1 , L 2 , L 3 are linearised polynomials. To our knowledge, the given class has not been studied previously in the literature. We identify several new types of permutation polynomials of this class. While most of the newly identified polynomials are linearly equivalent to permutation monomials, we show that there exist subclasses that are not affine equivalent to monomials, and we describe their forms. One of the newly identified classes contains a subclass of complete mappings. We use these complete mappings to define new sets of mutually orthogonal Latin squares, as well as new bent vectorial functions from the Maiorana-McFarland class. Moreover, the quasigroup polynomials obtained in the process are different and inequivalent to the previously known ones.