Modulo Pq Research Articles

Linear Complexity of r-Ary Sequences Derived from Euler Quotient Modulo pq

Linear Complexity of r-Ary Sequences Derived from Euler Quotient Modulo pq

On Divisibility Property of Type 2 (p, q)-Analogue of r-Whitney Numbers of the Second Kind

In this paper, the divisibility property of the type 2 (p; q)-analogue of the r-Whitney numbers of the second kind is established. More precisely, a congruence relation modulo pq for this (p; q)-analogue is derived.

New Binary Sequences Derived from Euler Quotients Modulo pq and Their Generalizations

Pseudorandom sequences with large linear complexity can resist the linear attack. The trace representation plays an important role in analysis and design of pseudorandom sequences. In this letter, we present the construction of a family of new binary sequences derived from Euler quotients modulo pq, where pq is a product of two primes and p divides q-1. Firstly, the linear complexity of the sequences are investigated. It is proved that the sequences have larger linear complexity and can resist the attack of Berlekamp-Massey algorithm. Then, we give the trace representation of the proposed sequences by determining the corresponding defining pair. Moreover, we generalize the result to the Euler quotients modulo pmqn with m≤n. Results indicate that the generalized sequences still have high linear complexity. We also give the trace representation of the generalized sequences by determining the corresponding defining pair. The result will be helpful for the implementation and the pseudorandom properties analysis of the sequences.

Notes about symmetric m-adic complexity of generalized cyclotomic sequences of order two with period pq

In this paper, we consider binary generalized cyclotomic sequences with period pq, where p and q are two distinct odd primes. These sequences derive from generalized cyclotomic classes of order two modulo pq. We investigate the generalized binary cyclotomic sequences as the sequences over the ring of integers modulo m for a positive integer m and study m-adic complexity of sequences. We show that they have high symmetric m-adic complexity. Our results generalize well-known statements about 2-adic complexity of these sequences.

Linear Complexity of Quaternary Sequences over Z<sub>4</sub> Based on Ding-Helleseth Generalized Cyclotomic Classes

A family of quaternary sequences over Z4 is defined based on the Ding-Helleseth generalized cyclotomic classes modulo pq for two distinct odd primes p and q. The linear complexity is determined by computing the defining polynomial of the sequences, which is in fact connected with the discrete Fourier transform of the sequences. The results show that the sequences possess large linear complexity and are good sequences from the viewpoint of cryptography.

Linear complexity and trace representation of quaternary sequences over ℤ 4 $\mathbb {Z}_{4}$ based on generalized cyclotomic classes modulo pq

We define a family of quaternary sequences over the residue class ring modulo $4$ of length $pq$, a product of two distinct odd primes, using the generalized cyclotomic classes modulo $pq$ and calculate the discrete Fourier transform (DFT) of the sequences. The DFT helps us to determine the exact values of linear complexity and the trace representation of the sequences.

The sizes of the intersections of two unitals in [formula omitted

We show that the size of the intersection of a Hermitian variety in PG(n,q2), and any set satisfying an r-dimensional-subspace intersection property, is congruent to 1 modulo a power of p. In particular, in the case where n=2, if the two sets are a Hermitian unital and any other unital, the size of the intersection is congruent to 1 modulo q or modulo pq. If the second unital is a Buekenhout–Metz unital, we show that the size is congruent to 1 modulo q.

A new result on the distinctness of primitive sequences over [formula omitted] modulo 2

Let Z / ( p q ) be the integer residue ring modulo pq with odd prime numbers p and q. This paper studies the distinctness problem of modulo 2 reductions of two primitive sequences over Z / ( p q ) , which has been studied by H.J. Chen and W.F. Qi in 2009. First, it is shown that almost every element in Z / ( p q ) occurs in a primitive sequence of order n > 2 over Z / ( p q ) . Then based on this element distribution property of primitive sequences over Z / ( p q ) , previous results are greatly improved and the set of primitive sequences over Z / ( p q ) that are known to be distinct modulo 2 is further enlarged.

Linear complexity and autocorrelation values of a polyphase generalized cyclotomic sequence of length pq

A construction of a family of generalized polyphase cyclotomic sequences of length pq is presented in terms of the generalized cyclotomic classes modulo pq. Their linear complexity and corresponding minimal polynomials are deduced. Some upper bounds on periodic and aperiodic autocorrelation values of resulting sequences are also estimated by using certain exponential sums.

Solvability of the problem of taking the discrete logarithm

It is proved that consideration of the solvability problem for taking the discrete logarithm in a residue ring modulo composite M can be reduced to consideration of a similar problem in residue rings modulo pq, where p and q are prime divisors of M. For moduli of form pq, necessary and sufficient conditions for solvability checking are obtained in some cases. In addition, the problem of raising a solution of an exponential comparison in a residue ring modulo p α is solved.

Symmetric polynomials over [formula omitted] and simultaneous communication protocols

We study the problem of representing symmetric Boolean functions as symmetric polynomials over Z m . We prove an equivalence between representations of Boolean functions by symmetric polynomials and simultaneous communication protocols. We show that computing a function f on 0–1 inputs with a polynomial of degree d modulo pq is equivalent to a two player simultaneous protocol for computing f where one player is given the first ⌈ log p d ⌉ digits of the weight in base p and the other is given the first ⌈ log q d ⌉ digits of the weight in base q . This equivalence allows us to show degree lower bounds by using techniques from communication complexity. For example, we show lower bounds of Ω ( n ) on symmetric polynomials weakly representing classes of Mod r and Threshold functions. Previously the best known lower bound for such representations of any function modulo pq was Ω ( n 1 2 ) [D.A. Barrington, R. Beigel, S. Rudich, Representing Boolean functions as polynomials modulo composite numbers, Comput. Complexity 4 (1994) 367–382]. The equivalence also allows us to use results from number theory to prove upper bounds for Threshold- k functions. We show that proving bounds on the degree of symmetric polynomials strongly representing the Threshold- k function is equivalent to counting the number of solutions to certain Diophantine equations. We use this to show an upper bound of O ( nk ) 1 2 + ɛ for Threshold- k assuming the abc-conjecture. We show the same bound unconditionally for k constant. Prior to this, non-trivial upper bounds were known only for the OR function [D.A. Barrington, R. Beigel, S. Rudich, Representing Boolean functions as polynomials modulo composite numbers, Comput. Complexity 4 (1994) 367–382]. We show an almost tight lower bound of Ω ( nk ) 1 2 , improving the previously known bound of Ω ( max ( k , n ) ) [S.-C. Tsai, Lower bounds on representing Boolean functions as polynomials in Z m , SIAM J. Discrete Math. 9 (1996) 55–62].