On character sums with determinants

We estimate weighted character sums with determinants ad-bc of 2×2 matrices modulo a prime p with entries a,b,c and d vary ing over the interval [1,N].Our goal is to obtain non-trivial bounds for values of N as small as possible.In particular,we achieve this goal,with a power saving,for N≥p^(1/8+ε) with any fixed ε>0,which is very likely to be the best possible unless the celebrated Burgess bound is improved,By other techniques,we also treat more general sums but sometimes for larger values of N.

The Hybrid Power Mean Involving the Character Sum of Polynomials and a Sum Analogous to Kloosterman Sum

The main purpose of this paper is using the properties of the classical Gauss sum and the analytic methods to study the computational problem of one kind of hybrid power mean involving the character sum of polynomials and a sum analogous to Kloosterman sum mod p,an odd prime,and give two sharp asymptotic formulae for them.

On the Fourth Power Mean of the Character Sums Over Short Intervals

The main purpose of this paper is to study the mean value properties of the character sums over the interval [1,p/8） by using the mean value theorems of the Dirichlet L-functions, and give an interesting mean value formula for this study.

Bounds for Certain Character Sums

This paper shows a connection between exponential sums and character sums. In particular, we introduce a character sum that is an analog of the classical Kloosterman sums and establish the analogous Weil-Estermann's upper bound for it. The paper also analyzes a generalized Hardy-Littlewood example for character sums, which shows that the upper bounds given here are the best possible. The analysis makes use of local bounds for the exponential sums and character sums. The basic theorems have been previously established.

Upper Bounds on Character Sums with Rational Function Entries

We obtain formulae and estimates for character sums of the type $S\left( {\chi ,f,p^m } \right) = \sum\nolimits_{x = 1}^{p^m } {\chi \left( {f\left( x \right)} \right),} $ where pm is a prime power with m S 2, L is a multiplicative character (mod p^m), and f=f1/f2 is a rational function over ê. In particular, if p is odd, d=deg(f1)+deg(f2) and d* = max(deg(f1), deg(f2)) then we obtain $\left| {S\left( {\chi ,f,p^m } \right)} \right| \le \left( {d - 1} \right)p^{m\left( {1 - {1 \over {d*}}} \right)}$ for any non-constant f (mod p) and primitive character L. For p = 2 an extra factor of $2\sqrt 2$ is needed.

Local and Global Character Sums

Hua’s estimate is established for character sums in a number field.A relationship between liftings of a character sum in a local field is also studied.

On the Character Sum of Polynomials and the Two-term Exponential Sums

The main purpose of this paper is using the analytic method and the properties of the character sums to study the computational problem of one kind hybrid power mean involving the character sums of polynomials and the two-term exponential sums,and give several interesting identities and asymptotic formulae for them.

Constructing quasi-random subsets of Z_N by using elliptic curves

Let ε ： y^2 = x3 ＋ Ax ＋ B be an elliptic curve defined over the finite field Zp（p 〉 3） and G be a rational point of prime order N on ε. Define a subset of ZN, the residue class ring modulo N, asS：={n：n∈ZN,n≠0,（X（nG）/p）=1} where X（nG） denotes the x-axis of the rational points nC and （＊/P） is the Legendre symbol. Some explicit results on quasi-randomness of S are investigated. The construction depends on the intrinsic group structures of elliptic curves and character sums along elliptic curves play an important role in the proofs.

GENERATORS BY THE QUADRATIC EXPONENTIAL METHOD

Based on the quadratic exponential method, this paper constructs two types of generators over finite field Fq, the digital quadratic exponential generator and quadratic exponential pseudorandom vector generator. We investigate the distribution of the sequence generated by the generators, and present results about their one dimensional discrepancy. The proofs are based on the estimate of certain character sum over Fq. Ift is the least period of the sequence and t≥q^1/2＋2c, then the bound of the discrepancy is O（t^-1/4q^1/8＋τ logq） for any ε 〉 0. It shows that the sequence is asymptotically uniformly distributed.

On the Mean Value of General k-th Gauss Sums

The main purpose of this paper is using residue system and character sums methods to investigate the mean value properties of general k-th Gauss sums,and two exact calculating formulas are given.

New classes of sequence families with low correlation by using multiplicative and additive characters

For an odd prime p, a new sequence family of period prom- 1, size （M-1）pmr is proposed using multi-plicative and additive characters. The upper bound for the maximum magnitude of nontrivial correlations of the sequence family is derived using well-known character sums. The upper bound is shown to be （r ＋ 1）√pm ＋ 3, which meets the Welch bound asymptotically.

On the pseudorandom properties of d-ary generalized two-prime Sidelnikov sequences

Let p and q be two distinct odd primes and let d=(p-1,q-1).In this paper,we construct d-ary generalized two-prime Sidelnikov sequences and study the autocorrelation values and linear complexity.

Pseudo-Randomness of Certain Sequences of k Symbols with Length pq

The theory of finite pseudo-random binary sequences was built by C. Mauduit and A. Sarkozy and later extended to sequences of k symbols （or k-ary sequences）. Certain constructions of pseudo-random sequences of k symbols were presented over finite fields in the literature. In this paper, two families of sequences of k symbols are constructed by using the integers modulo pq for distinct odd primes p and q. The upper bounds on the well-distribution measure and the correlation measure of the families sequences are presented in terms of certain character sums over modulo pq residue class rings. And low bounds on the linear complexity profile are also estimated.

Distribution of Elements in Primitive Sequences over Z_(p^e)

Using the estimates of character sums over Galoi8 rings and the trace de-scription of primitive sequences over Z_(p^e), we obtain an estimate for the frequency of theoccurrences of any element in Z_(p^e) in one period of a primitive sequence, which is betterthan Kuzmin's results if n >4e, where n is the degree of the generating polynomial ofthe primitive sequence.

Construction of k-ary Pseudorandom Elliptic Curve Sequences

We present a method for constructing k-ary sequences over elliptic curves. Using the multiplicative character of order k of finite fields, we construct a family of k-ary pseudorandom elliptic curve sequences. The pseudorandom measures, such as the well-distribution measure, the correlation measure of order e, and the linear complexity are estimated by using certain character sums. Such sequences share the same order of magnitude on the well-distribution measure, the correlation measure of order e as the ＇truly＇ random sequences. The method indicates that it is possible to construct ＇good＇ pseudorandom sequences over elliptic curves widely used in public key cryptography.

Small Solutions of the Congruence a_1x_1~2+a_2x_2~2+a_3x_3~2+a_4x_4~2≡c(mod p)

For any integers a1,a2,a3,a4 and c with a1a2a3a40（mod p）,this paper shows that there exists a solution X=（x1,x2,x3,x4）∈Z4 of the congruence a1x12+a2x22+a3x32+a4x42≡ c（mod p）such that ‖X‖=max{|x1|,|x2|,|x3|,|x4|}《p1/2logp.

On error distance of Reed-Solomon codes

The complexity of decoding the standard Reed-Solomon code is a well known open prob-lem in coding theory. The main problem is to compute the error distance of a received word. Using the Weil bound for character sum estimate, we show that the error distance can be determined precisely when the degree of the received word is small. As an application of our method, we give a significant improvement of the recent bound of Cheng-Murray on non-existence of deep holes (words with maximal error distance).

Some Notes on Generalized Cyclotomic Sequences of Length pq

We review the constructions of two main kinds of generalized cyclotomic binary sequences with length pq （the product with two distinct primes）. One is the White-generalized cyclotomic sequences, the other is the Ding-Helleseth（DH, for short）-generalized cyclotomic sequences. We present some new pseudo-random properties of DH-generalized cyclotomic sequences using the theory of character sums instead of the theory of cyclotomy, which is a conventional method for investigating generalized cyclotomic sequences.

High-dimensional D.H.Lehmer Problem over Short Intervals

Letk be a positive integer and n a nonnegative integer,0 〈 λ1,...,λk＋1 ≤ 1 be real numbers and w =（λ1,λ2,...,λk＋1）.Let q ≥ max{[1/λi ]：1 ≤ i ≤ k ＋ 1} be a positive integer,and a an integer coprime to q.Denote by N（a,k,w,q,n） the 2n-th moment of（b1··· bk c） with b1··· bk c ≡ a（mod q）,1 ≤ bi≤λiq（i = 1,...,k）,1 ≤ c ≤λk＋1 q and 2（b1＋ ··· ＋ bk ＋ c）.We first use the properties of trigonometric sum and the estimates of n-dimensional Kloosterman sum to give an interesting asymptotic formula for N（a,k,w,q,n）,which generalized the result of Zhang.Then we use the properties of character sum and the estimates of Dirichlet L-function to sharpen the result of N（a,k,w,q,n） in the case ofw =（1/2,1/2,...,1/2） and n = 0.In order to show our result is close to the best possible,the mean-square value of N（a,k,q） φk（q）/2k＋2and the mean value weighted by the high-dimensional Cochrane sum are studied too.

Some new optimal quaternary constant weight codes

Constant weight codes (CWCs) are an important class of codes in coding theory. Generalized Steiner systems GS(2, k, v, g) were first introduced by Etzion and used to construct optimal nonlinear CWCs over an alphabet of size g+1 with minimum Hamming distance 2k - 3, in which each codeword has length v and weight k. In this paper, Weil's theorem on character sum estimates is used to show that there exists a GS(2,4, v, 3) for any prime v≡1 (mod 4) and v > 13. From the coding theory point of view, an optimal nonlinear quaternary (v, 5,4) CWC exists for such a prime v.