Browsing Mathematics publications (MU) by Author "Shparlinski, Igor E."
Now showing items 120 of 26

Arithmetic properties of φ(n)/λ(n) and the structure of the multiplicative group modulo n
Banks, William David, 1964; Luca, Florian; Shparlinski, Igor E. (European Mathematical Society, 2006)For a positive integer n, we let φ(n) and λ(n) denote the Euler function and the Carmichael function, respectively. We define ξ(n) as the ratio φ(n)/λ(n) and study various arithmetic properties of ξ(n). 
Average Normalisations of Elliptic Curves
Banks, William David, 1964; Shparlinski, Igor E. (Australian Mathematical Society, 2002)Ciet, Quisquater, and Sica have recently shown that every elliptic curve E over a finite field Fp is isomorphic to a curve y2 = x3 +ax+b with a and b of size O(p3/4). In this paper, we show that almost all elliptic curves ... 
Character Sums over Integers with Restricted gary Digits
Banks, William David, 1964; Conflitti, Alessandro; Shparlinski, Igor E. (2002)We establish upper bounds for multiplicative character sums and exponential sums over sets of integers that are described by various properties of their digits in a fixed base g ≥ 2. Our main tools are the Weil and ... 
Coincidences in the values of the Euler and Carmichael functions
Banks, William David, 1964; Friedlander, J. B. (John B.); Luca, Florian; Pappalardi, Francesco; Shparlinski, Igor E. (Polish Academy of Sciences, Institute of Mathematics, 2006)The Euler function has long been regarded as one of the most basic of the arithmetic functions. More recently, partly driven by the rise in importance of computational number theory, the Carmichael function has drawn an ... 
Congruences and Exponential Sums with the Euler Function
Banks, William David, 1964; Shparlinski, Igor E. (2004)We give upper bounds for the number of solutions to congruences with the Euler function φ(n) and with the Carmichael function λ(n). We also give nontrivial bounds for certain exponential sums involving φ(n). Analogous ... 
Cryptographic applications of sparse polynomials over finite rings
Banks, William David, 1964; Lieman, Daniel, 1965; Shparlinski, Igor E.; To, Van Thuong (2001)This paper gives new examples that exploit the idea of using sparse polynomials with restricted coefficients over a finite ring for designing fast, reliable cryptosystems and identification schemes. 
Distribution of Inverses in Polynomial Rings
Banks, William David, 1964; Shparlinski, Igor E. (2001)Let IFp be the finite field with p elements, and let F(X) ∈ IFp[X] be a squarefree polynomial. We show that in the ring R = IFp[X]/F(X), the inverses of polynomials of small height are uniformly distributed. We also show ... 
Distributional Properties of the Largest Prime Factor
Banks, William David, 1964; Harman, G. (Glyn), 1956; Shparlinski, Igor E. (University of Michigan, 2005)Let P(n) denote the largest prime factor of an integer n ≥ 2, and put P(1) = 1. In this paper, we study the distribution of the sequence {P(n) : n ≥ 1} over the set of congruence classes modulo an integer q ≥ 2, and we ... 
Exponential Sums over Mersenne Numbers
Banks, William David, 1964; Conflitti, Alessandro; Friedlander, J. B. (John B.); Shparlinski, Igor E. (2004)We give estimates for exponential sums of the form Σn≤N Λ(n) exp(2πiagn/m), where m is a positive integer, a and g are integers relatively prime to m, and Λ is the von Mangoldt function. In particular, our results yield ... 
An extremely small and efficient identification scheme
Banks, William David, 1964; Lieman, Daniel, 1965; Shparlinski, Igor E. (2000)We present a new identification scheme which is based on Legendre symbols modulo a certain hidden prime and which is naturally suited for low power, low memory applications. 
An identification scheme based on sparse polynomials
Banks, William David, 1964; Lieman, Daniel, 1965; Shparlinski, Igor E. (2000)This paper gives a new example of exploiting the idea of using polynomials with restricted coefficients over finite fields and rings to construct reliable cryptosystems and identification schemes. 
Incomplete exponential sums and DiffieHellman triples
Banks, William David, 1964; Friedlander, J. B. (John B.); Koniagin, S. V. (Sergeĭ Vladimirovich); Shparlinski, Igor E. (Cambridge University Press, 2006)Let p be a prime and 79 an integer of order t in the multiplicative group modulo p. In this paper, we continue the study of the distribution of DiffieHellman triples (Vx, Vy, Vxy) by considering the closely related ... 
Irrationality of Power Series for Various Number Theoretic Functions
Banks, William David, 1964; Luca, Florian; Shparlinski, Igor E. (200510)We study formal power series whose coefficients are taken to be a variety of number theoretic functions, such as the Euler, Möbius and divisor functions. We show that these power series are irrational over ℤ [X], and we ... 
Multiplicative Structure of Values of the Euler Function
Banks, William David, 1964; Friedlander, J. B. (John B.); Pomerance, Carl; Shparlinski, Igor E. (2004)We establish upper bounds for the number of smooth values of the Euler function. In particular, although the Euler function has a certain “smoothing” effect on its integer arguments, our results show that, in fact, most ... 
Nonresidues and primitive roots in Beatty sequences
Banks, William David, 1964; Shparlinski, Igor E. (Australian Mathematical Society, 2006)We study multiplicative character sums taken on the values of a nonhomogeneous Beatty sequence Bα,β = {⌊αn + β⌋ : n = 1,2,3,…}, where α,β ∈ R, and α is irrational. In particular, our bounds imply that for every fixed ε > ... 
Nonlinear complexity of the NaorReingold pseudorandom function
Banks, William David, 1964; Griffin, Frances; Lieman, Daniel, 1965; Shparlinski, Igor E. (2000)We obtain an exponential lower bound on the nonlinear complexity of the new pseudorandom function, introduced recently by M. Naor and O. Reingold. This bound is an extension of the lower bound on the linear complexity ... 
Number Theoretic Designs for Directed Regular Graphs of Small Diameter
Banks, William David, 1964; Conflitti, Alessandro; Shparlinski, Igor E. (Society for Industrial and Applied Mathematics, 2004)In 1989, F. R. K. Chung gave a construction for certain directed hregular graphs of small diameter. Her construction is based on finite fields, and the upper bound on the diameter of these graphs is derived from bounds ... 
On the average value of divisor sums in arithmetic progressions
Banks, William David, 1964; HeathBrown, D. R.; Shparlinski, Igor E. (2005)We consider very short sums of the divisor function in arithmetic progressions prime to a fixed modulus and show that "on average" these sums are close to the expected value. We also give applications of our result to sums ... 
On the Number of Sparse RSA Exponents
Banks, William David, 1964; Shparlinski, Igor E. (2002)An RSA modulus is a product M = pl of two primes p and l. We show that for almost all RSA moduli M, the number of sparse exponents e (which allow for fast RSA encryption) with the property that gcd(e,φ(M)) = 1 (hence RSA ... 
On the Value Set of n! Modulo a Prime
Banks, William David, 1964; Luca, Florian; Shparlinski, Igor E.; Stichtenoth, H. (Henning), 1944 (2005)We show that for infinitely many prime numbers p there are at least log log p/ log log log p distinct residue classes modulo p that are not congruent to n! for any integer n.