Results 1  10
of
21
Elliptic Curve Systems
 IEEE P1363, Part 4: Elliptic Curve Systems
, 1995
"... . This standard describes a method for data encryption and for digital signatures using the elliptic curve analogue of the ElGamal publickey cryptosystem. Elliptic curve systems are publickey (asymmetric) cryptographic algorithms, typically used in conjunction with a hash algorithm to create digit ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
. This standard describes a method for data encryption and for digital signatures using the elliptic curve analogue of the ElGamal publickey cryptosystem. Elliptic curve systems are publickey (asymmetric) cryptographic algorithms, typically used in conjunction with a hash algorithm to create digital signatures, and for the secure distribution of secret keys for use in symmetrickey cryptosystems. Elliptic curve systems may also be used to transmit confidential information. Introduction The algebraic system defined on the points of an elliptic curve provides an alternate means to implement the ElGamal and ElGamallike public key encryption and signature protocols. These protocols are typically described in the literature in the algebraic system Z p , the integers modulo p, where p is a prime. For example, the NIST Digitial Signature Algorithm (DSA) is an ElGamallike signature scheme defined over Z p . Precisely the same protocol for signing could be defined over the points on an ell...
Applications of Exponential Sums in Communications Theory
, 1999
"... We provide an introductory overview of how exponential sums, and bounds for them, have been exploited by coding theorists and communications engineers. ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
We provide an introductory overview of how exponential sums, and bounds for them, have been exploited by coding theorists and communications engineers.
On Boolean Functions with Generalized Cryptographic Properties
 Properties, Indocrypt 2004, LNCS 3348
, 2004
"... By considering a new metric, we generalize cryptographic properties of Boolean functions such as resiliency and propagation characteristics. ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
By considering a new metric, we generalize cryptographic properties of Boolean functions such as resiliency and propagation characteristics.
The Number of Irreducible Polynomials over GF(2) with Given Trace and Subtrace
, 1999
"... The trace of a degree n polynomial p(x) over GF(2) is the coefficient of x n\Gamma1 and the subtrace is the coefficient of x n\Gamma2 . We derive an explicit formula for the number of irreducible degree n polynomials over GF(2) that have a given trace and subtrace. The trace and subtrace of an e ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
The trace of a degree n polynomial p(x) over GF(2) is the coefficient of x n\Gamma1 and the subtrace is the coefficient of x n\Gamma2 . We derive an explicit formula for the number of irreducible degree n polynomials over GF(2) that have a given trace and subtrace. The trace and subtrace of an element fi 2 GF(2 n ) are defined to be the coefficients of x n\Gamma1 and x n\Gamma2 , respectively, in the polynomial q(x) = n\Gamma1 Y i=0 (x + fi 2 i ). We also derive an explicit formula for the number of elements of GF(2 n ) of given trace and subtrace. Moreover, a new two equation Mobiustype inversion formula is proved. Keywords: Irreducible polynomial, minimal polynomial, trace, subtrace, Mobius inversion. 1 Introduction The trace of a degree n polynomial p(x) over GF(2) is the coefficient of x n\Gamma1 and the subtrace is the coefficient of x n\Gamma2 . It is well known that the formula L(n) = 1 n X djn (d)2 n=d (1) HewlettPackard Labs, Santa Rosa ema...
Computing Composed Products of Polynomials
 IN FINITE FIELDS: THEORY, APPLICATIONS, AND ALGORITHMS
, 1999
"... If f(x) and g(x) are polynomials in Fq [x] of degrees m and n respectively, then the composed sum of f and g, denoted f g, is the degree mn polynomial whose roots are all sums of roots of f with roots of g. Likewise, the composed multiplication of f and g, denoted f ffi g, is the degree mn poly ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
If f(x) and g(x) are polynomials in Fq [x] of degrees m and n respectively, then the composed sum of f and g, denoted f g, is the degree mn polynomial whose roots are all sums of roots of f with roots of g. Likewise, the composed multiplication of f and g, denoted f ffi g, is the degree mn polynomial whose roots are all products of roots of f with roots of g. In 1987, Brawley and Carlitz defined a more general notion of polynomial composition, denoted by f \Pi g, for which f g and f ffi g are special cases. They prove that when f and g are irreducible with degrees m and n coprime, then f \Pi g is irreducible of degree mn. This gives us a way to obtain irreducibles of relatively large degree using irreducibles of smaller degrees. In this paper, we describe several methods of computing polynomial compositions of the above form and compare their time complexities.
HAMMING DISTANCE FROM IRREDUCIBLE POLYNOMIALS OVER F2
"... Abstract. We study the Hamming distance from polynomials to classes of polynomials that share certain properties of irreducible polynomials. The results give insight into whether or not irreducible polynomials can be effectively modeled by these more general classes of polynomials. For example, we p ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Abstract. We study the Hamming distance from polynomials to classes of polynomials that share certain properties of irreducible polynomials. The results give insight into whether or not irreducible polynomials can be effectively modeled by these more general classes of polynomials. For example, we prove that the number of degree n polynomials of Hamming distance one from a randomly chosen set of ⌊2 n /n ⌋ odd density polynomials is asymptotically (1 − e −4)2 n−1, and this appears to be inconsistent with the numbers for irreducible polynomials. We also conjecture that there is a constant c such that every polynomial has Hamming distance at most c from an irreducible polynomial. Using exhaustive lists of irreducible polynomials over F2 for degrees 1 ≤ n ≤ 32, we count the number of polynomials with a given Hamming distance to some irreducible polynomial of the same degree. Our work is based on this “empirical ” study. 1.
COUNTING STRINGS WITH GIVEN ELEMENTARY SYMMETRIC FUNCTION EVALUATIONS II: CIRCULAR STRINGS
"... Abstract. Let α be a string over an alphabet that is a finite ring, R. The kth elementary symmetric function evaluated at α is denoted Tk(α). In a companion paper we studied the properties of SR(n; τ1, τ2,..., τk), the set of of length n strings for which Ti(α) = τi. Here we consider the set, LR(n ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. Let α be a string over an alphabet that is a finite ring, R. The kth elementary symmetric function evaluated at α is denoted Tk(α). In a companion paper we studied the properties of SR(n; τ1, τ2,..., τk), the set of of length n strings for which Ti(α) = τi. Here we consider the set, LR(n; τ1, τ2,..., τk), of equivalence classes under rotation of aperiodic strings in SR(n; τ1, τ2,..., τk), sometimes called Lyndon words. General formulae are established, and then refined for the cases where R is the ring of integers Zq or the finite field Fq.
Random Number Generators: A Survival Guide for Large Scale Simulations
"... U.S.A. Abstract. Monte Carlo simulations are an important tool in statistical physics, complex systems science, and many other fields. An increasing number of these simulations is run on parallel systems ranging from multicore desktop computers to supercomputers with thousands of CPUs. This raises t ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
U.S.A. Abstract. Monte Carlo simulations are an important tool in statistical physics, complex systems science, and many other fields. An increasing number of these simulations is run on parallel systems ranging from multicore desktop computers to supercomputers with thousands of CPUs. This raises the issue of generating large amounts of random numbers in a parallel application. In this lecture we will learn just enough of the theory of pseudo random number generation to make wise decisions on how to choose and how to use random number generators when it comes to large scale, parallel simulations.
Computing special powers in finite fields
, 2003
"... We study exponentiation in nonprime finite fields with very special exponents such as they occur, for example, in inversion, primitivity tests, and polynomial factorization. Our algorithmic approach improves the corresponding exponentiation problem from about quadratic to about linear time. ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We study exponentiation in nonprime finite fields with very special exponents such as they occur, for example, in inversion, primitivity tests, and polynomial factorization. Our algorithmic approach improves the corresponding exponentiation problem from about quadratic to about linear time.