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. This standard describes a method for data encryption and for digital signatures using the elliptic curve analogue of the ElGamal public-key cryptosystem. Elliptic curve systems are public-key (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 symmetric-key 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 ElGamal-like 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 ElGamal-like signature scheme defined over Z p . Precisely the same protocol for signing could be defined over the points on an ell...

We provide an introductory overview of how exponential sums, and bounds for them, have been exploited by coding theorists and communications engineers.

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.

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 Mobius-type 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) Hewlett-Packard Labs, Santa Rosa e-ma...

Abstract. 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. 1.

The trace of a degree n polynomial f(x) over GF (q) is the coefficient of x . Carlitz [Proc. AMS, 3 (1952) 693-700] obtained an expression Iq (n; t), for the number of monic irreducible polynomials over GF (q) of degree n and trace t. Using a different approach, we derive a simple explicit expression for Iq (n; t). If t > 0, Iq (n; t) = ( )=(qn), where the sum is over all divisors d of n which are relatively prime to q. This same approach is used to count Lq (n; t), the number of q-ary Lyndon words whose characters sum to t mod q. This number is given by Lq (n; t) = ( )=(qn), where the sum is over all divisors d of n for which gcd(d; q)jt. Both results rely on a new form of Möbius inversion.

By considering a new metric, we generalize cryptographic properties of Boolean functions such as resiliency and propagation characteristics.

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.

. 1 This standard describes a method for data encryption and for digital signatures using the elliptic curve analogue of the ElGamal public-key cryptosystem, and a method for key agreement using a variant of the Diffie-Hellman key agreement protocol. Elliptic curve systems are public-key (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 symmetric-key cryptosystems. Elliptic curve systems may also be used to transmit confidential information. Introduction Many public-key cryptographic systems are based on exponentiation operations in large finite mathematical groups. The cryptographic strength of these systems is derived from the believed computational intractability of computing logarithms in these groups. The most commonly seen groups are the multiplicative group of Z p (the integers modulo a prime p) and F 2 m (characteristic 2 finite fields). The prima...