Abstract not found

There are well-known techniques for message authentication using universal hash functions. This approach seems very promising, as it provides schemes that are both efficient and provably secure under reasonable assumptions. This paper contributes to this line of research in two ways. First, it analyzes the basic construction and some variants under more realistic and practical assumptions. Second, it shows how these schemes can be efficiently implemented, and it reports on the results of empirical performance tests that demonstrate that these schemes are competitive with other commonly employed schemes whose security is less well-established. 1 Introduction Message Authentication. Message authentication schemes are an important security tool. As more and more data is being transmitted over networks, the need for secure, high-speed, software-based message authentication is becoming more acute. The setting for message authentication is the following. Two parties A and B agree on a secre...

We consider the problem of factoring univariate polynomials over a finite field. We demonstrate that the new baby step/giant step factoring method, recently developed by Kaltofen & Shoup, can be made into a very practical algorithm. We describe an implementation of this algorithm, and present the results of empirical tests comparing this new algorithm with others. When factoring polynomials modulo large primes, the algorithm allows much larger polynomials to be factored using a reasonable amount of time and space than was previously possible. For example, this new software has been used to factor a "generic" polynomial of degree 2048 modulo a 2048-bit prime in under 12 days on a Sun SPARC-station 10, using 68 MB of main memory. 1 Introduction We consider the problem of factoring a univariate polynomial of degree n over the field F p of p elements, where p is prime. This problem has been well-studied, and many algorithms for its solution have been proposed. In general, the running tim...

New algorithms are presented for computing the minimal polynomial over a finite field K of a given element in an algebraic extension of K of the form K[ff] or K[ff][fi]. The new algorithms are explicit and can be implemented rather easily in terms of polynomial multiplication, and are much more efficient than other algorithms in the literature. 1 Introduction In this paper, we consider the problem of computing the minimal polynomial over a finite field K of a given element oe in an algebraic extension of K of the form K[ff] or K[ff][fi]. The minimal polynomial of oe is defined to be the unique monic polynomial OE oe=K 2 K[x] of least degree such that OE oe=K (oe) = 0. In the first case, we assume that the ring K[ff] is given as K[x]=(f) where f 2 K[x] is a monic polynomial of degree n, and that elements in K[ff] are represented in the natural way as elements of K[x] !n (the set of polynomials of degree less than n). Similarly, in the second case, we assume that K[ff] is given as a...

Several algorithms for factoring polynomials over finite fields are compared from the point of view of asymptotic complexity, and from a more realistic point of view: how well actual implementations perform on "moderately" sized inputs. 1 Introduction The purpose of this paper is to examine several algorithms for factoring polynomials over finite fields, from both the point of view of asymptotic complexity, and from a more realistic point of view: how well actual implementations perform on "moderately" sized inputs. We restrict our attention to factoring in Z p [x], where p is prime. The algorithms we consider are the algorithms of Berlekamp [B], Cantor & Zassenhaus [CZ], and von zur Gathen & Shoup [GS]. 2 Asymptotic Complexity Let n be the degree of the polynomial f 2 Z p [x] to be factored. It is natural to measure the running times of factorization algorithms in terms of the number of operations in Z p (additions, subtractions, multiplications, divisions, and zero-tests). All of t...

solving linear congruences integer, 19 polynomial, 358 Sophie Germain prime, 94 splitting field, 364 square root (modular), 275 algorithm for computing, 284 square-free integer, 12 polynomial, 431 square-free decomposition algorithm, 457, 467 standard basis, 298 statistical distance, 125 Stein, C., 273 Stein, J., 74 Strassen, V., 55, 258, 289 strict polynomial time, 142 subalgebra, 347 subfield, 212 subgroup, 177 generated by, 194 submodule, 293 generated (or spanned) by, 294 subring, 211 subspace, 300 surjective, 2 theta function of Chebyshev, 77 total degree, 221, 222 trace, 440 transcendental element, 363 transpose, 310 trial division, 236 trivial ring, 206 Pomerance, C., 55, 96, 258, 259, 344, 345, 482 de la Vallee Poussin, C.-J., 95, 96 power map, 186 pre-image, 2 pre-period, 72 prefix free, 143 primality test deterministic, 471 probabilistic, 236 prime ideal, 226 in an integral domain, 371 number, 5 prime number theorem,

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