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PRIMES is in P
 Ann. of Math
, 2002
"... We present an unconditional deterministic polynomialtime algorithm that determines whether an input number is prime or composite. 1 ..."
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Cited by 26 (2 self)
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We present an unconditional deterministic polynomialtime algorithm that determines whether an input number is prime or composite. 1
Embedding Dynamics for RoundOff Errors Near a Periodic Orbit
"... We study the propagation of roundo errors near the periodic orbits of a linear map conjugate to a planar rotation with rational rotation number. We embed the twodimensional discrete phase space (a lattice) in a higherdimensional torus, where points sharing the same roundo error are uniformly dist ..."
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Cited by 7 (3 self)
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We study the propagation of roundo errors near the periodic orbits of a linear map conjugate to a planar rotation with rational rotation number. We embed the twodimensional discrete phase space (a lattice) in a higherdimensional torus, where points sharing the same roundo error are uniformly distributed within nitely many convex polyhedra. The embedding dynamics is linear and discontinuous, with algebraic integer coecients. This representation aords ecient algorithms for classifying and computing the orbits and their densities, which we apply to the case of rational rotation number with denominator 7, corresponding to certain algebraic integers of degree three. We provide evidence that the hierarchical arrangement of orbits previously detected in quadratic cases [7] disappears, and that the growth of the number of orbits with the period is algebraic. 1 Introduction The study of roundo errors in computer representations of dynamical systems has attracted considerable attention...
Efficient Implementation of Rijndael Encryption With Composite Field Arithmetic
"... We explore the use of subfield arithmetic for efficient implementations Galois Field arithmetic in the context of Rijndael cipher. ..."
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Cited by 4 (2 self)
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We explore the use of subfield arithmetic for efficient implementations Galois Field arithmetic in the context of Rijndael cipher.
Efficient Galois Field Arithmetic on SIMD Architectures
"... We propose techniques to utilize the data parallelism capabilities of a SIMD architecture in computations involving Galois Field arithmetic. Galois Field arithmetic nds wide use in engineering applications, including errorcorrecting codes and cryptography. Often these applications involve exten ..."
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Cited by 3 (0 self)
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We propose techniques to utilize the data parallelism capabilities of a SIMD architecture in computations involving Galois Field arithmetic. Galois Field arithmetic nds wide use in engineering applications, including errorcorrecting codes and cryptography. Often these applications involve extensive arithmetic on small (8bit) numbers, and straightforward implementations may highly underutilize the wideword capabilities of a SIMD processor.
Computing Shifts in 90/150 Cellular Automata Sequences
 Applications, Volume 9, Issue
, 2001
"... Sequences produced by cellular automata (CA) are studied algebraically. A suitable kcell 90/150 CA over F q generates a sequence of length q k 1. The temporal sequence of any cell of such a CA can be obtained by shifting the temporal sequence of any other cell. We obtain a general algorithm to c ..."
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Cited by 3 (1 self)
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Sequences produced by cellular automata (CA) are studied algebraically. A suitable kcell 90/150 CA over F q generates a sequence of length q k 1. The temporal sequence of any cell of such a CA can be obtained by shifting the temporal sequence of any other cell. We obtain a general algorithm to compute these relative shifts. This is achieved by developing the proper algebraic framework for the study of CA sequences. 1
Computational Methods in Public Key Cryptology
, 2002
"... These notes informally review the most common methods from computational number theory that have applications in public key cryptology. ..."
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Cited by 1 (1 self)
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These notes informally review the most common methods from computational number theory that have applications in public key cryptology.
Walecki Tournaments: Part I
, 2001
"... . Walecki tournaments were dened by Alspach in 1966. They form a class of regular tournaments that posses a natural Hamilton directed cycle decomposition. It has been conjectured by Kelly in 1964 that every regular tournament possesses such a decomposition. Therefore Walecki tournaments speak in fav ..."
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. Walecki tournaments were dened by Alspach in 1966. They form a class of regular tournaments that posses a natural Hamilton directed cycle decomposition. It has been conjectured by Kelly in 1964 that every regular tournament possesses such a decomposition. Therefore Walecki tournaments speak in favor of the conjecture. A second interest in Walecki tournaments arises from the mapping between cycles of the complementing circular shift register and isomorphism classes of Walecki tournaments. An upper bound on the number of isomorphism classes of Walecki tournaments was determined by Alspach. It was conjectured that the bound is tight. The problem of enumerating Walecki tournaments has not been solved to date. However, it was published as an open problem in a paper by Alspach in 1989. In an attempt to prove this 34 years old conjecture, we rst determine the arc structure of Walecki tournaments for all initial cases and those whose corresponding binary sequences have zero pattern. Subsequent papers deal with more general cases. Techniques used in
Characterization of cyclotomic schemes on a finite field and normal Schur rings over a cyclic group
, 2001
"... It is well known that in general a cyclotomic scheme C on a finite field F cannot be characterized up to isomorphism by its intersection numbers. We show that the set of intersection numbers of some scheme b C (b) on the bfold Cartesian product of F where b is the base number of the group Aut(C) fo ..."
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It is well known that in general a cyclotomic scheme C on a finite field F cannot be characterized up to isomorphism by its intersection numbers. We show that the set of intersection numbers of some scheme b C (b) on the bfold Cartesian product of F where b is the base number of the group Aut(C) forms a full set of invariants of C. A key point here is that the scheme b C (b) can be de ned for an arbitrary scheme C in a purely combinatorial way. The proof is based on the complete description of normal Cayley and Schur rings (introduced in this paper) over a finite cyclic group. The developed technique enables us to show that a Schur ring over a cyclic group that is different from the group ring has a nontrivial automorphism.
Efficient Parallel Exponentiation in GF(2^n) Using Normal Basis Representations
, 2001
"... Von zur Gathen proposed an ecient parallel exponentiation algorithm in nite elds using normal basis representations. In this paper we present a processorecient parallel exponentiation ) which improves upon von zur Gathen's algorithm. We also show that exponentiation ) can be done in O(log n) ..."
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Von zur Gathen proposed an ecient parallel exponentiation algorithm in nite elds using normal basis representations. In this paper we present a processorecient parallel exponentiation ) which improves upon von zur Gathen's algorithm. We also show that exponentiation ) can be done in O(log n) time using n=(log n) processors. Hence we get processor time bound O(n= log n), which is optimal. Finally, we present an online processor assignment scheme which was missing in von zur Gathen's algorithm, and show that its time complexity is negligible.