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PolynomialTime Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
 SIAM J. on Computing
, 1997
"... A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. ..."
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Cited by 882 (2 self)
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A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and which have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.
Feedback shift registers, 2adic span, and combiners with memory
 Journal of Cryptology
, 1997
"... Feedback shift registers with carry operation (FCSR’s) are described, implemented, and analyzed with respect to memory requirements, initial loading, period, and distributional properties of their output sequences. Many parallels with the theory of linear feedback shift registers (LFSR’s) are presen ..."
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Cited by 50 (7 self)
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Feedback shift registers with carry operation (FCSR’s) are described, implemented, and analyzed with respect to memory requirements, initial loading, period, and distributional properties of their output sequences. Many parallels with the theory of linear feedback shift registers (LFSR’s) are presented, including a synthesis algorithm (analogous to the BerlekampMassey algorithm for LFSR’s) which, for any pseudorandom sequence, constructs the smallest FCSR which will generate the sequence. These techniques are used to attack the summation cipher. This analysis gives a unified approach to the study of pseudorandom sequences, arithmetic codes, combiners with memory, and the MarsagliaZaman random number generator. Possible variations on the FCSR architecture are indicated at the end. Index Terms – Binary sequence, shift register, stream cipher, combiner with memory, cryptanalysis, 2adic numbers, arithmetic code, 1/q sequence, linear span. 1
Accurate and efficient evaluation of Schur and Jack functions
 Math. Comp
, 2006
"... Abstract. We present new algorithms for computing the values of the Schur sλ(x1,x2,...,xn)andJackJ α λ (x1,x2,...,xn) functions in floating point arithmetic. These algorithms deliver guaranteed high relative accuracy for positive data (xi,α>0) and run in time that is only linear in n. 1. ..."
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Cited by 7 (4 self)
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Abstract. We present new algorithms for computing the values of the Schur sλ(x1,x2,...,xn)andJackJ α λ (x1,x2,...,xn) functions in floating point arithmetic. These algorithms deliver guaranteed high relative accuracy for positive data (xi,α>0) and run in time that is only linear in n. 1.
A MULTIMODULAR ALGORITHM FOR COMPUTING BERNOULLI NUMBERS
"... Abstract. We describe an algorithm for computing Bernoulli numbers. Using a parallel implementation, we have computed Bk for k = 108, a new record. Our method is to compute Bk modulo p for many small primes p, and then reconstruct Bk via the Chinese Remainder Theorem. The asymptotic time complexity ..."
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Cited by 3 (1 self)
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Abstract. We describe an algorithm for computing Bernoulli numbers. Using a parallel implementation, we have computed Bk for k = 108, a new record. Our method is to compute Bk modulo p for many small primes p, and then reconstruct Bk via the Chinese Remainder Theorem. The asymptotic time complexity is O(k2 log2+ε k), matching that of existing algorithms that exploit the relationship between Bk and the Riemann zeta function. Our implementation is significantly faster than several existing implementations of
Can We Learn Algorithms from People Who Compute Fast: An Indirect Analysis in the Presence of Fuzzy Descriptions
"... Abstract — In the past, mathematicians actively used the ability of some people to perform calculations unusually fast. With the advent of computers, there is no longer need for human calculators – even fast ones. However, recently, it was discovered that there exist, e.g., multiplication algorithms ..."
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Abstract — In the past, mathematicians actively used the ability of some people to perform calculations unusually fast. With the advent of computers, there is no longer need for human calculators – even fast ones. However, recently, it was discovered that there exist, e.g., multiplication algorithms which are much faster than standard multiplication. Because of this discovery, it is possible than even faster algorithm will be discovered. It is therefore natural to ask: did fast human calculators of the past use faster algorithms – in which case we can learn from their experience – or they simply performed all operations within a standard algorithm much faster? This question is difficult to answer directly, because the fast human calculators ’ selfdescription of their algorithm is very fuzzy. In this paper, we use an indirect analysis to argue that fast human calculators most probably used the standard algorithm.
A CACHEFRIENDLY TRUNCATED FFT
, 810
"... Abstract. We describe a cachefriendly version of van der Hoeven’s truncated FFT and inverse truncated FFT, focusing on the case of ‘large ’ coefficients, such as those arising in the Schönhage–Strassen algorithm for multiplication in Z[x]. We describe two implementations and examine their performan ..."
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Abstract. We describe a cachefriendly version of van der Hoeven’s truncated FFT and inverse truncated FFT, focusing on the case of ‘large ’ coefficients, such as those arising in the Schönhage–Strassen algorithm for multiplication in Z[x]. We describe two implementations and examine their performance. 1.
IRREGULAR PRIMES TO 163 MILLION
"... Abstract. We compute all irregular primes less than 163 577 856. For all of these primes we verify that the Kummer–Vandiver conjecture holds and that the λinvariant is equal to the index of irregularity. 1. ..."
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Abstract. We compute all irregular primes less than 163 577 856. For all of these primes we verify that the Kummer–Vandiver conjecture holds and that the λinvariant is equal to the index of irregularity. 1.
AN Õ(log2 (N)) TIME PRIMALITY TEST FOR GENERALIZED CULLEN NUMBERS
"... Abstract. Generalized Cullen Numbers are positive integers of the form Cb(n):=nbn + 1. In this work we generalize some known divisibility properties of Cullen Numbers and present two primality tests for this family of integers. The first test is based in the following property of primes from this fa ..."
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Abstract. Generalized Cullen Numbers are positive integers of the form Cb(n):=nbn + 1. In this work we generalize some known divisibility properties of Cullen Numbers and present two primality tests for this family of integers. The first test is based in the following property of primes from this family: nbn ≡ (−1) b (mod nbn + 1). It is stronger and has less computational cost than Fermat’s test (to bases b and n) and than MillerRabin’s test (if b is odd, to base n). Pseudoprimes for this new test seem to be very scarce, only 4 pseudoprimes have been found among the many millions of Generalized Cullen Numbers tested. We also present a second, more demanding, test for which no pseudoprimes have been found. These tests lead to an algorithm, running in Õ(log2 (N)) time, which might be very useful in the search of Generalized Cullen Primes. 1.