Results 1  10
of
19
Removing redundancy in highprecision Newton iteration
, 2004
"... This paper speeds up Brent's algorithms for various highprecision computations in the power series ring C[[t]]. If it takes time 3 to compute a product then it takes time roughly 5:6 to compute a reciprocal; roughly 8:2 to compute a quotient or a logarithm; roughly 6:5 to compute a square root; r ..."
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Cited by 32 (6 self)
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This paper speeds up Brent's algorithms for various highprecision computations in the power series ring C[[t]]. If it takes time 3 to compute a product then it takes time roughly 5:6 to compute a reciprocal; roughly 8:2 to compute a quotient or a logarithm; roughly 6:5 to compute a square root; roughly 9 to compute both a square root and a reciprocal square root; and roughly 10:4 to compute an exponential. The same ideas apply to approximate computations in R, Q p, etc.
FloatingPoint Arithmetic And Message Authentication
, 2000
"... There is a wellknown class of message authentication systems guaranteeing that attackers will have a negligible chance of successfully forging a message. This paper shows how one of these systems can hash messages at extremely high speed  much more quickly than previous systems at the same securi ..."
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Cited by 28 (8 self)
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There is a wellknown class of message authentication systems guaranteeing that attackers will have a negligible chance of successfully forging a message. This paper shows how one of these systems can hash messages at extremely high speed  much more quickly than previous systems at the same security level  using IEEE floatingpoint arithmetic. This paper also presents a survey of the literature in a unified mathematical framework.
Fast Multiplication And Its Applications
"... This survey explains how some useful arithmetic operations can be sped up from quadratic time to essentially linear time. ..."
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Cited by 20 (4 self)
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This survey explains how some useful arithmetic operations can be sped up from quadratic time to essentially linear time.
Generalizations of the Karatsuba Algorithm for Efficient Implementations
 Department of
, 2006
"... In this work we generalize the classical Karatsuba Algorithm (KA) for polynomial multiplication to (i) polynomials of arbitrary degree and (ii) recursive use. We determine exact complexity expressions for the KA and focus on how to use it with the least number of operations. We develop a rule for th ..."
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Cited by 19 (0 self)
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In this work we generalize the classical Karatsuba Algorithm (KA) for polynomial multiplication to (i) polynomials of arbitrary degree and (ii) recursive use. We determine exact complexity expressions for the KA and focus on how to use it with the least number of operations. We develop a rule for the optimum order of steps if the KA is used recursively. We show how the usage of dummy coe#cients may improve performance. Finally we provide detailed information on how to use the KA with least cost, and also provide tables that describe the best possible usage of the KA for polynomials up to a degree of 127. Our results are especially useful for e#cient implementations of cryptographic and coding schemes over fixedsize fields like GF (p ).
Composing Power Series over a Finite Ring in Essentially Linear Time
, 1998
"... This paper presents an algorithm that computes the first n terms of the composition u(v), given the first n terms of u and v,inn ring operations. The algorithm is very fast in practice when R has small characteristic ..."
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Cited by 14 (0 self)
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This paper presents an algorithm that computes the first n terms of the composition u(v), given the first n terms of u and v,inn ring operations. The algorithm is very fast in practice when R has small characteristic
A GMPbased implementation of SchönhageStrassen’s large integer multiplication algorithm
 In Proceedings of ISSAC’07
, 2007
"... Abstract. SchönhageStrassen’s algorithm is one of the best known algorithms for multiplying large integers. Implementing it efficiently is of utmost importance, since many other algorithms rely on it as a subroutine. We present here an improved implementation, based on the one distributed within th ..."
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Cited by 12 (4 self)
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Abstract. SchönhageStrassen’s algorithm is one of the best known algorithms for multiplying large integers. Implementing it efficiently is of utmost importance, since many other algorithms rely on it as a subroutine. We present here an improved implementation, based on the one distributed within the GMP library. The following ideas and techniques were used or tried: faster arithmetic modulo 2 n + 1, improved cache locality, Mersenne transforms, Chinese Remainder Reconstruction, the √ 2 trick, Harley’s and Granlund’s tricks, improved tuning. We also discuss some ideas we plan to try in the future.
Asymmetric squaring formulae
, 2006
"... We present efficient squaring formulae based on the ToomCook multiplication algorithm. The latter always requires at least one nontrivial constant division in the interpolation step. We show such nontrivial divisions are not needed in the case two operands are equal for three, four and fiveway s ..."
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Cited by 11 (0 self)
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We present efficient squaring formulae based on the ToomCook multiplication algorithm. The latter always requires at least one nontrivial constant division in the interpolation step. We show such nontrivial divisions are not needed in the case two operands are equal for three, four and fiveway squarings. Our analysis shows that our 3way squaring algorithms have much less overhead than the best known 3way ToomCook algorithm. Our experimental results show that one of our new 3way squaring methods performs faster than mpz_mul() in GNU multiple precision library (GMP) for squaring integers of approximately 2400–6700 bits on Pentium IV Prescott 3.2GHz. For squaring in Z[x], our 3way squaring algorithms are much superior to other known squaring algorithms for small input size. In addition, we present 4way and 5way squaring formulae which do not require any constant divisions by integers other than a power of 2. Under some reasonable assumptions, our 5way squaring formula is faster than the recently proposed Montgomery’s 5way Karatsubalike formulae. Keywords: Squaring, Karatsuba algorithm, Toom
Faster Square Roots in Annoying Finite Fields
"... Let q be an odd prime number. There are several methods known to compute square roots in Z=q: the quadraticextension methods of Legendre, Pocklington, Cipolla, Lehmer, et al., and the discretelogarithm methods of Tonelli, Shanks, et al. The quadraticextension methods use (3 + o(1)) lg q multiplic ..."
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Cited by 10 (0 self)
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Let q be an odd prime number. There are several methods known to compute square roots in Z=q: the quadraticextension methods of Legendre, Pocklington, Cipolla, Lehmer, et al., and the discretelogarithm methods of Tonelli, Shanks, et al. The quadraticextension methods use (3 + o(1)) lg q multiplications and, on average, 2 + o(1) Jacobisymbol computations mod q. The discretelogarithm methods use only (1 + o(1)) lg q multiplications, after an easy precomputation of one element of Z=q, if ord2 (q 1) 2 o( p lg q). This paper presents an algorithm that uses only (1 + o(1)) lg q multiplications, after an easy precomputation of (lg q) O(1) elements of Z=q, if ord2 (q 1) 2 o( p lg q lg lg q). For example, the new algorithm can compute square roots in Z=q for q = 2 224 2 96 + 1 using 364 multiplications in Z=q and 1024 precomputed elements of Z=q. The same technique speeds up the SilverPohligHellman algorithm for computing discrete logarithms in any cyclic group of smooth order.
Generalizations of the Karatsuba Algorithm for Polynomial Multiplication
"... In this work we generalize the classical Karatsuba Algorithm (KA) for polynomial multiplication to (i) polynomials of arbitrary degree and (ii) recursive use. We determine exact complexity expressions for the KA and focus on how to use it with the least number of operations. We develop a rule for th ..."
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Cited by 4 (1 self)
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In this work we generalize the classical Karatsuba Algorithm (KA) for polynomial multiplication to (i) polynomials of arbitrary degree and (ii) recursive use. We determine exact complexity expressions for the KA and focus on how to use it with the least number of operations. We develop a rule for the optimum order of steps if the KA is used recursively. We show how the usage of dummy coefficients may improve performance. Finally we provide detailed information on how to use the KA with least cost, and also provide tables that describe the best possible usage of the KA for polynomials up to a degree of 127.
Cryptography meets voting
, 2005
"... We survey the contributions of the entire theoretical computer science/cryptography community during 19752002 that impact the question of how to run verifiable elections with secret ballots. The approach based on homomorphic encryptions is the most successful; one such scheme is sketched in detail ..."
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Cited by 4 (0 self)
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We survey the contributions of the entire theoretical computer science/cryptography community during 19752002 that impact the question of how to run verifiable elections with secret ballots. The approach based on homomorphic encryptions is the most successful; one such scheme is sketched in detail and argued to be feasible to implement. It is explained precisely what these ideas accomplish but also what they do not accomplish, and a short history of election fraud throughout history is included.