Fast Integer Multiplication Using Modular Arithmetic

De, Anindya; Kurur, Piyush P.; Saha, Chandan; Saptharishi, Ramprasad

doi:10.1137/100811167

Cited by 40 publications

(41 citation statements)

References 7 publications

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“…By using fast Fourier transforms, M (n) = O( n w log( n w ) log log( n w )) [24]. The best known bound is [7,12]. It is conjectured that M (n) = Ω( n w log( n w )) [24].…”

Section: Bibliographymentioning

confidence: 99%

Exact and Efficient Generation of Geometric Random Variates and Random Graphs

Bringmann

Friedrich

2013

Automata, Languages, and Programming

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Abstract. The standard algorithm for fast generation of Erdős-Rényi random graphs only works in the Real RAM model. The critical point is the generation of geometric random variates Geo(p), for which there is no algorithm that is both exact and efficient in any bounded precision machine model. For a RAM model with word size w = Ω(log log(1/p)), we show that this is possible and present an exact algorithm for sampling Geo(p) in optimal expected time O(1 + log(1/p)/w). We also give an exact algorithm for sampling min{n, Geo(p)} in optimal expected time O(1+log(min{1/p, n})/w). This yields a new exact algorithm for sampling Erdős-Rényi and Chung-Lu random graphs of n vertices and m (expected) edges in optimal expected runtime O(n + m) on a RAM with word size w = Θ(log n).

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“…By using fast Fourier transforms, M (n) = O( n w log( n w ) log log( n w )) [24]. The best known bound is [7,12]. It is conjectured that M (n) = Ω( n w log( n w )) [24].…”

Section: Bibliographymentioning

confidence: 99%

Exact and Efficient Generation of Geometric Random Variates and Random Graphs

Bringmann

Friedrich

2013

Automata, Languages, and Programming

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“…Shortly after Fürer's algorithm appeared, De et al [15] presented a variant based on modular arithmetic that also achieves the complexity bound I(n) = O(n log n K log n ) for some K > 1. Roughly speaking, they replace the coecient ring C with the eld Q p of p-adic numbers, for a suitable prime p. In this context, working to nite precision means performing computations in Z/ p Z, where > 1 is a precision parameter.…”

Section: Fast Multiplication Using Modular Arithmeticmentioning

confidence: 99%

Even faster integer multiplication

Harvey

Hoeven

Lecerf

2016

Journal of Complexity

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We give a new proof of Fürer's bound for the cost of multiplying n-bit integers in the bit complexity model. Unlike Fürer, our method does not require constructing special coecient rings with fast roots of unity. Moreover, we prove the more explicit bound O(n logn K log n ) with K = 8. We show that an optimised variant of Fürer's algorithm achieves only K = 16, suggesting that the new algorithm is faster than Fürer's by a factor of 2 log n . Assuming standard conjectures about the distribution of Mersenne primes, we give yet another algorithm that achieves K = 4.

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“…Then the maximal coefficient in any polynomial -the value of A -can be upper-bounded by nc 2 max + c max . Further, for integer matrices, and only for integer matrices, for any i, the absolute value of the leading coefficient in front of the highest power of r in P i (r) can be lower-bounded by 1 since the leading coefficient must be an integer.…”

Section: Preliminariesmentioning

confidence: 99%

“…Let M (q) be the complexity of multiplying q-bit numbers on a log-cost RAM. The best known upper bound M (q) = log q 2 Θ(log * q) is due to Fürer [12] and De, Saha, Kurur and Saptharishi [2]. When measured in bit operations, O(log 3 p 2 Θ(log * q) ) steps suffice to check a proof of p s primality in Step 4.…”

Section: Non-deterministic Multiplication Of Real Matricesmentioning

confidence: 99%

Fast Nondeterministic Matrix Multiplication via Derandomization of Freivalds’ Algorithm

Wiedermann

2014

Advanced Information Systems Engineering

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We design two nondeterministic algorithms for matrix multiplication. Both algorithms are based on derandomization of Freivalds' algorithm for verification of matrix products. The first algorithm works with real numbers and its time complexity on Real RAMs is O(n 2 log n). The second one is of the same complexity, works with integer matrices on a unit cost RAM with numbers whose size is proportional to the size of the largest entry in the underlying matrices. Our algorithms bring new ideas into the design of matrix multiplication algorithms and open new avenues for their further development. The results pose exciting questions concerning the relation of the complexity of deterministic versus nondeterministic algorithms for matrix multiplication, and complexity of integer versus real matrices multiplication.

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Fast Integer Multiplication Using Modular Arithmetic

Cited by 40 publications

References 7 publications

Exact and Efficient Generation of Geometric Random Variates and Random Graphs

Exact and Efficient Generation of Geometric Random Variates and Random Graphs

Even faster integer multiplication

Fast Nondeterministic Matrix Multiplication via Derandomization of Freivalds’ Algorithm

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