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Even faster integer multiplication [article]

David Harvey, Joris van der Hoeven, Grégoire Lecerf
2014 arXiv   pre-print
We give a new proof of Fürer's bound for the cost of multiplying n-bit integers in the bit complexity model.  ...  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).  ...  David Harvey, Joris van der Hoeven, Grégoire Lecerf Even faster integer multiplication  ... 
arXiv:1407.3360v1 fatcat:ldl7p5e6gvfvbdoy4pbe4tftgy

Even faster integer multiplication

David Harvey, Joris van der Hoeven, Grégoire Lecerf
2016 Journal of Complexity  
We give a new proof of Fürer's bound for the cost of multiplying n-bit integers in the bit complexity model.  ...  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 .  ...  Even faster multiplication In this section, we present an optimised version of the new integer multiplication algorithm.  ... 
doi:10.1016/j.jco.2016.03.001 fatcat:maj3qbr7mrdi3ovtok2pglauei

Montgomery Multiplier for Faster Cryptosystems

Nitha Thampi, Meenu Elizabath Jose
2016 Procedia Technology - Elsevier  
This paper presents a 32-bit implementation of a Faster Montgomery algorithm for performing modular multiplication.  ...  Many hardware and software implementations for faster modular multiplication have been proposed, Montgomery Multiplication Algorithm is recognized as the most efficient among these.  ...  Modular Multiplication Modular multiplication problem is defined as the computation of P = A × B (mod m), given the integers A, B and m.  ... 
doi:10.1016/j.protcy.2016.08.123 fatcat:dlvldhtomzhgnij2utaot5gmrm

Fast Library for Number Theory: An Introduction [chapter]

William B. Hart
2010 Lecture Notes in Computer Science  
Magma is already up to five times faster than NTL even for polynomial multiplication. The Pari library is not usually asymptotically fast and NTL is not threadsafe.  ...  This implementation is often faster than GMP [5] (which is used for smaller multiplications), by as much as 30%.  ... 
doi:10.1007/978-3-642-15582-6_18 fatcat:i2legnxi3fe2fjjt2msbu5c3hm

More Generalized Mersenne Numbers [chapter]

Jaewook Chung, Anwar Hasan
2004 Lecture Notes in Computer Science  
We also show that it is possible to perform long integer modular arithmetic without using multiple precision operations when t is chosen properly.  ...  It is shown that such p's lead to fast modular reduction methods which use only a few integer additions and subtractions. We further generalize this idea by allowing any integer for t.  ...  Even without the KOA, our methods are still faster than the classical and the Montgomery algorithms.  ... 
doi:10.1007/978-3-540-24654-1_24 fatcat:nbfyp7dmrrfatagquqtvg6k4ae

Flexural reticular pigmentation and pruritic papules in a 33-year-old woman

A. C. Sherley-Dale, R. A. Carr, J. E. Gach
2011 Clincal and Experimental Dermatology  
Magma is already up to five times faster than NTL even for polynomial multiplication. The Pari library is not usually asymptotically fast and NTL is not threadsafe.  ...  This implementation is often faster than GMP [5] (which is used for smaller multiplications), by as much as 30%.  ... 
doi:10.1111/j.1365-2230.2010.03913.x pmid:21323938 fatcat:o2wi6ja7ubcfrpolzdlqugku34

High-Performance Modular Multiplication on the Cell Processor [chapter]

Joppe W. Bos
2010 Lecture Notes in Computer Science  
Typically, the multiplication and special reduction are performed sequentially. For the separated multiplication step we consider schoolbook and Karatsuba multiplication [18] techniques.  ...  multiple computations concurrently; e.g. graphics processing units.  ...  The SPEs are equipped with a 4-way SIMD multiplier (even instruction) which can compute four 16bit integer multiplications simultaneously per clock cycle.  ... 
doi:10.1007/978-3-642-13797-6_2 fatcat:sw6allqzozdpjn7qh3lgyrjhfi

Strongly Universal String Hashing is Fast

D. Lemire, O. Kaser
2013 Computer journal  
Moreover, conventional wisdom is that hash functions with fewer multiplications are faster. Yet we find that they may fail to be faster due to operation pipelining.  ...  Our tests include hash functions designed for processors with the Carry-Less Multiplication (CLMUL) instruction set. We also prove, using accessible proofs, the strong universality of our families.  ...  Yet another alternative, Multilinear (2-by-2), was slightly faster (≈ 15%) for 32-bit hashing on the mobile ARM-based processors even though it requires twice as many multiplications as Multilinear-HM.  ... 
doi:10.1093/comjnl/bxt070 fatcat:ihslalyvrbc63njznfrhcbtxgy

Fast Implementation of 4-bit Convolutional Neural Networks for Mobile Devices [article]

Anton Trusov, Elena Limonova, Dmitry Slugin, Dmitry Nikolaev, Vladimir V. Arlazarov
2020 arXiv   pre-print
It shows 2.9 times speedup compared to standard floating-point multiplication and is 1.5 times faster than 8-bit quantized one.  ...  However, their advantages are apparent for FPGA and ASIC devices, while general-purpose processor architectures are not always able to perform low-bit integer computations efficiently.  ...  We can see that multiplication of 32-bit integers and floating point values takes almost the same time, multiplication of 4-bit unsigned integers works approximately 2.4 times faster than that of 32-bit  ... 
arXiv:2009.06488v2 fatcat:pojrbhq6cnearmrheyxi7tmxni

Software Optimization of Decorrelation Module [chapter]

Fabrice Noilhan
2000 Lecture Notes in Computer Science  
This paper investigates software optimization of special multiplication. In particular we concentrate on ax + b mod 2 64 + 13 mod 2 64 which is the bottleneck operation in the DFC cipher.  ...  These methods are faster than using four integer multiplications. The only problem is to convert from integers to floats.  ...  Even hand-written bytecode, which should produce faster code, does not have a noticeable speedup.  ... 
doi:10.1007/3-540-46513-8_13 fatcat:iwfephzrxfh5rbrhcl3wbxe76m

Faster remainder by direct computation: Applications to compilers and software libraries

Daniel Lemire, Owen Kaser, Nathan Kurz
2019 Software, Practice & Experience  
On common processors, integer multiplication is many times faster than integer division.  ...  If the divisor is known in advance---or if repeated integer divisions will be performed with the same divisor---it can be beneficial to substitute a less costly multiplication for an expensive division  ...  Given an integer numerator and an integer divisor , the quotient ( ) and the remainder ( ) are always integers even when the fraction ∕ is not an integer.  ... 
doi:10.1002/spe.2689 fatcat:ujlp4nxjzjhctgmq32oreanpsa

Number Theoretic Transforms for Fast Digital Computation

Salila Hegde,, Rohini Nagapadma
2019 International Journal of Research in Advent Technology  
The aim of this study is to show that Number Theoretic Transforms (NTTs) can be really beneficial in terms of error free and faster computation.  ...  This is much simpler than complex domain multiplications; hence FNT is faster than FFT.  ...  If multiplicative inverse exists for all nonzero integers in then becomes field. [4] [5] .  ... 
doi:10.32622/ijrat.752019106 fatcat:s6crqwmhlrdftgx7gg2pjcn6na

How Fast Can We Multiply Large Integers on an Actual Computer? [article]

Martin Fürer
2014 arXiv   pre-print
We provide two complexity measures that can be used to measure the running time of algorithms to compute multiplications of long integers.  ...  The random access machine with unit or logarithmic cost is not adequate for measuring the complexity of a task like multiplication of long integers.  ...  Even without multiplication, length T (n) integers can be produced in time T (n), resulting in unrealistically cheap additions.  ... 
arXiv:1402.1811v1 fatcat:73lp23v6enevrfnyj52iy6jj5y

Efficient Big Integer Multiplication and Squaring Algorithms for Cryptographic Applications

Shahram Jahani, Azman Samsudin, Kumbakonam Govindarajan Subramanian
2014 Journal of Applied Mathematics  
The proposed multiplication algorithm is also 2.3 to 3.9 and 7 to 2.4 times faster for multiplying 32-bit and 8-Kbit numbers, respectively.  ...  We present a modified version of the classical multiplication and squaring algorithms based on the Big-ones to improve the efficiency of big integer multiplication and squaring in number theory based cryptosystems  ...  The proposed multiplication algorithm is about 2.3 times faster than CM MUL for multiplying 32-bit numbers and about 3 times faster for multiplying 64-bit numbers.  ... 
doi:10.1155/2014/107109 fatcat:xnciccnokvgtliq5sk2n74kw7e

Efficient Multiple-Precision Evaluation of Elementary Functions

David M. Smith
1989 Mathematics of Computation  
Because this improvement is fairly simple, the resulting algorithms are faster than those in [3] even at low precision. Exponential and Related Functions.  ...  . , Sjpl requires one multiplication to get the next power of xi, j divisions by an integer, and j additions.  ... 
doi:10.2307/2008657 fatcat:wqyzslhqrfa6dfvolicoh2diy4
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