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Fast Integer Multiplication using Modular Arithmetic [article]

Anindya De, Piyush P Kurur, Chandan Saha, Ramprasad Saptharishi
2008 arXiv   pre-print
Thus, we show that the two seemingly different approaches to integer multiplication, modular and complex arithmetic, are similar.  ...  We give an O(N· N· 2^O(^*N)) algorithm for multiplying two N-bit integers that improves the O(N· N· N) algorithm by Schönhage-Strassen. Both these algorithms use modular arithmetic.  ...  Schönhage and Strassen introduced two seemingly different approaches to integer multiplication -using complex and modular arithmetic.  ... 
arXiv:0801.1416v3 fatcat:7vvehkcxmnferjdimsqdz2s2gi

Fast Integer Multiplication Using Modular Arithmetic

Anindya De, Piyush P. Kurur, Chandan Saha, Ramprasad Saptharishi
2013 SIAM journal on computing (Print)  
We give an N · log N · 2 O(log * N ) time algorithm to multiply two N -bit integers that uses modular arithmetic for intermediate computations instead of arithmetic over complex numbers as in Fürer's algorithm  ...  The previous best algorithm using modular arithmetic (by Schönhage and Strassen) has complexity O(N · log N · log log N ).  ...  The Motivation Schönhage and Strassen introduced two seemingly different approaches to integer multiplication -using complex and modular arithmetic.  ... 
doi:10.1137/100811167 fatcat:7jy5vrfat5gmjbuuhfeu3cm5xu

Fast integer multiplication using modular arithmetic

Anindya De, Piyush P. Kurur, Chandan Saha, Ramprasad Saptharishi
2008 Proceedings of the fourtieth annual ACM symposium on Theory of computing - STOC 08  
We give an N · log N · 2 O(log * N ) time algorithm to multiply two N -bit integers that uses modular arithmetic for intermediate computations instead of arithmetic over complex numbers as in Fürer's algorithm  ...  The previous best algorithm using modular arithmetic (by Schönhage and Strassen) has complexity O(N · log N · log log N ).  ...  The Motivation Schönhage and Strassen introduced two seemingly different approaches to integer multiplication -using complex and modular arithmetic.  ... 
doi:10.1145/1374376.1374447 dblp:conf/stoc/DeKSS08 fatcat:y2dslsj3jfewjav4ulseq4hm3u

Automatic Generation of Vectorized Montgomery Algorithm [article]

Lingchuan Meng
2016 arXiv   pre-print
Modular arithmetic is widely used in crytography and symbolic computation.  ...  This paper presents a vectorized Montgomery algorithm for modular multiplication, the key to fast modular arithmetic, that fully utilizes the SIMD instructions.  ...  with integer blend Fast Algorithms In this section, we review the fast algorithms for modular arithmetic operations, with a particular focus on the pivotal modular multiplication, since the modular  ... 
arXiv:1609.00999v1 fatcat:nnenh5jk6rg6vmtwrc26bzdpki

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.  ...  Such representations lead to fast modular reduction which uses only a few integer additions and subtractions.  ... 
doi:10.1007/978-3-540-24654-1_24 fatcat:nbfyp7dmrrfatagquqtvg6k4ae

Optimal extension fields for fast arithmetic in public-key algorithms [chapter]

Daniel V. Bailey, Christof Paar
1998 Lecture Notes in Computer Science  
This contribution introduces a class of Galois eld used to achieve fast nite eld arithmetic which we call an Optimal Extension Field (OEF).  ...  Our construction employs well-known techniques for fast nite eld arithmetic which fully exploit the fast integer arithmetic found on these processors.  ...  It is well known that fast modular reduction is possible with moduli of the form 2 n c, where c is a \small" integer. Integers of this form allow modular reduction without division.  ... 
doi:10.1007/bfb0055748 fatcat:7fclfpqsyrej3jqy4h6excogdi

A Review on Fast FPGA Development of RSD based ECC Processor

V. Surega
2018 International Journal of Trend in Scientific Research and Development  
Furthermore, an pwerfull smodular adder without comparison and a highthroughput modular divider, which results in a short datapath for maximized frequency, are implemented.  ...  The processor employs extensive pipelining techniques for Karatsuba-Ofman method to achieve high throughput multiplication.  ...  Left-right point multiplication method: The RSD delgation, first popularized by Avizienisis a carry free arithmetic where integers are expressed by the difference of two other integers.  ... 
doi:10.31142/ijtsrd7166 fatcat:fak5lseqizbi7okb24firkhqay

Parallel computation of residue number system

C.C. Chang, Y.T. Kuo, Y.P. Lai
2006 2006 International Conference on Computing & Informatics  
Some recently proposed modular arithmetic operations are stated and employed to accelerate the computation.  ...  Chinese remainder theorem (CRT), an old and famous theorem, is widely used in many modern computer applications.  ...  Montgomery Reduction and the Fast Modular Arithmetic The division operation is intuitionally necessary for modular arithmetic. However, a division is much more expansive than a multiplication.  ... 
doi:10.1109/icoci.2006.5276434 fatcat:5vpjn3m6njdyvo4hquftq55qzu

CNOT-count optimized quantum circuit of the Shor's algorithm [article]

Xia Liu, Huan Yang, Li Yang
2021 arXiv   pre-print
We present improved quantum circuit for modular exponentiation of a constant, which is the most expensive operation in Shor's algorithm for integer factorization.  ...  First, we give the implementation of basic arithmetic with known lowest number of CNOT gate and the construction of improved modular exponentiation of a constant by accumulating intermediate date and windowing  ...  Roetteler et al.[22] showed the quantum cir- cuits of two approaches to compute modular multiplication of two factors both in the form of quantum state: Fast modular multiplication[24] and Montgomery modular  ... 
arXiv:2112.11358v1 fatcat:yzzjnpv6e5dpthmkjn6hkg7uni

Special issue in honor of Peter Lawrence Montgomery

Francisco Rodríguez-Henríquez, Erkay Savaş
2018 Journal of Cryptographic Engineering  
Cheung present a tutorial on spectral arithmetic using Montgomery modular multiplication.  ...  Such algorithms require integer and/or modular arithmetic on large integer operands, and this means that the computational efficiency of public-key cryptography heavily depends on the computational cost  ... 
doi:10.1007/s13389-017-0168-3 fatcat:4a5fgu6bezepjna4nd6lodvkgy

Modular arithmetic and finite field theory

E. Horowitz
1971 Proceedings of the second ACM symposium on Symbolic and algebraic manipulation - SYMSAC '71  
A second use of modular arithmetic has been in the area of polynomial factorization over the field of rationals. However, the advantage gained here is not the ability for fast multiplication.  ...  for constructing fast algorithms. Related to the theory of modular arithmetic is the theory of finite fields.  ... 
doi:10.1145/800204.806287 fatcat:dhyhlimhefcshjsgacfnbmptie

Efficient FPGA-based ECDSA Verification Engine for Permissioned Blockchains [article]

Rashmi Agrawal, Ji Yang, Haris Javaid
2021 arXiv   pre-print
We propose several optimizations for modular arithmetic (e.g., custom multipliers and fast modular reduction) and point arithmetic (e.g., reduced number of point double and addition operations, and optimal  ...  Based on these optimized modular and point arithmetic modules, we propose an ECDSA verification engine that can be used by any application for fast verification of ECDSA signatures.  ...  8: end if FPGA, we implement all modular arithmetic modules using multi- 9: Return(𝑐) word integer arithmetic [12].  ... 
arXiv:2112.02229v1 fatcat:r3bw4kn4tnhzvhh3cjqugnj65m

Low-Power Elliptic Curve Cryptography Using Scaled Modular Arithmetic [chapter]

E. Öztürk, B. Sunar, E. Savaş
2004 Lecture Notes in Computer Science  
The scaling technique may be used to improve multiplication and inversion in finite fields. We present an efficient inversion algorithm that utilizes the structure of scaled modulus.  ...  We introduce new modulus scaling techniques for transforming a class of primes into special forms which enables efficient arithmetic.  ...  To alleviate the reduction problem in integer modular multiplications, Crandall proposed [3] using special primes, primes of the form p = 2 k − u, where u is a small integer constant.  ... 
doi:10.1007/978-3-540-28632-5_7 fatcat:m434epm46bgxlmdp7it4mogxdy

Computing A*B (mod N) efficiently in ANSI C

Henry G. Baker
1992 SIGPLAN notices  
The modular product computation A*B (mod N) is a bottleneck for some public-key encryption algorithms, as well as many exact computations implemented using the Chinese Remainder Theorem.  ...  INTRODUCTION Many exact integer computations, rather than being performed using multiple-precision arithmetic, are performed instead over a number of single-precision modular rings or fields, with the  ...  However, since 0≤R<N<W/2<W, we can more efficiently calculate R using single-precision ANSI C unsigned (modular) arithmetic.  ... 
doi:10.1145/130722.130735 fatcat:5rmmpzh3g5fzjlbt36i6rug5ou

Optimal Extension Fields for XTR [chapter]

Dong-Guk Han, Ki Soon Yoon, Young-Ho Park, Chang Han Kim, Jongin Lim
2003 Lecture Notes in Computer Science  
in GF (p 2m ) to achieve fast finite field arithmetic in GF (p 2m ).  ...  In order to select such fields, we introduce a new notion of Generalized Optimal Extension Fields(GOEFs) and suggest a condition of prime p, a defining polynomial of GF (p 2m ) and a fast method of multiplication  ...  Fast Subfield Multiplication with Modular Reduction In general, fast subfield multiplication is essential for fast multiplication in GF (p m ).  ... 
doi:10.1007/3-540-36492-7_24 fatcat:qn2xh5z3h5fs7bjgk4pgd3u3uu
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