Fast Polynomials Multiplication Using FFT

Transcription

1 Li Chen lichen.xd at gmail.com Xidian University January 17, 2014

2 Outline 1 Discrete Fourier Transform (DFT) 2 Discrete Convolution 3 Fast Fourier Transform (FFT) 4 umber Theoretic Transform (TT) 5 More Optimize & Application Scenario Li Chen Xidian University 2/18

3 References [1] Doz Dr A Schönhage and Volker Strassen. Schnelle multiplikation grosser zahlen. Computing, 7(3-4): , [2] Martin Fürer. Faster integer multiplication. In Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pages ACM, [3] Martin Fürer. Faster integer multiplication. SIAM Journal on Computing, 39(3): , [4] Anindya De, Piyush P Kurur, Chandan Saha, and Ramprasad Saptharishi. Fast integer multiplication using modular arithmetic. In Proceedings of the 40th annual ACM symposium on Theory of computing, pages ACM, [5] Anindya De, Piyush P Kurur, Chandan Saha, and Ramprasad Saptharishi. Fast integer multiplication using modular arithmetic. SIAM Journal on Computing, 42(2): , Claim: Our slides are based on reference [1], [2], [3], [4], [5]. Li Chen Xidian University 3/18

4 Discrete Fourier Transform 1 Discrete Fourier Transform (DFT) Definition 1.1 (Discrete Fourier Transform (DFT)) Let X = (x 0, x 1,, x 1 ) be a -length sequence, the discrete fourier transform of X is defined as a -length sequences F (X) = (f 1, f 2,, f ), where 1 f k = x nω nk, k = 0, 1,, 1. n=0 where ω is the a principal th root of unity in a ring R (with unity). Li Chen Xidian University 4/18

5 Discrete Fourier Transform ote1: Let R is a ring with unity, α R is called a principal th root of unity, if α = 1 and 1 α kn = 0, 1 k <. (1) n=0 ote2: If R is a integral domain, it is sufficient to choose α as a primitive th root of unity, which replaces the condition (1) by α k 1, 1 k <. (2) Proof: Take β = α k, 1 k <, since α = 1, β = α k = 1, thus we have β 1 1 = (β 1) β n 1 = 0, which implies β n = 0. n=1 n=1 Li Chen Xidian University 5/18

6 Discrete Fourier Transform Theorem 1.2 (Inverse of Discrete Fourier Transform (IDFT)) Let X = (x 0, x 1,, x 1 ) be a -length sequence, F (X) = (f 1, f 2,, f ) is the discrete fourier transform of X, then x n = 1 1 k=0 f k ω kn, k = 0, 1,, 1. where ω is the a principal -th root of unity in a ring R, and 1 is multiplicative inverse of in R (if this inverse does not exist, the DFT cannot be inverted). Proof Li Chen Xidian University 6/18

7 Discrete Fourier Transform 1 1 f k ω kn k=0 = 1 = 1 = 1 1 ( 1 k=0 1 1 k=0 1 x j ) x j ω jk ω kn x j ω (j n)k 1 k=0 ω (j n)k = 1 1 xn ω = x n + 0 k=0 j n x j 1 k=0 ω (j n)k Li Chen Xidian University 7/18

8 Discrete Convolution 2 Discrete Convolution Definition 2.1 (Acyclic Convolution) Let X = (x 0, x 1,, x 1 ), Y = (y 0, y 1,, y 1 ) be two -length sequences, then the acyclic convolution or linear convolution is defined as a length sequences Z = (z 0, z 1,, z 2( 1) ), where z i = j+k=i j [0, 1] k [0, 1] x j y k, i = 0, 1,, 2( 1). And the cyclic convolution or wrapped convolution is defined as a -length sequences Z = ( z 0, z 1,, z 1 ), where or equivalent, z i = z i + z i+, i = 0, 1,, 1, (pad z 2 1 = 0). 1 z i = x j y i j, i = 0, 1,, 1, (y k = y k, k = 0, 1,, 1). Li Chen Xidian University 8/18

9 Discrete Convolution Definition 2.2 (egacyclic Convolution) Let X = (x 0,, x 1 ), Y = (y 0,, y 1 ) be two -length sequences, then the negacyclic convolution is defined as a -length sequences Z = ( z 0, z 1,, z 1 ), where or equivalent, z i = z i z i+, i = 0, 1,, 1, (pad z 2 1 = 0). 1 z i = x j y i j, i = 0, 1,, 1, (y k = y k, k = 0, 1,, 1). Li Chen Xidian University 9/18

10 Discrete Convolution Theorem 2.3 (Convolution Theorem) Let X = (x 0,, x 1 ), Y = (y 0,, y 1 ) be two -length sequences, take F ( ) as the discrete fourier transform on a -length sequences ( ), then F (X Y ) = F (X) F (Y ) where indicates cyclic convolution, and indicates component-wise multiplication. Proof Let z 0,, z 1 be the cyclic convolution of X, Y, i.e., 1 z j = x j y n j Li Chen Xidian University 10/18

11 Discrete Convolution and let F (x Y ) = (f 0, f 1,, f 1 ). For each k [0, 1], f k = 1 z nω nk = n=0 1 ( 1 = n=0 1 ( 1 = n=0 1 = x j ω jk 1 = 1 ( 1 n=0 x j y n j ) ω jk+(n j)k x j y n j ) ω nk ) x j ω jk y n jω (n j)k 1 1 n=0 1 x j ω jk x j y n j ω (n j)k y j ω jk Li Chen Xidian University 11/18

12 Discrete Convolution Lemma 2.4 Let X = (x 0,, x 1 ), Y = (y 0,, y 1 ) be two -length sequences, let X be a 2 1-length sequences (x 0,, x 1, 0,, 0), i.e. padding X with 0, similarly, let Y be a 2 1-length sequences (y 0,, y 1, 0,, 0), i.e. padding Y with 0, then AcyclicConvolution(X, Y ) = CyclicConvolution(X, Y ) Proof Li Chen Xidian University 12/18

13 Fast Fourier Transform 3 Fast Fourier Transform (FFT) Cooley-Tukey Algorithm (radix-2) Let X = (x 0,, x 1 ) be a sequence with length = 2M, and F (X) = (f 0,, f 1 ) be the discrete fourier transform of X, i.e. 1 f k = x nω nk n=0 M 1 = m=0 M 1 = m=0 M 1 x 2mω 2mk + x 2mω mk 2 + ω k m=0 x 2m+1ω (2m+1)k M 1 m=0 x 2m+1ω mk 2 ote: ω is a principle th root of unity ω 2 is a principle 2 th root of unity. Li Chen Xidian University 13/18

14 Fast Fourier Transform M 1 Thus, let E k = m=0 x 2mω mk M 1, and D k = 2 m=0 x 2m+1ω mk, we can write f k = E k + ω k D k. 2 Since E k+ = E k, D 2 k+ 2 for 0 k M 1 = D k, k = 0,, M 1, and ω k+ 2 = ω k, finally we have, f k+ 2 f k = E k + ω k D k = E k ω k D k And ow we can view E k as the kth term of the discrete fourier transform on the 2 -length sequence (x 0, x 2,, x 2 ), and view D k as the kth term of the discrete fourier transform on the 2 -length sequence (x1, x3,, x 1). Suppose = 2 m, by reduction, the asymptotic complexity of FFT is: O( log ) Li Chen Xidian University 14/18

15 umber Theoretic Transform 4 umber Theoretic Transform (TT) The number theoretic transform (TT) is obtained by specializing the discrete Fourier transform on a special ring Z p, the integers modulo a prime p, which is a finite field. Lemma 4.1 Given a prime integer p, there exist a primitive nth root of unity in Z p if and only if n p 1. Li Chen Xidian University 15/18

16 umber Theoretic Transform Li Chen Xidian University 16/18

17 umber Theoretic Transform 5 More Optimize & Application Scenario Internal Discussed... Li Chen Xidian University 17/18

18 Thanks! & Questions?