Integer and polynomial multiplication
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(2) Context of today’s talk. In 1 second, we can multiply integers of 30 000 000 digits and polynomials of degree 500 000.. É Mini-course (L2, M2 & bonus) É Theoretical and practical aspects É Presentation of team research topics É Link with current research É Relation with implementations (LinBox). 2/33.
(3) Application / Motivation É Exact computation: É Many operations reduced to multiplication: exponentiation, division, pgcd, factorization, . . . É Mathematical software: GMP, Sage, Matlab, Maple, . . .. É Other domains using exact computation: É Cryptography: [Discrete logarithm computation in a 180-digit prime field, 2014] É Combinatorics, Number Theory, . . .. É Numerical computation: É Trillions of digits of π. [YEE, KONDO ’11]. É Robotics: Equilibrium of cable driven parallel robots. 3/33.
(4) From polynomial to integer multiplication To multiply. (794x2 + 983x + 523) Evaluate at x = 107. ×. (564x2 + 637x + 185). (794 · (107 )2 + 983 · 107 + 523) 79400009830000523. × (564 · (107 )2 + 637 · 107 + 185) × 56400006370000185 = 4478161060190106803305150060096755. "Interpolate" at x = 107 447816x4 + 1060190x3 + 1068033x2 + 515006x + 96755 Remarks: É Technique called Kronecker substitution (1882) É 107 is the minimal power of 10 greater than all coefficients 4/33.
(5) From integer to polynomial multiplication To multiply 794983523 "Interpolation" at x = 103. ×. 564637185. (794x2 + 983x + 523). × (564x2 + 637x + 185) = 447816x4 + 1060190x3 + 1068033x2 + 515006x + 96755. Evaluate at x = 103 447816 · 1012 + 1060190 · 109 + 1068033 · 106 + 515006 · 103 + 96755 = 448877258548102755 Remarks: É Addition with carry É Base 264 instead of base 10 5/33.
(6) Outline. Polynomial multiplication algorithms: 1. 2. 3. 4.. Karatsuba Toom-Cook Fast Fourier Tranform (FFT) Truncated FFT (TFT). 6/33.
(7) Polynomial data structures É Dense polynomials = list of all coefficients. x11 + 5x10 + 9x8 + 4x7 + 7x6 + x2 + 8 1. 5. 0. 9. 4. 7. 0. 0. 0. 1. 0. 8. 7/33.
(8) Polynomial data structures É Dense polynomials = list of all coefficients É Sparse polynomials = list of all non-zero coefficients. x29 + 9x12 + 4x11 + 2x2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 4 0 0 0 0 0 0 0 0 2 0 0 29 12 11. 2. 9. 2. 1. 4. 7/33.
(9) Polynomial data structures É Dense polynomials = list of all coefficients É Sparse polynomials = list of all non-zero coefficients É Straight-line programs. X 4 +4X 3 Y +6X 2 Y 2 +4XY 3 +Y 4 +X 2 Z+2XYZ+Y 2 Z+X 2 +2XY +Y 2 +Z2 +2Z+1. 7/33.
(10) Polynomial data structures É Dense polynomials = list of all coefficients É Sparse polynomials = list of all non-zero coefficients É Straight-line programs. ((X + Y )2 )2 + (X + Y )2 · (Z + 1) + (Z + 1)2 X. Y. Z. 1. 7/33.
(11) Polynomial data structures É Dense polynomials = list of all coefficients É Sparse polynomials = list of all non-zero coefficients É Straight-line programs. 7/33.
(12) Naive polynomial multiplication Example: + a0 + a1 x + a2 x2 + a3 x3 + × = + b0 + b1 x + b2 x2 + b3 x3 + + +. ( ( ( ( ( ( (. a0 b3 a0 b2 a0 b1 a0 b0. + + +. a1 b3 a1 b2 a1 b1 a1 b0. + + +. a2 b3 a2 b2 a2 b1 a2 b0. + + +. a3 b3 a3 b2 a3 b1 a3 b0. ) ) ) ) ) ) ). x6 x5 x4 x3 x2 x1 x0. 8/33.
(13) Naive polynomial multiplication Example: + a0 + a1 x + a2 x2 + a3 x3 + × = + b0 + b1 x + b2 x2 + b3 x3 + + +. In general:. ( ( ( ( ( ( (. a0 b3 a0 b2 a0 b1 a0 b0. + + +. a1 b3 a1 b2 a1 b1 a1 b0. + + +. a2 b3 a2 b2 a2 b1 a2 b0. + + +. a3 b3 a3 b2 a3 b1 a3 b0. ) ) ) ) ) ) ). x6 x5 x4 x3 x2 x1 x0. Multiplication of degree n polynomials in O(n2 ) arithmetic operations (+, −, ×). 8/33.
(14) Naive polynomial multiplication Example: + a0 + a1 x + a2 x2 + a3 x3 + × = + b0 + b1 x + b2 x2 + b3 x3 + + +. In general:. ( ( ( ( ( ( (. a0 b3 a0 b2 a0 b1 a0 b0. + + +. a1 b3 a1 b2 a1 b1 a1 b0. + + +. a2 b3 a2 b2 a2 b1 a2 b0. + + +. a3 b3 a3 b2 a3 b1 a3 b0. ) ) ) ) ) ) ). x6 x5 x4 x3 x2 x1 x0. Multiplication of degree n polynomials in O(n2 ) arithmetic operations (+, −, ×). Lower bound: The multiplication costs at least n arith. operations. 8/33.
(15) Naive polynomial multiplication Example: + a0 + a1 x + a2 x2 + a3 x3 + × = + b0 + b1 x + b2 x2 + b3 x3 + + +. In general:. ( ( ( ( ( ( (. a0 b3 a0 b2 a0 b1 a0 b0. + + +. a1 b3 a1 b2 a1 b1 a1 b0. + + +. a2 b3 a2 b2 a2 b1 a2 b0. + + +. a3 b3 a3 b2 a3 b1 a3 b0. ) ) ) ) ) ) ). x6 x5 x4 x3 x2 x1 x0. Multiplication of degree n polynomials in O(n2 ) arithmetic operations (+, −, ×). Lower bound: The multiplication costs at least n log n arith. operations. 8/33.
(16) Naive polynomial multiplication Example: + a0 + a1 x + a2 x2 + a3 x3 + × = + b0 + b1 x + b2 x2 + b3 x3 + + +. In general:. ( ( ( ( ( ( (. a0 b3 a0 b2 a0 b1 a0 b0. + + +. a1 b3 a1 b2 a1 b1 a1 b0. + + +. a2 b3 a2 b2 a2 b1 a2 b0. + + +. a3 b3 a3 b2 a3 b1 a3 b0. ) ) ) ) ) ) ). x6 x5 x4 x3 x2 x1 x0. Multiplication of degree n polynomials in O(n2 ) arithmetic operations (+, −, ×). Lower bound: The multiplication costs at least n log n arith. operations What is the best complexity of multiplication ? 8/33.
(17) Outline. Polynomial multiplication algorithms: 1. 2. 3. 4.. Karatsuba Toom-Cook Fast Fourier Tranform (FFT) Truncated FFT (TFT). 9/33.
(18) Karatsuba. [KARATSUBA, OFMAN ’68]. (a0 + a1 x) · (b0 + b1 x) = c0 + c1 x + c2 x2 Naive algorithm: 4 multiplications c0 = a0 b0 c1 = a0 b1 + a1 b0 c =a b 2 1 1 Karatsuba: 3 multiplications by writing c0 = a0 b0 c1 = (a0 + a1 ) · (b0 + b1 ) − a0 b0 − a1 b1 c =a b 2 1 1. 10/33.
(19) Karatsuba - Divide and conquer algorithm What about multiplication a(x) · b(x) of polynomials of degree n?. 11/33.
(20) Karatsuba - Divide and conquer algorithm What about multiplication a(x) · b(x) of polynomials of degree n?. Recursive multiplication algorithm: P 1. a(x) = 0≤i<n ai xi. Split in 2 parts. 11/33.
(21) Karatsuba - Divide and conquer algorithm What about multiplication a(x) · b(x) of polynomials of degree n?. Recursive multiplication algorithm: P P 1. a(x) = 0≤i<n/2 ai xi + xn/2 0≤i<n/2 ai+n/2 xi. Split in 2 parts. 11/33.
(22) Karatsuba - Divide and conquer algorithm What about multiplication a(x) · b(x) of polynomials of degree n?. Recursive multiplication algorithm: 1. a(x) = al (x) + xn/2 ah (x). Split in 2 parts. 11/33.
(23) Karatsuba - Divide and conquer algorithm What about multiplication a(x) · b(x) of polynomials of degree n?. Recursive multiplication algorithm: 1. a(x) = al (x) + xn/2 ah (x). 2. b(x) = bl (x) + x. n/2. Split in 2 parts. bh (x). 11/33.
(24) Karatsuba - Divide and conquer algorithm What about multiplication a(x) · b(x) of polynomials of degree n?. Recursive multiplication algorithm: 1. a(x) = al (x) + xn/2 ah (x) 2. 3. 4. 5.. b(x) = bl (x) + x bh (x) cl (x) = al (x) · bl (x) cm (x) = (al + ah ) · (bl + bh ) ch (x) = ah (x) · bh (x). Split in 2 parts. n/2. Recursive call of size n/2 Recursive call of size n/2 Recursive call of size n/2. 11/33.
(25) Karatsuba - Divide and conquer algorithm What about multiplication a(x) · b(x) of polynomials of degree n?. Recursive multiplication algorithm: 1. a(x) = al (x) + xn/2 ah (x) 2. 3. 4. 5. 6.. Split in 2 parts. b(x) = bl (x) + x bh (x) cl (x) = al (x) · bl (x) Recursive call of size n/2 cm (x) = (al + ah ) · (bl + bh ) Recursive call of size n/2 ch (x) = ah (x) · bh (x) Recursive call of size n/2 return c(x) = cl (x) + (cm (x) − cl (x) − ch (x)) xn/2 + ch (x)xn n/2. 11/33.
(26) Karatsuba - Divide and conquer algorithm What about multiplication a(x) · b(x) of polynomials of degree n?. Recursive multiplication algorithm: 1. a(x) = al (x) + xn/2 ah (x) 2. 3. 4. 5. 6.. Split in 2 parts. b(x) = bl (x) + x bh (x) cl (x) = al (x) · bl (x) Recursive call of size n/2 cm (x) = (al + ah ) · (bl + bh ) Recursive call of size n/2 ch (x) = ah (x) · bh (x) Recursive call of size n/2 return c(x) = cl (x) + (cm (x) − cl (x) − ch (x)) xn/2 + ch (x)xn n/2. Remarks: É É É É. Complexity: K (n) = 3K (n/2) + O(n) = O(nlog2 (3) ) = O(n1,59 ) Karatsuba K (n) O(n2 ) naive In practice, hybrid Karatsuba / naive algorithm Need careful memory management (one memory allocation, in-place algorithms) 11/33.
(27) Outline. Polynomial multiplication algorithms: 1. 2. 3. 4.. Karatsuba Toom-Cook Fast Fourier Tranform (FFT) Truncated FFT (TFT). 12/33.
(28) Karatsuba revisited. Karatsuba: ¨ a0 + a1 x b0 + b1 x yMult.. c(x) = a(x) · b(x). 13/33.
(29) Karatsuba revisited. Karatsuba: ¨ a0 + a1 x b0 + b1 x yMult.. Evaluation. −−−−−→. ¨. a(0), a(1), a(∞). b(0), b(1), b(∞). c(x) = a(x) · b(x). 13/33.
(30) Karatsuba revisited. Karatsuba: ¨ a0 + a1 x b0 + b1 x yMult.. c(x) = a(x) · b(x). Evaluation. −−−−−→. ¨. a(0), a(1), a(∞). b(0), b(1), b(∞) Pointwise y mult.. c(0) = a(0) · b(0) c(1) = a(1) · b(1) c(∞) = a(∞) · b(∞). 13/33.
(31) Karatsuba revisited. Karatsuba: ¨ a0 + a1 x b0 + b1 x yMult.. c(x) = a(x) · b(x). Evaluation. −−−−−→. Interpolation. ←−−−−−−−. ¨. a(0), a(1), a(∞). b(0), b(1), b(∞) Pointwise y mult.. c(0) = a(0) · b(0) c(1) = a(1) · b(1) c(∞) = a(∞) · b(∞). 13/33.
(32) Interpolation Interpolation: Recover the polynomial from evaluations of it.. 14/33.
(33) Interpolation Interpolation: Recover the polynomial from evaluations of it.. 14/33.
(34) Interpolation Interpolation: Recover the polynomial from evaluations of it.. 14/33.
(35) Interpolation Interpolation: Recover the polynomial from evaluations of it.. 14/33.
(36) Evaluation/interpolation algorithms Karatsuba: [KARATSUBA, OFMAN 1963] Polynomials a, b with 2 coefficients Evaluation in 3 points, e.g. 0, 1, ∞ Complexity: K (n) = 3K (dn/2e) + O(n) = O(nlog2 (3) ) = O(n1,59 ). Toom-3: [TOOM ’63] Polynomials a, b with 3 coefficients Evaluation in 5 points, e.g. −1, 0, 1, 2, ∞ Complexity: T (n) = 5K (dn/3e) + O(n) = O(nlog3 (5) ) = O(n1,47 ) Toom-Cook:. [COOK ’66]. É Generalization that reaches complexity O(n1+" ) for all " > 0 É Less and less pratical (big constant in O). 15/33.
(37) Outline. Polynomial multiplication algorithms: 1. 2. 3. 4.. Karatsuba Toom-Cook Fast Fourier Tranform (FFT) Truncated FFT (TFT). 16/33.
(38) Evaluation / Interpolation algorithms. Monomial representation ¨ a(x) degree n b(x) degree n yMult. c(x) = a(x) · b(x) of degree 2n. Evaluation. −−−−−→. Interpolation. ←−−−−−−−. Evaluation representation ¨ a(0), a(1), . . . , a(2n + 1) b(0), b(1), . . . , b(2n + 1) Pointwise mult. y. c(0) = a(0) · b(0) ... c(2n + 1) = a(2n + 1) · b(2n + 1). 17/33.
(39) Evaluation / Interpolation algorithms Karatsuba, Toom-3, Toom-Cook: Fixed size eval./interp. scheme + recursion Fast Fourier Transform: Full size eval./interp. scheme + no recursion. Monomial representation ¨ a(x) degree n b(x) degree n yMult. c(x) = a(x) · b(x) of degree 2n. Evaluation. −−−−−→. Interpolation. ←−−−−−−−. Evaluation representation ¨ a(0), a(1), . . . , a(2n + 1) b(0), b(1), . . . , b(2n + 1) Pointwise mult. y. c(0) = a(0) · b(0) ... c(2n + 1) = a(2n + 1) · b(2n + 1). 17/33.
(40) Evaluation / Interpolation algorithms Karatsuba, Toom-3, Toom-Cook: Fixed size eval./interp. scheme + recursion Fast Fourier Transform: Full size eval./interp. scheme + no recursion. Monomial representation ¨ a(x) degree n b(x) degree n yMult. c(x) = a(x) · b(x) of degree 2n. Evaluation. −−−−−→. Interpolation. ←−−−−−−−. Evaluation representation ¨ a(0), a(1), . . . , a(2n + 1) b(0), b(1), . . . , b(2n + 1) Pointwise mult. yCost : O(n). c(0) = a(0) · b(0) ... c(2n + 1) = a(2n + 1) · b(2n + 1). 17/33.
(41) Evaluation / Interpolation algorithms Karatsuba, Toom-3, Toom-Cook: Fixed size eval./interp. scheme + recursion Fast Fourier Transform: Full size eval./interp. scheme + no recursion. Monomial representation ¨ a(x) degree n b(x) degree n yMult. c(x) = a(x) · b(x) of degree 2n. Evaluation. −−−−−→ Cost?. Interpolation. ←−−−−−−− Cost?. Evaluation representation ¨ a(0), a(1), . . . , a(2n + 1) b(0), b(1), . . . , b(2n + 1) Pointwise mult. yCost : O(n). c(0) = a(0) · b(0) ... c(2n + 1) = a(2n + 1) · b(2n + 1). From now on, we will focus on the cost of evaluation / interpolation. 17/33.
(42) Evaluation / Interpolation algorithms Karatsuba, Toom-3, Toom-Cook: Fixed size eval./interp. scheme + recursion Fast Fourier Transform: Full size eval./interp. scheme + no recursion. Monomial representation ¨ a(x) degree n b(x) degree n yMult. c(x) = a(x) · b(x) of degree 2n. Evaluation. −−−−−→ Cost?. Interpolation. ←−−−−−−− Cost?. Evaluation representation ¨ a(0), a(1), . . . , a(2n + 1) b(0), b(1), . . . , b(2n + 1) Pointwise mult. yCost : O(n). c(0) = a(0) · b(0) ... c(2n + 1) = a(2n + 1) · b(2n + 1). From now on, we will focus on the cost of evaluation / interpolation. Note: Interpolation with errors. 17/33.
(43) Discrete Fourier Transform (DFT). Evaluation / interpolation is generally costly. But if evaluation points are specific, it can be very efficient: É Evaluate at ξ0 , ξ1 , ξ2 , . . . , ξn−1 É where ξ is a primitive root of unity. Discrete Fourier Transform = Evaluation on roots of unity DFTξ (a(x)) := (a(ξ0 ), . . . , a(ξn−1 )) where ξ is a n-th primitive root of unity and deg a(x) < n.. 18/33.
(44) Complex root of unity y 2iπ. ξ5 ξ. ξ. 4. ξ = e 16 ∈ C is a 16-th primitive root of unity: É ξ16 = 1. ξ3 ξ2. 6. ξ7. ξ1 2π 16. ξ8 ξ9. ξ0 ξ15. ξ10. ξ14 ξ. 11. ξ12. ξ. 13. x. É ξi 6= 1 for 0 < i < 16 Remark: É 1 = ξ0 = ξ16 = ξ32 = . . . É ξ16/2 = −1. Pros & Cons: Fast floating point arithmetic, but precision issues. 19/33.
(45) Modular root of unity Modular integers: Z/pZ if p prime (a = a + p = a + 2p = . . . modulo p) Example Z/17Z: 13. 5. ξ4. ξ5. 15. 10 ξ3 ξ2. ξ6 11. 16 ξ. 3. ξ1. ξ7. Z/17Z. 8. ξ. ξ9. 0. 1. ξ15. 14. 6 ξ. ξ = 3 ∈ Z/17Z is a 16-th primitive root of unity: É ξ16 = 1. 9. 10. ξ. 8. ξ11 7. ξ12 4. 14. 2. ξ13 12. É ξi 6= 1 for 0 < i < 16 Remark: É 1 = ξ0 = ξ16 = ξ32 = . . . É ξ16/2 = −1 20/33.
(46) Fast Fourier Transform Goal: Given a(x), compute (a(ξ0 ), . . . , a(ξn−1 )) (when n = 2k ) If a(x) = al (x) + xn/2 ah (x) then a(ξj ) = al (ξj ) + (ξj )n/2 ah (ξj ). 21/33.
(47) Fast Fourier Transform Goal: Given a(x), compute (a(ξ0 ), . . . , a(ξn−1 )) (when n = 2k ) If a(x) = al (x) + xn/2 ah (x) then a(ξj ) = al (ξj ) + (ξn/2 )j ah (ξj ). 21/33.
(48) Fast Fourier Transform Goal: Given a(x), compute (a(ξ0 ), . . . , a(ξn−1 )) (when n = 2k ) If a(x) = al (x) + xn/2 ah (x) then a(ξj ) = al (ξj ) + (−1)j ah (ξj ). 21/33.
(49) Fast Fourier Transform Goal: Given a(x), compute (a(ξ0 ), . . . , a(ξn−1 )) (when n = 2k ) If a(x) = al (x) + xn/2 ah (x) then a(ξj ) = al (ξj ) + (−1)j ah (ξj ) ⇒. a(ξ2i. §. a(ξ. 2i+1. ) = al (ξ2i ) + ah (ξ2i ) ) = al (ξ2i+1 ) − ah (ξ2i+1 ). 21/33.
(50) Fast Fourier Transform Goal: Given a(x), compute (a(ξ0 ), . . . , a(ξn−1 )) (when n = 2k ) If a(x) = al (x) + xn/2 ah (x) then a(ξj ) = al (ξj ) + (−1)j ah (ξj ) ⇒. a(ξ2i. §. a(ξ. 2i+1. ) = al (ξ2i ) + ah (ξ2i ) ) = al (ξ2i+1 ) − ah (ξ2i+1 ). Define r̄(x) = al (x) + ah (x), r0 (x) = al (x) − ah (x). ¯. 21/33.
(51) Fast Fourier Transform Goal: Given a(x), compute (a(ξ0 ), . . . , a(ξn−1 )) (when n = 2k ) If a(x) = al (x) + xn/2 ah (x) then a(ξj ) = al (ξj ) + (−1)j ah (ξj ) ⇒. a(ξ2i. §. a(ξ. 2i+1. ) = al (ξ2i ) + ah (ξ2i ) = r̄(ξ2i ) ) = al (ξ2i+1 ) − ah (ξ2i+1 ) = r0 (ξ2i+1 ) ¯. Define r̄(x) = al (x) + ah (x), r0 (x) = al (x) − ah (x). ¯. 21/33.
(52) Fast Fourier Transform Goal: Given a(x), compute (a(ξ0 ), . . . , a(ξn−1 )) (when n = 2k ) If a(x) = al (x) + xn/2 ah (x) then a(ξj ) = al (ξj ) + (−1)j ah (ξj ) ⇒. a(ξ2i. §. a(ξ. 2i+1. ) = al (ξ2i ) + ah (ξ2i ) = r̄(ξ2i ) ) = al (ξ2i+1 ) − ah (ξ2i+1 ) = r0 (ξ2i+1 ) ¯. Define r̄(x) = al (x) + ah (x), r0 (x) = al (x) − ah (x) and r(x) = r0 (ξx). ¯ ¯ ¯. 21/33.
(53) Fast Fourier Transform Goal: Given a(x), compute (a(ξ0 ), . . . , a(ξn−1 )) (when n = 2k ) If a(x) = al (x) + xn/2 ah (x) then a(ξj ) = al (ξj ) + (−1)j ah (ξj ) ⇒. a(ξ2i. §. a(ξ. 2i+1. ) = al (ξ2i ) + ah (ξ2i ) = r̄(ξ2i ) ) = al (ξ2i+1 ) − ah (ξ2i+1 ) = r(ξ2i ) ¯. Define r̄(x) = al (x) + ah (x), r0 (x) = al (x) − ah (x) and r(x) = r0 (ξx). ¯ ¯ ¯. 21/33.
(54) Fast Fourier Transform Goal: Given a(x), compute (a(ξ0 ), . . . , a(ξn−1 )) (when n = 2k ) If a(x) = al (x) + xn/2 ah (x) then a(ξj ) = al (ξj ) + (−1)j ah (ξj ) ⇒. a(ξ2i. §. a(ξ. 2i+1. ) = al (ξ2i ) + ah (ξ2i ) = r̄(ξ2i ) ) = al (ξ2i+1 ) − ah (ξ2i+1 ) = r(ξ2i ) ¯. Define r̄(x) = al (x) + ah (x), r0 (x) = al (x) − ah (x) and r(x) = r0 (ξx). ¯ ¯ ¯ Finally (a(ξ0 ), a(ξ1 ), a(ξ2 ), a(ξ3 ), . . . ) = (r̄(ξ0 ), r(ξ0 ), r̄(ξ2 ), r(ξ2 ), . . . ) ¯ ¯. 21/33.
(55) Fast Fourier Transform Goal: Given a(x), compute (a(ξ0 ), . . . , a(ξn−1 )) (when n = 2k ) If a(x) = al (x) + xn/2 ah (x) then a(ξj ) = al (ξj ) + (−1)j ah (ξj ) ⇒. a(ξ2i. §. a(ξ. 2i+1. ) = al (ξ2i ) + ah (ξ2i ) = r̄(ξ2i ) ) = al (ξ2i+1 ) − ah (ξ2i+1 ) = r(ξ2i ) ¯. Define r̄(x) = al (x) + ah (x), r0 (x) = al (x) − ah (x) and r(x) = r0 (ξx). ¯ ¯ ¯ Finally (a(ξ0 ), a(ξ1 ), a(ξ2 ), a(ξ3 ), . . . ) = (r̄(ξ0 ), r(ξ0 ), r̄(ξ2 ), r(ξ2 ), . . . ) ¯ ¯ FFT Algorithm: 1. 2. 3. 4. 5. 6. 7.. [COOLEY, TUKEY ’65]. Write a(x) = al (x) + xn/2 ah (x) Split in 2 parts Compute r̄(x) = al (x) + ah (x) Compute r0 (x) = al (x) − ah (x) Compute ¯r(x) = r0 (ξx) Evaluate r̄¯(ξ0 ), r̄¯ (ξ2 ), r̄(ξ4 ), . . . Recursive call in size n/2 0 2 4 Evaluate r(ξ ), r(ξ ), r(ξ ), . . . Recursive call in size n/2 return r̄(¯ξ0 ), r(¯ξ0 ), r̄(¯ξ2 ), r(ξ2 ), r̄(ξ4 ), r(ξ4 ), . . . ¯ ¯ ¯ 21/33.
(56) FFT butterflies a0 a1 a2 a3 al (x) a4 a5 a6 a7 a8 a9 a10 a11 ah (x) a12 a13 a14 a15. + + + + + + + + − − − − − − − −. ξ0. × 1 ξ × 2 ξ × 3 ξ × 4 ξ × 5 ξ × 6 ξ × 7 ξ ×. r̄0 r̄1 r̄2 r̄3 r̄(x) r̄4 r̄5 r̄6 r̄7 r0 ¯ r1 ¯ r2 ¯ r3 ¯ r(x) r4 ¯ ¯ r5 ¯ r6 ¯ r7 ¯. É r̄(x) = al (x) + ah (x) É r0 (x) = al (x) − ah (x) ¯ É r(x) = r0 (ξx) ¯ ¯. 22/33.
(57) FFT butterflies a0 a1 a2 a3 al (x) a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15. r̄0 r̄1 r̄2 r̄3 r̄4 r̄5 r̄6 r̄7 r0 ¯ r1 ¯ r2 ¯ r3 ¯ r4 ¯ r5 ¯ r6 ¯ r7 ¯ 22/33.
(58) FFT butterflies a0 a1 a2 a3 al (x) a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15. a(ξ0 ) a(ξ8 ) a(ξ4 ) a(ξ12 ) a(ξ2 ) a(ξ10 ) a(ξ6 ) a(ξ14 ) a(ξ1 ) a(ξ9 ) a(ξ5 ) a(ξ13 ) a(ξ3 ) a(ξ11 ) a(ξ7 ) a(ξ15 ) 22/33.
(59) FFT butterflies a00002 a00012 a00102 a00112 a01002 a01012 a01102 a01112 a10002 a10012 a10102 a10112 a11002 a11012 a11102 a11112. 2. a(ξ00002 ) a(ξ10002 ) a(ξ01002 ) a(ξ11002 ) a(ξ00102 ) a(ξ10102 ) a(ξ01102 ) a(ξ11102 ) a(ξ00012 ) a(ξ10012 ) a(ξ01012 ) a(ξ11012 ) a(ξ00112 ) a(ξ10112 ) a(ξ01112 ) a(ξ1111 ) 22/33.
(60) FFT timings É Evaluation or interpolation in 3/2n log n arithmetic operations É Multiplication in ∼ 9n log n É But only for degree n = 2k , pad with zeroes otherwise, loose factor 2 600. FFT. Multiplication time. 500. 400. 300. 200. 100. 0 2000. 4000. 6000. 8000 10000 Polynomial degree. 12000. 14000. 16000. Figure 1: Fast Fourier Transform timings 23/33.
(61) Outline. Polynomial multiplication algorithms: 1. 2. 3. 4.. Karatsuba Toom-Cook Fast Fourier Tranform (FFT) Truncated FFT (TFT). 24/33.
(62) Truncated Fourier Transform. [HOEVEN ’04]. Goal: Save computations when n = deg a(x) is not 2k : É compute only the first n evaluates of a(x) É get a cost ∼ 23 n log n for all degrees n. É save up to a factor 2. (instead of ∼ 32 2k log 2k ). 25/33.
(63) Truncated Fourier Transform a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 0 0 0 0 0. a(ξ0 ) a(ξ8 ) a(ξ4 ) a(ξ12 ) a(ξ2 ) a(ξ10 ) a(ξ6 ) a(ξ14 ) a(ξ1 ) a(ξ9 ) a(ξ5 ) a(ξ13 ) a(ξ3 ) a(ξ11 ) a(ξ7 ) a(ξ15 ) 26/33.
(64) Truncated Fourier Transform a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 0 0 0 0 0. a(ξ0 ) a(ξ8 ) a(ξ4 ) a(ξ12 ) a(ξ2 ) a(ξ10 ) a(ξ6 ) a(ξ14 ) a(ξ1 ) a(ξ9 ) a(ξ5 ) a(ξ13 ) a(ξ3 ) a(ξ11 ) a(ξ7 ) a(ξ15 ) 26/33.
(65) Truncated Fourier Transform a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 0 0 0 0 0. a(ξ0 ) a(ξ8 ) a(ξ4 ) a(ξ12 ) a(ξ2 ) a(ξ10 ) a(ξ6 ) a(ξ14 ) a(ξ1 ) a(ξ9 ) a(ξ5 ) a(ξ13 ) a(ξ3 ) a(ξ11 ) a(ξ7 ) a(ξ15 ) 26/33.
(66) Truncated Fourier Transform a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 0 0 0 0 0. a(ξ0 ) a(ξ8 ) a(ξ4 ) a(ξ12 ) a(ξ2 ) a(ξ10 ) a(ξ6 ) a(ξ14 ) a(ξ1 ) a(ξ9 ) a(ξ5 ) a(ξ13 ) a(ξ3 ) a(ξ11 ) a(ξ7 ) a(ξ15 ) 26/33.
(67) Truncated Fourier Transform a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 0 0 0 0 0. a(ξ0 ) a(ξ8 ) a(ξ4 ) a(ξ12 ) a(ξ2 ) a(ξ10 ) a(ξ6 ) a(ξ14 ) a(ξ1 ) a(ξ9 ) a(ξ5 ) a(ξ13 ) a(ξ3 ) a(ξ11 ) a(ξ7 ) a(ξ15 ) 26/33.
(68) Inverse Truncated Fourier Transform Goal: É É É É. recover the polynomial a(x) from only its first n evaluates knowing that deg a(x) < n (instead of deg a(x) < 2k ) 3 get a cost ∼ 2 n log n (instead of ∼ 32 2k log 2k ) save up to a factor 2. Operations required: É FFT butterfly:. −→. É Inverse FFT butterfly:. −→. É Crossed butterfly:. −→. 27/33.
(69) Inverse TFT - Recursive Algorithm : Case 1. 28/33.
(70) Inverse TFT - Recursive Algorithm : Case 1. 28/33.
(71) Inverse TFT - Recursive Algorithm : Case 1. 28/33.
(72) Inverse TFT - Recursive Algorithm : Case 1. 28/33.
(73) Inverse TFT - Recursive Algorithm : Case 1. 28/33.
(74) Inverse TFT - Recursive Algorithm : Case 1. 28/33.
(75) Inverse TFT - Recursive Algorithm : Case 1. 28/33.
(76) Inverse TFT - Recursive Algorithm : Case 1. 28/33.
(77) Inverse TFT - Recursive Algorithm : Case 1. 28/33.
(78) Inverse TFT - Recursive Algorithm : Case 1. 28/33.
(79) Inverse TFT - Recursive Algorithm : Case 2. 29/33.
(80) Inverse TFT - Recursive Algorithm : Case 2. 29/33.
(81) Inverse TFT - Recursive Algorithm : Case 2. 29/33.
(82) Inverse TFT - Recursive Algorithm : Case 2. 29/33.
(83) Inverse TFT - Recursive Algorithm : Case 2. 29/33.
(84) Inverse TFT - Recursive Algorithm : Case 2. 29/33.
(85) Inverse TFT - Recursive Algorithm : Case 2. 29/33.
(86) Inverse TFT - Recursive Algorithm : Case 2. 29/33.
(87) Complexity and timings Evaluation or interpolation in 3/2n log n for all degrees n 600. FFT TFT. Multiplication time. 500. 400. 300. 200. 100. 0 2000. 4000. 6000. 8000 10000 Polynomial degree. 12000. 14000. 16000. Figure 2: Fast Fourier Transform vs Truncated FFT timings. 30/33.
(88) Link with team research É Implementation of polynomial matrix multiplication in LinBox É FFT for lattice and symmetric polynomials. [HOEVEN, L., SCHOST ’14]. 31/33.
(89) Link with team research É Implementation of polynomial matrix multiplication in LinBox É FFT for lattice and symmetric polynomials. [HOEVEN, L., SCHOST ’14]. 31/33.
(90) Conclusion É Open questions on multiplication of polynomials of degree n:. 32/33.
(91) Conclusion É Open questions on multiplication of polynomials of degree n: É Complexity O(n log n) when coefficients in Z/pZ but n < p. 32/33.
(92) Conclusion É Open questions on multiplication of polynomials of degree n: É Complexity O(n log n) when coefficients in Z/pZ but n < p É Complexity O(n log n log log n) in general [SCHÖNHAGE, STRASSEN ’71]. 32/33.
(93) Conclusion É Open questions on multiplication of polynomials of degree n: É Complexity O(n log n) when coefficients in Z/pZ but n < p É Complexity O(n log n log log n) in general [SCHÖNHAGE, STRASSEN ’71] ∗ É Complexity O(n log n8log n ) when coefficients in Z/pZ [HARVEY, HOEVEN, LECERF ’17]. 32/33.
(94) Conclusion É Open questions on multiplication of polynomials of degree n: É Complexity O(n log n) when coefficients in Z/pZ but n < p É Complexity O(n log n log log n) in general [SCHÖNHAGE, STRASSEN ’71] ∗ É Complexity O(n log n8log n ) when coefficients in Z/pZ [HARVEY, HOEVEN, LECERF ’17]. É In practice :. 32/33.
(95) Conclusion É Open questions on multiplication of polynomials of degree n: É Complexity O(n log n) when coefficients in Z/pZ but n < p É Complexity O(n log n log log n) in general [SCHÖNHAGE, STRASSEN ’71] ∗ É Complexity O(n log n8log n ) when coefficients in Z/pZ [HARVEY, HOEVEN, LECERF ’17]. É In practice : É Complementary algorithms: Naive 0. FFT / TFT. Karatsuba 20. 100. n. 32/33.
(96) Conclusion É Open questions on multiplication of polynomials of degree n: É Complexity O(n log n) when coefficients in Z/pZ but n < p É Complexity O(n log n log log n) in general [SCHÖNHAGE, STRASSEN ’71] ∗ É Complexity O(n log n8log n ) when coefficients in Z/pZ [HARVEY, HOEVEN, LECERF ’17]. É In practice : É Complementary algorithms: Naive 0. FFT / TFT. Karatsuba 20. 100. n. É Will [HHL ’17] become practical in the future ?. 32/33.
(97) Conclusion É Open questions on multiplication of polynomials of degree n: É Complexity O(n log n) when coefficients in Z/pZ but n < p É Complexity O(n log n log log n) in general [SCHÖNHAGE, STRASSEN ’71] ∗ É Complexity O(n log n8log n ) when coefficients in Z/pZ [HARVEY, HOEVEN, LECERF ’17]. É In practice : É Complementary algorithms: Naive 0. FFT / TFT. Karatsuba 20. 100. n. É Will [HHL ’17] become practical in the future ? É Technical aspects not discussed: SIMD, cache, . . .. 32/33.
(98) Conclusion É Open questions on multiplication of polynomials of degree n: É Complexity O(n log n) when coefficients in Z/pZ but n < p É Complexity O(n log n log log n) in general [SCHÖNHAGE, STRASSEN ’71] ∗ É Complexity O(n log n8log n ) when coefficients in Z/pZ [HARVEY, HOEVEN, LECERF ’17]. É In practice : É Complementary algorithms: Naive 0. FFT / TFT. Karatsuba 20. 100. n. É Will [HHL ’17] become practical in the future ? É Technical aspects not discussed: SIMD, cache, . . .. É Thank you for your attention !. 32/33.
(99) Different Fourier transforms Classical Fourier transform É Decomposition in the frequency domain É Integral formula: Z f̂ (ξ) =. f (x)e−2iπxξ dx. É Multiplicativity: ĥ(ξ) = f̂ (ξ) · ĝ(ξ) when h = Convolution(f , g). Discrete Fourier transform É Discrete formula:. f̂k =. X n. fn e−. 2iπ N nk. P 2iπ É Link with evaluation: p̂k = P(e− N k ) where P(x) = n pn xn É Multiplicativity: Let c(x) = a(x) · b(x) then ĉk = âk · b̂k 33/33.
(100)
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