Faster Polynomial Multiplication via    Discrete Fourier Transforms                   Alexey Pospelov    Computer Science ...
Polynomial multiplication   Given    a(x) = a0 + a1 x + · · · + an x n , b(x) = b0 + b1 x + · · · + bn x n ,   Compute    ...
Polynomial multiplication   Given     a(x) = a0 + a1 x + · · · + an x n , b(x) = b0 + b1 x + · · · + bn x n ,   Compute   ...
In what model?      Arithmetic circuits with binary “+”, “−”, “·”      Each binary gate has unit cost      No divisions   ...
History and state of the art   School method: O(n2 )   Karatsuba 1960: O(nlog2 3 ) = O(n1.585 )                        √  ...
History and state of the art   School method: O(n2 )   Karatsuba 1960: O(nlog2 3 ) = O(n1.585 )                        √  ...
History and state of the art   School method: O(n2 )   Karatsuba 1960: O(nlog2 3 ) = O(n1.585 )                        √  ...
Multiplication in O(n log n)   Given                  a(x) = a0 + a1 x + · · · + an−1 x n−1 ,                  b(x) = b0 +...
Multiplication in O(n log n)          a0   a1           . . . an−1                                         b0           b1...
Discrete Fourier transform
Discrete Fourier transform       Maps a degree n − 1 polynomial to its values at n distinct nth       roots of unity:     ...
Discrete Fourier transform         a0   a1           . . . an−1                                         b0           b1   ...
Discrete Fourier transform       Lk (n): the complexity of degree n polynomial multiplication       over a field k       Dk...
What if roots of unity are not available?
What if roots of unity are not available?   Attach them!
What if roots of unity are not available?   Attach them!       Switch from the field k to its algebraic extension Am where ...
What if roots of unity are not available?   Attach them!       Switch from the field k to its algebraic extension Am where ...
What if roots of unity are not available?   Attach them!       Switch from the field k to its algebraic extension Am where ...
What if roots of unity are not available?   Attach them!       Switch from the field k to its algebraic extension Am where ...
Fast polynomial multiplication            a0 . . . a m −1 . . . an−1                       2                              ...
Fast polynomial multiplication   In this case     Lk (n) ≤ 2 Lk (m)              + 3DAm (n) · complexity of aritrhmetics i...
Does it work?   Yes!!!   Sch¨nhage-Strassen 1971:      o                       =m   Sch¨nhage 1977: 3 = 2m (+ a little tri...
Slow fields
Slow fields   Recall:     Lk (n) ≤ 2 Lk (m)              + 3DAm (n) · complexity of arithmetics in Am                     +...
Slow fields   Recall:     Lk (n) ≤ 2 Lk (m)               + 3DAm (n) · complexity of arithmetics in Am                     ...
Slow fields   Recall:     Lk (n) ≤ 2 Lk (m)               + 3DAm (n) · complexity of arithmetics in Am                     ...
More details      To attach an th root of unity we need an extension of degree      at least fk ( )      The degree of the...
More details      To attach an th root of unity we need an extension of degree      at least fk ( )      The degree of the...
“Lower bound”  For the rational field Q, for all n                                                n                      fQ...
“Lower bound”  For the rational field Q, for all n                                                n                      fQ...
Summary     Uniform treatment of all known asymptotically fastest     polynomial multiplication algorithms w.r.t. the tota...
Summary     Uniform treatment of all known asymptotically fastest     polynomial multiplication algorithms w.r.t. the tota...
Thank you for attention!
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Csr2011 june14 17_00_pospelov

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Alexey Pospelov. Faster Polynomial Multiplication via Discrete Fourier Transforms

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Csr2011 june14 17_00_pospelov

  1. 1. Faster Polynomial Multiplication via Discrete Fourier Transforms Alexey Pospelov Computer Science Department, Saarland UniversityCluster of Excellence Multimodal Computing and Interaction June 14th, 2011
  2. 2. Polynomial multiplication Given a(x) = a0 + a1 x + · · · + an x n , b(x) = b0 + b1 x + · · · + bn x n , Compute c(x) = c0 + c1 x + · · · + c2n x 2n = a(x)b(x).
  3. 3. Polynomial multiplication Given a(x) = a0 + a1 x + · · · + an x n , b(x) = b0 + b1 x + · · · + bn x n , Compute c(x) = c0 + c1 x + · · · + c2n x 2n = a(x)b(x). For all 0 ≤ i ≤ 2n, compute a0 bi + a1 bi−1 + · · · + ai b0 , 0 ≤ i ≤ n, ci = ai−n bn + ai−n+1 bn−1 + · · · + an bi−n , n < i ≤ 2n.
  4. 4. In what model? Arithmetic circuits with binary “+”, “−”, “·” Each binary gate has unit cost No divisions Constants from the field available at no cost Inputs are the coefficients of the polynomials to be multiplied Outputs are the coefficients of the product polynomial Interested in a circuit for degree n polynomial multiplication of the minimal size
  5. 5. History and state of the art School method: O(n2 ) Karatsuba 1960: O(nlog2 3 ) = O(n1.585 ) √ Toom 1963: n1+O(1/ log n) = O(n1+ ), for any fixed >0 Over infinite fields
  6. 6. History and state of the art School method: O(n2 ) Karatsuba 1960: O(nlog2 3 ) = O(n1.585 ) √ Toom 1963: n1+O(1/ log n) = O(n1+ ), for any fixed >0 Over infinite fields Sch¨nhage-Strassen 1971: O(n log n log log n) o Didn’t work for fields of char = 2
  7. 7. History and state of the art School method: O(n2 ) Karatsuba 1960: O(nlog2 3 ) = O(n1.585 ) √ Toom 1963: n1+O(1/ log n) = O(n1+ ), for any fixed >0 Over infinite fields Sch¨nhage-Strassen 1971: O(n log n log log n) o Didn’t work for fields of char = 2 Sch¨nhage 1977: O(n log n log log n), over field of char = 2 o Kaminski 1988, Cantor-Kaltofen 1991: O(n log n log log n), over arbitrary algebras Over C or R: O(n log n) All general lower bounds: Ω(n).
  8. 8. Multiplication in O(n log n) Given a(x) = a0 + a1 x + · · · + an−1 x n−1 , b(x) = b0 + b1 x + · · · + bn−1 x n−1 , Compute c(x) = c0 + c1 x + · · · + xn−1 x n−1 = a(x)b(x) (mod x n − 1). (Can always choose a larger n and pad polynomials with zeroes to reduce the ordinary polynomial multiplication to the product in k[x]/(x n − 1).)
  9. 9. Multiplication in O(n log n) a0 a1 . . . an−1 b0 b1 . . . bn−1 8 x 8 x DFTω n DFTω n . . . ˜n−1 a ˜ ˜ b0 b1 . . . ˜0 ˜1 a a ˜n−1 b 5 { 5 { 5 { · · · c0 ˜ c1 . . . cn−1 ˜ ˜ ω ∈ k is 8 x an nth IDFTω n primitive root of 1 c0 c1 . . . cn−1
  10. 10. Discrete Fourier transform
  11. 11. Discrete Fourier transform Maps a degree n − 1 polynomial to its values at n distinct nth roots of unity: n−1 ˜i := a(ω i ) = a aj ω ij , 0≤i ≤n−1 j=0 DFTω : (a0 , a1 , . . . , an−1 ) → (˜0 , ˜1 , . . . , ˜n−1 ) n a a a (ω is a primitive nth root of unity) Linear transform: DFTω : k[x] → k n n Isomorphism: DFTω : k[x]/(x n − 1) → k n n Can be often computed in O(n log n) The inverse isomorphism is almost a DFT again: 1 n−1 DFTω n : (˜0 , ˜1 , . . . , ˜n−1 ) → (a0 , a1 , . . . , an−1 ) a a a n
  12. 12. Discrete Fourier transform a0 a1 . . . an−1 b0 b1 . . . bn−1 8 x 8 x DFTω n DFTω n . . . ˜n−1 a ˜ ˜ b0 b1 . . . ˜0 ˜1 a a ˜n−1 b 5 { 5 { 5 { · · · c0 ˜ c1 . . . cn−1 ˜ ˜ ω ∈ k is 8 x an nth IDFTω n primitive root of 1 c0 c1 . . . cn−1
  13. 13. Discrete Fourier transform Lk (n): the complexity of degree n polynomial multiplication over a field k Dk (n): the complexity of computing length n DFT over k Lk (n) ≤ 3Dk (n) + 2n = O(n log n). Note: we need roots of unity.
  14. 14. What if roots of unity are not available?
  15. 15. What if roots of unity are not available? Attach them!
  16. 16. What if roots of unity are not available? Attach them! Switch from the field k to its algebraic extension Am where roots of unity of sufficiently large order exist.
  17. 17. What if roots of unity are not available? Attach them! Switch from the field k to its algebraic extension Am where roots of unity of sufficiently large order exist. More precisely: take a (ring) extension Am of k of degree m over k with a 2 th root of unity ω ∈ Am : For example, Am = k[x]/pm (x), pm (x) ∈ k[x] is a polynomial of degree m, pm (x) vanishes on ω2 , (ω2 is a primitive 2 th root of unity in the algebraic closure of the field k.)
  18. 18. What if roots of unity are not available? Attach them! Switch from the field k to its algebraic extension Am where roots of unity of sufficiently large order exist. More precisely: take a (ring) extension Am of k of degree m over k with a 2 th root of unity ω ∈ Am : For example, Am = k[x]/pm (x), pm (x) ∈ k[x] is a polynomial of degree m, pm (x) vanishes on ω2 , (ω2 is a primitive 2 th root of unity in the algebraic closure of the field k.) How can it help?
  19. 19. What if roots of unity are not available? Attach them! Switch from the field k to its algebraic extension Am where roots of unity of sufficiently large order exist. More precisely: take a (ring) extension Am of k of degree m over k with a 2 th root of unity ω ∈ Am : For example, Am = k[x]/pm (x), pm (x) ∈ k[x] is a polynomial of degree m, pm (x) vanishes on ω2 , (ω2 is a primitive 2 th root of unity in the algebraic closure of the field k.) How can it help? See next slide.
  20. 20. Fast polynomial multiplication a0 . . . a m −1 . . . an−1 2 b0 . . . b m −1 . . . bn−1 2 % } % % } % → Am ... → Am → Am ... → Am A0 7 y A2 −1 B0 7 y B2 −1 DFTω 2 DFTω 2 ˜ A2 −1 ˜ B0 ˜ A0 5 y 7 { B2 ˜ · ... · −1 2 ·m =n ˜ C0 7 y ˜ C2 −1 Am = k[x]/pm (x) IDFTω 2 C0 C2 −1 ω ∈ Am is a 2 th primitive ← Am ... ← Am root of 1 Õ 3 Ù c0 . . . cm−1 . . . cn−1
  21. 21. Fast polynomial multiplication In this case Lk (n) ≤ 2 Lk (m) + 3DAm (n) · complexity of aritrhmetics in Am + cost of embedding and unembedding in Am Our contribution #1: Formalize this kind of algorithms The relation between m and 2 is a barrier for the algorithm’s performance This relation depends heavily on the field properties The cost of the DFT can usually be made O(n log n) Embedding and unembedding run usually in linear time, e.g., if pm (x) is sparse
  22. 22. Does it work? Yes!!! Sch¨nhage-Strassen 1971: o =m Sch¨nhage 1977: 3 = 2m (+ a little trick) o Kaminski 1988: = φ(m) (Euler’s totient function) Cantor-Kaltofen 1991: = m (and Am is a little more complicated than k[x]/pm (x))
  23. 23. Slow fields
  24. 24. Slow fields Recall: Lk (n) ≤ 2 Lk (m) + 3DAm (n) · complexity of arithmetics in Am + cost of embedding and unembedding in Am Ideally we want m to be small and to be large.
  25. 25. Slow fields Recall: Lk (n) ≤ 2 Lk (m) + 3DAm (n) · complexity of arithmetics in Am + cost of embedding and unembedding in Am Ideally we want m to be small and to be large. Definition For a field k, and n, s.t. char k n, let fk (n) be [k(ωn ) : k], the degree function of k.
  26. 26. Slow fields Recall: Lk (n) ≤ 2 Lk (m) + 3DAm (n) · complexity of arithmetics in Am + cost of embedding and unembedding in Am Ideally we want m to be small and to be large. Definition For a field k, and n, s.t. char k n, let fk (n) be [k(ωn ) : k], the degree function of k. Our contribution #2: If fk (n) = o(log log n) for some not too sparse set of n then k is fast and Lk (n) = o(n log n log log n) If fk (n) = Ω(n1− ) for any fixed 0, then k is slow and any algorithm of that kind runs in Ω(n log n log log n)
  27. 27. More details To attach an th root of unity we need an extension of degree at least fk ( ) The degree of the polynomial is then ∼ · fk ( ) √ For the least solution i0 of i · fk (i) ≥ n, fk (n) := fk (i0 ) The number√of recursive steps is at least the number of √ √ fk (fk (· · · fk (n) · · · )), until the value becomes O(1) √ This superposition depth will be denoted (fk )∗ (n) The cost of all steps on a single recursion level is determined by the complexity of the DFTs, and is Θ(n log n)
  28. 28. More details To attach an th root of unity we need an extension of degree at least fk ( ) The degree of the polynomial is then ∼ · fk ( ) √ For the least solution i0 of i · fk (i) ≥ n, fk (n) := fk (i0 ) The number√of recursive steps is at least the number of √ √ fk (fk (· · · fk (n) · · · )), until the value becomes O(1) √ This superposition depth will be denoted (fk )∗ (n) The cost of all steps on a single recursion level is determined by the complexity of the DFTs, and is Θ(n log n) The total cost is estimated as √ Ω(n log n) · (fk )∗ (n)
  29. 29. “Lower bound” For the rational field Q, for all n n fQ (n) = φ(n) ≥ c · , log log n and √ (fQ )∗ (n) = Ω(log log n).
  30. 30. “Lower bound” For the rational field Q, for all n n fQ (n) = φ(n) ≥ c · , log log n and √ (fQ )∗ (n) = Ω(log log n). Complexity of any DFT-based multiplication algorithm is then Ω(n log n log log n). It follows that over Q we need another kind of an algorithm.
  31. 31. Summary Uniform treatment of all known asymptotically fastest polynomial multiplication algorithms w.r.t. the total complexity A way to improve the total complexity upper bounds over certain fields Impossibility to improve Sch¨nhage-Strassen over any fields o (and rings or algebras) of characteristic 0 In particular, no light at the end of the tunnel for polynomial multiplication over Q
  32. 32. Summary Uniform treatment of all known asymptotically fastest polynomial multiplication algorithms w.r.t. the total complexity A way to improve the total complexity upper bounds over certain fields Impossibility to improve Sch¨nhage-Strassen over any fields o (and rings or algebras) of characteristic 0 In particular, no light at the end of the tunnel for polynomial multiplication over Q Over fields of positive characteristic,
  33. 33. Thank you for attention!

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