1. Fast Polynomial Factorization and Modular Composition
Chris Umans
Caltech
joint work with Kiran Kedlaya (MIT)
[Umans STOC 08] + [Kedlaya-Umans FOCS 08]

2. Introduction A basic problem: given: degree n polynomial A(X)
output: factorization into irreducible polynomials
Example:
given: A(X) = x3 – 1
output: (x2 + x + 1)(x – 1)
Nov. 18, 2009

3. Introduction
factoring a degree n polynomial A(X) with coefficients in Fq is easy
Why? can easily compute
(Xqi – X) mod A(X)
(contains all potential factors of degree dividing i; use GCD, and i = 1,2,3…, n)
Nov. 18, 2009

4. product of degree i polynomials
Introduction
polynomial-time factoring in Fq[X] [Berlekamp, Cantor-Zassenhaus]:
make A(X) square-free
distinct degree factorization:
A(X) = A1(X)A2(X)...Ai(X)…An(X)
equal-degree factorization:
Ai(X) = g1(X)g2(X)…gk(X)
product of degree i polynomials
irreducible factors
Nov. 18, 2009

5. Bottleneck in algorithms
how to compute this polynomial quickly:
(Xqi – X) mod A(X)
deg(A) = n; i ≤ n
Nov. 18, 2009

6. Bottleneck in algorithms
how to compute this polynomial quickly:
Xqi mod A(X)
repeated squaring: log(qi) = i log q operations
deg(A) = n; i ≤ n
“operations” = modular addition, multiplication, composition of degree n polynomials
Nov. 18, 2009

7. Bottleneck in algorithms
how to compute this polynomial quickly:
Xqi mod A(X)
repeated squaring: log(qi) = i log q operations
modular composition: log q + log i operations
compute Xq mod A(X) using repeated squaring
compose it with itself (Xq)q = Xq2
compose it with itself again (Xq2)q2 = Xq4
deg(A) = n; i ≤ n
von zur Gathen + Shoup
1992
mod A(X)
“operations” = modular addition, multiplication, composition of degree n polynomials
Nov. 18, 2009

8. Operations on polynomials
degree n polynomials f(X), g(X), A(X)
Operation: Time:
f(X) + g(X) mod A(X) O’(n)
f(X)g(X) mod A(X) O’(n)
f(0), …, f(n) O’(n)
find f(X): f(0)=0, …, f(n) = n O’(n)
f(g(X)) mod A(X) O’(n1.667)
Nov. 18, 2009

9. Modular composition given deg. n polynomials f(X), g(X), A(X)
compute f(g(X)) mod A(X)
trivial in time O’(n2)
best known [Brent-Kung 1978; Huang-Pan 1997] O’(n1.667) (= O(n1.5 + n2/2) )
idea: reduce problem to matrix multiplication
This work: O’(n)
Nov. 18, 2009

10. Outline reduce to “multivariate multipoint evaluation”
new algorithm via multimodular reduction
) data structure for polynomial evaluation
) faster algorithms for polynomial factorization and other problems
Nov. 18, 2009

11. f(g(X)) ´ f(g0(X), …, glog n-1(X)) mod A(X)
The reduction
given deg. n=2m polynomials f(X), g(X), A(X)
compute f(g(X)) mod A(X)
convert f(X) to multilinear: f(X0,X1,…,Xlog n-1)
f(X) = f(X, X2, X4, X8, …, Xn/2)
compute g2i(X) mod A(X) (call this gi(X)) for i = 0,1,2,…, log n -1
note:
f(g(X)) ´ f(g0(X), …, glog n-1(X)) mod A(X)
Nov. 18, 2009

12. The reduction given deg. n=2m polynomials f(X), g(X), A(X)
compute f(g(X)) mod A(X)
f(X) = f(X, X2, X4, X8, …, Xn/2)
f(g(X)) ´ f(g0(X), …, glog n-1(X)) mod A(X)
idea: evaluate at n¢log n points;
evaluate each gi at n¢log n points
evaluate f at these n¢log n points in (Fq)log n
then interpolate; reduce modulo A(X)
degree n¢log n
Nov. 18, 2009

13. Multipoint evaluation
Recall univariate case:
given degree n polynomial f(X) 2 Fq[X] and points 0, 1, …, n
can compute f(0), …, f(n) in time O’(n)
Multivariate case harder:
given f(X1, X2, …, Xm) 2 Fq[X1, X2, …, Xm] with individual degrees · d, and 0, 1, …, N=dm
can compute f(0), …, f(N) in time O’(Nm) where < m < [Nüsken-Ziegler 2004]
Nov. 18, 2009

14. Multivariate multipoint evaluation
given f(X1,X2,…,Xm), ind. deg <d; 0, …, N=dm
compute f(0), f(1), …, f(N)
If N points are all of Fqm then computable in O’(N) time via (multidimensional, finite field) FFT
But we get un- structured points…
Fqm
Fqm
Nov. 18, 2009

15. Multivariate multipoint evaluation
Assume working over Fp
Lift coefficients of f and the coordinates of each ®i to {0,1, 2, …, p-1} µ Z
In integers, f(®i) · dmpdm = M
Solve problem mod primes p1, p2, …, pk with p1p2…pk ¸ M (so pj · O(log M))
reconstruct via Chinese Remainder Thm.
repeat; magnitude of the pj ! ¼ dm
Nov. 18, 2009

16. What happens to the eval. pts.?
(after few rounds) can afford to compute all evaluations at cost ¼ (dm)m (ideal cost: dm)
(mod 7)
0 1
0 1
(mod 2)
(mod 3)
(mod 5)
Nov. 18, 2009

17. Multivariate multipoint evaluation
given f(X1,X2,…,Xm), ind. deg <d; 0, …, N
compute f(0), f(1), …, f(N)
Theorem: for every const. ± > 0, can solve above problem in time
(dm + N)1+± log1+o(1)q
provided m · do(1).
Nov. 18, 2009

18. Data structure for poly. eval.
Observation: reduced f’s and tables of evaluations over entire domains don’t depend on the set of evaluation points
Theorem: given degree n poly f(X) over Fq, can produce a data structure in nearly-linear time that answers evaluation queries ® 2 Fq in time polylog(n)¢log1+o(1)q.
Nov. 18, 2009

19. Algorithmic improvements
modular composition in nearly-linear time
(as well as its “transpose” problem)
) faster algorithms for
polynomial factorization: O’(n1.5 + nlog q)¢log q
(best previous O’(n2 + n log q)¢log q or O’(n1.815 log q)¢log q )
von zur Gathen + Shoup ‘ Kaltofen + Shoup ‘98
irreducibility testing: O’(n log q)¢log q
finding minimal polynomials: O’(n log q)
(improved exponents in all cases)
Nov. 18, 2009

20. Open problems
Find an O’(n) algebraic algorithm for modular composition/multivariate multipoint evaluation in any characteristic
Find a fast algorithm for multivariate multipoint evaluation when m > do(1)
Find a nearly-linear time algorithm for polynomial factorization
Nov. 18, 2009