Download presentation

Presentation is loading. Please wait.

Published byBrisa Cogger Modified over 2 years ago

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 + n2/2) ) 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’(Nm) 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

Similar presentations

OK

Notation Intro. Number Theory Online Cryptography Course Dan Boneh

Notation Intro. Number Theory Online Cryptography Course Dan Boneh

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on history of olympics in canada Ppt on soft skills for teachers Ppt online templates Ppt on entrepreneurship development in india Seminar ppt on data mining Ppt on acc cement company Ppt on steve jobs as entrepreneur Ppt on semi solid casting process Ppt on vehicle security system Quiz ppt on india