Towards High Performance Multivariate Factorization. Michael Monagan. This is joint work with Baris Tuncer.

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1 Towards High Performance Multivariate Factorization Michael Monagan Center for Experimental and Constructive Mathematics Simon Fraser University British Columbia This is joint work with Baris Tuncer.

2 The Multivariate Diophantine Problem (MDP) Given A, B, C Z p [x 1, x 2,..., x n ] with gcd(a, B) = 1 solve σa + τb = C for σ, τ Z p [x 1,..., x n ] s.t. deg x1 σ < deg x1 B. Outline Context: factoring a Z[x 1,..., x n ] using Hensel lifting Wang s solution Kaltofen s sparse solution v. our sparse solution Eliminating multi-precision arithmetic Eliminating the error computation

3 Multivariate polynomial factorization Input a = fg Z[x 1,..., x n ] primitive and square-free. Output f, g. 1 Pick x 1 and factor LC(a) = h e1 1 he in Z[x 2,..., x n ]. 2 Pick α = α 2,..., α n Z such that LC(a)(α) 0 and gcd(a(α), a x 1 (α)) = 1 and h i (α) Z have distinct prime divisors. 3 Factor a(x 1, α) = f (x 1, α) g(x 1, α) over Z. 4 Distribute the factors of LC(a) on f (x 1, α) and g(x 1, α). 5 Pick a prime p > α i and L N s.t. p L > 2 f where f a and gcd(f (x 1, α), g(x 1, α)) = 1 in Z p [x 1 ]. Hensel lift x 2 then x 3 then... then x n doing arithmetic mod p L. P. Wang. An improved Multivariate Polynomial Factoring Algorithm. Mathematics of Computation, 32: AMS (1978) Algorithms for Computer Algebra, K.O. Geddes, S.R. Czapor, G. Labahn. Kluwer, 1992.

4 Wang s Multivariate Hensel Lifting (MHL) Input a Z[x 1,..., x n], α = (α 2,... α n), (f 0, g 0) Z[x 1] s.t. (i) a(x 1, α) f 0g 0 = 0 and (ii) gcd(f 0, g 0) = 1 in Z p[x 1]. Output f, g satifying a = fg or FAIL. 1 If n = 1 output (f 0, g 0). 2 Lift x 2,..., x n 1: (f 0, g 0) := MHL( a(x n = α n), α, (f 0, g 0) ) Idea: f = df k=0 f k(x n α n) k and g = dg k=0 g k(x n α n) k.

5 Wang s Multivariate Hensel Lifting (MHL) Input a Z[x 1,..., x n], α = (α 2,... α n), (f 0, g 0) Z[x 1] s.t. (i) a(x 1, α) f 0g 0 = 0 and (ii) gcd(f 0, g 0) = 1 in Z p[x 1]. Output f, g satifying a = fg or FAIL. 1 If n = 1 output (f 0, g 0). 2 Lift x 2,..., x n 1: (f 0, g 0) := MHL( a(x n = α n), α, (f 0, g 0) ) Idea: f = df k=0 f k(x n α n) k and g = dg k=0 g k(x n α n) k. 3 Initialize: (f, g) := (f 0, g 0); error := a fg. 4 For k = 1,..., deg x1 a while error 0 do T k := Taylor coeff( error, (x n α n) k ) If T k 0 then Solve f k g 0 + g k f 0 = T k for f k, g k Z p[x 1,..., x n 1]. Set f := f + f k (x n α n) k and g := g + g k (x n α n) k Set error := a fg 5 If error = 0 output (f, g) else output FAIL.

6 Wang s Multivariate Diophantine Lifting (MDP) Input A, B, C Z p[x 1,..., x n], α = α 2,... α n ) Output σ, τ Z p[x 1,..., x n] satisfying σa + τb = C 1 If n = 1 solve σa + τb = C using the Euclidean algorithm in Z p[x 1]. Let σ = df k=0 σ k(x n α n) k and τ = dg k=0 τ k(x n α n) k. 2 (σ 0, τ 0) := MultiDioLift( A(x n = α n), B(x n α n), C(x n = α n), α )

7 Wang s Multivariate Diophantine Lifting (MDP) Input A, B, C Z p[x 1,..., x n], α = α 2,... α n ) Output σ, τ Z p[x 1,..., x n] satisfying σa + τb = C 1 If n = 1 solve σa + τb = C using the Euclidean algorithm in Z p[x 1]. Let σ = df k=0 σ k(x n α n) k and τ = dg k=0 τ k(x n α n) k. 2 (σ 0, τ 0) := MultiDioLift( A(x n = α n), B(x n α n), C(x n = α n), α ) 3 Initialize: (σ, τ) := (σ 0, τ 0) error := C σa τb 4 For k = 1, 2,... while error 0 do T k := Taylor coeff( error, (x n α n) k ) If T k 0 then σ k, τ k := MultiDioLift( σ 0, τ 0, T k, α ) σ, τ := σ + σ k (x n α n) k, τ + τ k (x n α n) k ) error := error σ k (x n α n) k A τ k (x n α n) k B 5 output (σ, τ).

8 Wang s Multivariate Diophantine Lifting (MDP) Input A, B, C Z p[x 1,..., x n], α = α 2,... α n ) Output σ, τ Z p[x 1,..., x n] satisfying σa + τb = C 1 If n = 1 solve σa + τb = C using the Euclidean algorithm in Z p[x 1]. Let σ = df k=0 σ k(x n α n) k and τ = dg k=0 τ k(x n α n) k. 2 (σ 0, τ 0) := MultiDioLift( A(x n = α n), B(x n α n), C(x n = α n), α ) 3 Initialize: (σ, τ) := (σ 0, τ 0) error := C σa τb 4 For k = 1, 2,... while error 0 do T k := Taylor coeff( error, (x n α n) k ) If T k 0 then σ k, τ k := MultiDioLift( σ 0, τ 0, T k, α ) σ, τ := σ + σ k (x n α n) k, τ + τ k (x n α n) k ) error := error σ k (x n α n) k A τ k (x n α n) k B 5 output (σ, τ). Let M(n) count calls to the Euclidean algorithm and d max(deg xi f, deg xi g). Then M(1) = 1, M(n) dm(n 1) = M(n) d n 1.

9 Recall that f = df i=0 f i(x n α n) i and g = dg i=0 g i(x n α n) i. Solve the MDP f k g 0 + g k f 0 = T k for f k, g k Z p[x 1,..., x n 1]. Interpolate x 2,..., x n 1 from images in Z p[x 1] using sparse interpolation? We tried Zippel s variable at a time sparse interpolation from 1979.

10 Recall that f = df i=0 f i(x n α n) i and g = dg i=0 g i(x n α n) i. Solve the MDP f k g 0 + g k f 0 = T k for f k, g k Z p[x 1,..., x n 1]. Interpolate x 2,..., x n 1 from images in Z p[x 1] using sparse interpolation? We tried Zippel s variable at a time sparse interpolation from Theorem [MT] The Strong Sparse Hensel Assumption If α n is random then, with high probability supp(f 0) supp(f 1) supp(f df ) and supp(g 0) supp(g 1) supp(g dg ). Example (n = 8, #f = 10, 000, df = 16, α = 3) f i = 9877, 7043, 4932, 3374, 2310, 1545, 1001, 654, 418, 245, 141, 81, 34, 13, 5, 2, 1.

11 Recall that f = df i=0 f i(x n α n) i and g = dg i=0 g i(x n α n) i. Solve the MDP f k g 0 + g k f 0 = T k for f k, g k Z p[x 1,..., x n 1]. Interpolate x 2,..., x n 1 from images in Z p[x 1] using sparse interpolation? We tried Zippel s variable at a time sparse interpolation from Theorem [MT] The Strong Sparse Hensel Assumption If α n is random then, with high probability supp(f 0) supp(f 1) supp(f df ) and supp(g 0) supp(g 1) supp(g dg ). Example (n = 8, #f = 10, 000, df = 16, α = 3) f i = 9877, 7043, 4932, 3374, 2310, 1545, 1001, 654, 418, 245, 141, 81, 34, 13, 5, 2, 1. Kaltofen s Sparse Hensel Lifting. Let supp(f 0) = {M 1,..., M r ) and supp(g 0) = {N 1,..., N s} in (x 1,..., x n 1). Equate the coefficients c i of ( r ) ( s ) c i M i g 0 + c r+i N i f 0 = T k in Z p[x 1,..., x n 1] i=1 in monomials in (x 1,..., x n 1) and solve the (r + s) (r + s) linear system. i=1

12 MTSHL : Monagan and Tuncer s Sparse Hensel Lifting Solve f k g 0 + g k f 0 = T k for f k, g k Z p [x 1,..., x n 1 ]. Let f k 1 = df i=1 a i(x 2,..., x n 1)x1 i and supp(a i ) = {M i1,..., M iri }, and g k 1 = dg i=1 b i(x 2,..., x n 1)x1 i and supp(b i ) = {N i1,..., N isi }. Assume f k = ( df ri ) i=0 j=1 a ijm ij x1 i and g k = ( dg si ) i=0 j=1 b ijn ij x1. i

13 MTSHL : Monagan and Tuncer s Sparse Hensel Lifting Solve f k g 0 + g k f 0 = T k for f k, g k Z p [x 1,..., x n 1 ]. Let f k 1 = df i=1 a i(x 2,..., x n 1)x1 i and supp(a i ) = {M i1,..., M iri }, and g k 1 = dg i=1 b i(x 2,..., x n 1)x1 i and supp(b i ) = {N i1,..., N isi }. Assume f k = ( df ri ) i=0 j=1 a ijm ij x1 i and g k = ( dg si ) i=0 j=1 b ijn ij x1. i Let t = max(#a i, #b i ) #a + #b. Let β = (β 2,..., β n 1) Z n 2 p. Solve σ j g 0(β j ) + τ j f 0(β j ) = T k (β j ) for σ j, τ j Z p[x 1]. for 1 j t. Solve df + dg Vandermonde systems {coeff (f k (β j ), x i n) = coeff (σ j, x i n) for 1 j t} {coeff (g k (β j ), x i n) = coeff (τ, x i n) for 1 j t} Cost of MTSHL: number of arithmetic operations in Z p where d = max deg xi a (n 1)d O( #f #g + #a +(#a + #f + #g) t }{{}}{{}}{{} error Taylor eval + d 2 t }{{} UniDio + }{{} dt 2 ). Vandermonde

14 MTSHL with division f k g 0 + g k f 0 = T k = g k = (T k f k g 0 )/f 0. Let f k 1 = df i=1 a i(x 2,..., x n 1)x1 i and supp(a i ) = {M i1,..., M iri }, Assume f k = df i=0 ( ri j=1 a ijm ij ) x i 1. Let t = max(#a i, #b i ) and β = (β 2,..., β n 1) Z n 2 p. Solve σ j g 0(β j ) + τ j f 0(β j ) = T k (β j ) for σ j, τ j Z p[x 1]. for 1 j t. Solve df +dg Vandermonde systems obtained by equating coefficients of x 1 f k (β j ) = σ j and g k (β j ) = τ j for 1 j t Finally compute g k := (T k f k g 0)/f 0 in Z p[x 1,..., x n 1]. Cost of MTSHL: number of arithmetic operations in Z p where d = max deg xi a (n 1)d O( #f #g + #a +(#a + #f +#g) t }{{}}{{}}{{} error Taylor eval + d 2 t }{{} UniDio + }{{} dt 2 + #f #g ). }{{} Vandermonde g k

15 n/d/t Wang (MDP) Kaltofen (MDP) MTSHL (MDP) 4/35/ (11.95) 1.75 (1.18) 1.51 (0.24) 5/35/ (86.28) 3.75 (2.57) 1.16 (0.36) 7/35/ (797.0) 5.04 (4.08) 1.58 (0.59) 9/35/ (4449.4) 8.13 (6.22) 2.94 (0.56) 4/35/ (26.48) (635.1) (0.82) 5/35/ (402.5) (1899.6) 26.0 (4.86) 7/35/ (3842.2) (2305.5) 43.1 (6.84) 9/35/500 > (3805.9) 79.6 (9.71) > M := product( x i, i = 1..n ) ; > f := x d 1 + M randpoly( [x 1,..., x n], terms = T, degree = d ) ;

16 Chapter 6 Newton s Iteration and the Hensel Construction Run the entire Hensel lifting mod p L where p L > 2 max( a, f, g ). Why? To avoid the expression swell when solving σg 0(α) + τf 0(α) = T k (α) in Z[x 1] in MDP. Lemma [Gelfond] Let a Z[x 1,..., x n] with d i = deg xi a. If f a then f e d 1+d d n a where e =

17 Let p be a prime and let f = hf k=0 f kp k and g = hg k=0 g kp k. Example f = 2x 1 + (5 + 0 p + 2p 2 )x 2 + (7 + p)x 3 = (2x 1 + 5x 2 + 7x 3) + 3x 3 p + 2x }{{}}{{} 2 p }{{} f 0 f 1 f 2 2. supp(f 0) = {x 1, x 2, x 3} supp(f 1) = {x 3} supp(f 2) = {x 2}.

18 Let p be a prime and let f = hf k=0 f kp k and g = hg k=0 g kp k. Example f = 2x 1 + (5 + 0 p + 2p 2 )x 2 + (7 + p)x 3 = (2x 1 + 5x 2 + 7x 3) + 3x 3 p + 2x }{{}}{{} 2 p }{{} f 0 f 1 f 2 2. supp(f 0) = {x 1, x 2, x 3} supp(f 1) = {x 3} supp(f 2) = {x 2}. Theorem If p is chosen at random from [2 b 1, 2 b ] for b sufficiently large then supp(f 0) supp(f 1) supp(f 2)... supp(f hf ) and supp(g 0) supp(g 1) supp(g 2)... supp(g hg ) whp. Idea: Run the entire Hensel lifting modulo a machine prime p to obtain f 0, g 0 Z p[x 1,..., x n] satisfying a f 0g 0 mod p = 0. Now do a p-adic lift on f 0, g 0 to get f, g.

19 Multivariate p-adic Lifting Input a Z[x 1,..., x n], (f 0, g 0) Z p[x 1,..., x n] satisfying a f 0g 0 mod p = 0 prime p and a lifting bound L satisfying p L > f where f a. Output f, g Z[x 1,..., x n] satifying a = fg or FAIL. 1 Initialize: (f, g) := (f 0, g 0); error := a fg; 2 For k = 1,..., L while error 0 do T k := ( error p k ) mod p. If T k 0 then Solve the MDP f k g 0 + g k f 0 = T k for f k, g k Z p[x 1,..., x n]. Set f k := mods(σ, p) and g k := mods(τ, p). Set f := f + f k p k ; g := g + g k p k ; error := a fg. 3 If error = 0 output (f, g) else output FAIL. The MDP is solved in Z p[x 1,..., x n] where p is a machine prime!

20 n/d/t m L MTSHL mod p L MTSHL mod p + lift 5/10/ (5.10) /10/ (6.10) /10/ (7.596) /10/ (12.82) /10/ (15.18) /10/ (17.22) Table: CPU timings (seconds) for p adic lifting for p = > f := x d 1 + randpoly( [x 1,..., x n], terms = T, degree = d, coeffs = rand(p m ) ) ;

21 Eliminating the error multiplication a fg Let f (k) = k 1 i=0 f i(x n α n) i and g (k) = k 1 i=0 g i(x n α n) i. Main Idea: error(β j ) = a(β j ) f (k) (β j ) g (k) (β j ) in Z p[x 1, x n]. Compute a(β j ) for 1 j t before main loop. Compute f k 1 (β j ) and g k 1 (β j ) for 1 j t. Let f (k) (β j ) = k 1 i=0 a i(x n α n) i and g (k) (β j ) = k 1 i=0 b i(x n α n) i. Then T k (β j ) = coeff (a(β j ), (x n α) k ) k 1 i=1 }{{} b i(x 1)c k i (x 1). }{{} O(d 2 ) (n 1)d O( #f #g + #a }{{} + }{{}}{{} d 2 t + (#a/d + #f + #g) t }{{} error Taylor Taylor eval We eliminated the the error computation in Z p[x 1,..., x n 1]! O(d 2 ) + d 2 t }{{} UniDio + }{{} dt 2 ). Vandermonde

22 Factoring the determinants of Cyclic and Toeplitz matrices Let C n denote the n n cyclic matrix and let T n denote the n n symmetric Toeplitz matrix below. x 1 x 2... x n 1 x n x 1 x 2 x n 1 x n x n x 1... x n 2 x n 1 x 2 x 1 x n 2 x n 1 C n =..... T n = x 3 x 4... x 1 x 2 x n 1 x n 2 x 1 x 2 x 2 x 3... x n x 1 x n x n 1 x 2 x 1 The determinants of C n and T n are homogeneous polynomials in n variables. Example det T 4 = ( x 1 2 x 1x 2 x 1x 4 x x 2x 3 + x 2x 4 x 3 2 ) ( x1 2 + x 1x 2 + x 1x 4 x x 2x 3 + x 2x 4 x 3 2 ) det C 4 = (x 4 + x 3 + x 1 + x 2) (x 4 x 3 x 1 + x 2) ( x x 1x 3 + x x 2x 4 + x x 4 2 )

23 n #d n #factors Maple (MDP) MTSHL Magma Singular , % , % , % , % , % , % , % , % , % Table: Factorization timings (seconds) for det T n evaluated at x n = 1 Notes: Intel Core i GHz, 16 gigs RAM. Maple 17: kernelopts(numcpus=1), Magma , Singular 3 1 6,

24 n #d n #fmax Maple (MDP) MTSHL Magma Singular % % % % % % % ?? % > % Table: Factorization timings (seconds) for det C n evaluated at x n = 1 Notes:?? = I cannot compute det(c n) nor read in det(c n) nor it s factors.

Towards High Performance Multivariate Factorization. Michael Monagan. This is joint work with Baris Tuncer.

Towards High Performance Multivariate Factorization Michael Monagan Center for Experimental and Constructive Mathematics Simon Fraser University British Columbia This is joint work with Baris Tuncer. To