On the Complexity of Gröbner Basis Computation for Regular and Semi-Regular Systems

Transcription

1 On the Complexity of Gröbner Basis Computation for Regular and Semi-Regular Systems Algorithms Project, Inria Joint work with Magali Bardet & Jean-Charles Faugère September 21st, 2006

2 I Introduction

3 Don t Expect the Worst Complexity for m equations, degree d, n variables: worst case: 2 2O(n) [Mayr-Meyer82, Möller-Mora84] generically: m O(1) d O(n) [Lazard83, Giusti84] Questions: 1 how small can be the exponent in d O(n)? 2 what about overdetermined systems? does the extra information help?

4 Motivation from Cryptography Wanted: Solutions in F 2 of a system in F 2 [x 1,..., x n ]. Possible approach: add the equations xi 2 x i = 0 overdetermined system. Faugère-Joux 2003: break the HFE challenge thanks to a small regularity: How do these stairs grow?

5 Setting for the Talk homogeneous system f 1 = = f m = 0 in k[x 1,..., x n ], with char k = 0; deg f 1 = = deg f m = d; system regular if m n; semi-regular if n m; coordinates in simultaneous Noether position wrt the system; all bases are computed for the degree reverse lexicographical order (grevlex). See the forthcoming article for extensions to 1 formulæ with d i = deg f i and d 1 d m ; 2 non-homogeneous systems; 3 k = F 2.

6 Starting Point: the Macaulay Matrix M D all multiples of the f i of degree D m 1 f 1. m k f 1 m 2 f 2. m k f m Columns indexed by monomials of degree D (sorted by ) For D large enough Buchberger s algorithm (Structured) Gaussian elimination Reductions to 0 Useless lines Algorithm F 5 Construct matrices by increasing degrees [Faugère 02] avoiding useless lines coming from f i f j = f j f i.

7 Linear Algebra and its Complexity Proposition (General Upper Bound) The number of operations in k required to compute the GB up to degree D is bounded by ( ( ) n + D 1 ω ) O md, D 2 ω 3 is the complexity of matrix product. Strassen: ω < 2.81; Coppersmith-Winograd: ω < Needed: bounds on D. Regular system: D n(d 1) + 1 [Macaulay] ( d d bound (d 1) d 1 ) ωn

8 F 5 for Regular Systems in Simultaneous Noether Position Theorem (BaFaSa06, m = n) F 5 computes the GB in at most A(d) n n (C + O(1/n)) operations in k, n, with A(d) root of a simple polynomial of degree 2d 1. Quantifies how F 5 exploits the structure of the Macaulay matrix.

9 Semi-Regular Systems in Simultaneous Noether Position I. Theorem (BaFaSa06, m = n + k (k 1)) D i reg = d 1 2 d m α 2 1 k m +, n, 6 α k largest root of kth Hermite polynomial. Quantifies the gain brought by the extra equations.

10 Semi-Regular Systems in Simultaneous Noether Position II. Theorem (BaFaSa06, m = [αn] (α > 1)) D i reg = φ d (α)m + a 1 ψ d (α)m 1/3 +, n a 1 largest root of the Airy function, φ d & ψ d algebraic, φ d (α) = d 1 d 2 1 (α 1) 1/2 +, α

11 II Regular Systems

12 Hilbert: Function, Polynomial, Series Hilbert Function: I (m) := f 1,..., f m HF I (m)(d) := (dim k[x 1,..., x n ]/I (m) ) d. Using the Macaulay matrix M d : HF I (m)(d) = #cols(m d ) rank(m d ). For d large enough, this is a polynomial. The first such d is called the index of regularity (i reg (I (m) )). Hilbert series: H I (m)(z) := d 0 HF I (m)(d)z d = P(z) (1 z) δ.

13 Regular Systems Definition ((f 1,..., f m ) regular) For all i = 1,..., m, f i is not a zero-divisor in k[x 1,..., x n ]/I (i 1). (k[x 1,..., x n ]/I (i 1) ) d f i (k[x 1,..., x n ]/I (i 1) ) d+di injective d 0 HF I (i)(d + d i ) = HF I (i 1)(d + d i ) HF I (i 1)(d) for all d m j=1 H I (m)(z) = (1 zd j ) (1 z) n. Index of regularity: m (d i 1) + 1. i=1

14 F 5 for Regular Systems Proposition (Faugère02) For regular systems, F 5 constructs matrices M (i) d such that #cols( M (i) d ) #rows( M (i) d ) = HF I (i)(d). No reduction to 0 at all! Example: n = m = 4, d = 3

15 Noether Position Definition ((x 1,..., x m ) in Noether position wrt (f 1,..., f m )) 1 Their canonical images in k[x 1,..., x n ]/ f 1,..., f m are algebraic integers over k[x m+1,..., x n ]; 2 k[x m+1,..., x n ] f 1,..., f m = 0. Definition (Simultaneous Noether position) For i = 1,..., m, (x 1,..., x i ) in Noether position wrt (f 1,..., f i ). the leading terms of the elements of the grevlex GB do not depend on x m+1,..., x n.

16 Shape of a Gröbner Basis I Theorem (new?) (x 1,..., x n ) in simultaneous Noether position. G i reduced Gröbner basis of (f 1,..., f i ), 1 i m. The number of polynomials of degree d in G i \ G i 1 is bounded by b (i) d, where B i (z) = d=0 b (i) i 1 d zd = z d i k=1 1 z d k 1 z.

17 Shape of a Gröbner Basis II Number of operations for F 5 bounded by m i=1 i reg b (i) d d=d i ( )( ) n + d 1 i + d 1 d d

18 III Semi-Regular Systems

19 Definition Introduction Regular Systems Semi-Regular Systems Regular systems cannot be overdetermined. Definition ((f 1,..., f m ) semi-regular (m n)) HF I (i)(d) = HF I (i 1)(d) HF I (i 1)(d d i ), d i d < i reg (I (m) ) Proposition (Hilbert Series) [ m j=1 H I (m)(z) = (1 ] zd j ) (1 z) n. Notation: [ a i z i] = b i z i, with b i = i 0 i 0 { a i if a j > 0 for 0 j i 0 otherwise.

20 Asymptotics of the Index of Regularity Approach: 1 Compute an asymptotic approximation of the coefficient sequence in the neighborhood of the 0; 2 find its smallest zero.

21 Few More Equations than Unknowns m j=1 (1 zd j ) (1 z) n = (1 z) m n m j=1 1 z d j 1 z } {{ } F (z) Coefficients of F (z) approximated well by the saddle-point method. Cauchy: [z k ]F (z) = 1 2iπ F (z) dz (1 z) m n z k+1 z k+1 dz. 1 Saddle-point: F (ρ) = 0; 2 Locally: F (z) F (ρ)e λu2 (1 ρ iu) m n ; 3 coeff F (ρ) 2π e λu2 (1 ρ iu) m n du. }{{} π 2 k λ k+1 Hm n((1 ρ) λ)

22 Even More Equations 1 2iπ (1 z d ) αn (1 z) n dz zk+1 Small k Transition Larger k The coalescence of saddle-points is captured by the Airy function.

23 Coalescent Saddle-Points [Chester-Friedman-Ursell 57] Capture both saddle-points by a cubic change of variables. Leads to uniform asymptotic expansions involving Ai(z) = 1 e iπ/3 e t3 /3 zt dt. 2iπ e iπ/3 Airy z 0, two neighboring saddle-points Airy z = 0 A double saddle-point

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