Jean-Charles Faugre with many collaborators [in the talk] Workshop - PowerPoint PPT Presentation

Recent progress on computing Gröbner bases: theory and practice Jean-Charles Faugère with many collaborators [in the talk] Workshop 3: Computer Algebra and Polynomials Linz - Nov. 2013

Solving Polynomial Systems of Equations K a field, K r x 1 , . . . , x n s multivariate polynomials in n variables. $ f 1 p x 1 , . . . , x n q “ 0 & ¨ ¨ ¨ % f m p x 1 , . . . , x n q “ 0 In this talk: Zero-dimensional System = Finite Number of Solutions ☞ Reduce the difficult problem (several equations, deg ą 1) Ý Ñ easier case (several polynomials in one variable)

Solving Polynomial Systems of Equations K a field, K r x 1 , . . . , x n s multivariate polynomials in n variables. $ f 1 p x 1 , . . . , x n q “ 0 & ¨ ¨ ¨ % f m p x 1 , . . . , x n q “ 0 In this talk: Zero-dimensional System = Finite Number of Solutions ☞ Reduce the difficult problem (several equations, deg ą 1) Ý Ñ easier case (several polynomials in one variable) Tool: Gröbner bases [Buchberger] (rely heavily on linear algebra).

Applications: source of challenging problems Cryptology (finite fields) HFE, Minrank, IP, Discrete Logarithm Problem (finite fields or curves) Error Correcting Codes, (Mc Eliece ) ... Robotic Comp. Geometry Parallel Robots, Topology of ridges, Mecanisms, ... Voronoi, ..

Gröbner Bases: summary f 1 “ ¨ ¨ ¨ “ f m “ 0 Gaussian Elimination of Macaulay matrices up to degree d max Gröbner: total degree Linear algebra in K r x s{ I ù x i “ h i p x n q Gröbner: lexicographical

Gröbner Bases: summary f 1 “ ¨ ¨ ¨ “ f m “ 0 Macaulay Matrix in degree d Gaussian Elimination of m 1 ą m 2 ą ¨ ¨ ¨ ą m k Macaulay matrices up to ¨ ˛ t 1 , 1 f 1 degree d max . . . t 1 , 2 f 1 ˚ ‹ . . . ˚ ‹ . ˚ ‹ . ˚ ‹ M d “ . coeff p t f i , m j q ˚ ‹ ˚ ‹ t 2 , 1 f 2 . . . Gröbner: total degree ˝ ‚ . . . . . . Linear algebra in K r x s{ I ù x i “ h i p x n q Gröbner: lexicographical

Gröbner Bases: summary f 1 “ ¨ ¨ ¨ “ f m “ 0 terms of degree d Macaulay Matrix in degree d Gaussian Elimination of m 1 ą m 2 ą ¨ ¨ ¨ ą m k Macaulay matrices up to ¨ ˛ t 1 , 1 f 1 degree d max . . . t 1 , 2 f 1 ˚ ‹ . . . ˚ ‹ . ˚ ‹ . ˚ ‹ M d “ . coeff p t f i , m j q ˚ ‹ ˚ ‹ t 2 , 1 f 2 . . . Gröbner: total degree ˝ ‚ . . . . . . Linear algebra in K r x s{ I ù x i “ h i p x n q all products t f i , t P Monomials p d ´ deg p f i qq Gröbner: lexicographical

Gröbner Bases: summary f 1 “ ¨ ¨ ¨ “ f m “ 0 terms of degree d Macaulay Matrix in degree d Gaussian Elimination of m 1 ą m 2 ą ¨ ¨ ¨ ą m k Macaulay matrices up to ¨ ˛ t 1 , 1 f 1 degree d max . . . t 1 , 2 f 1 ˚ ‹ . . . ˚ ‹ . ˚ ‹ . ˚ ‹ M d “ . coeff p t f i , m j q ˚ ‹ ˚ ‹ t 2 , 1 f 2 . . . Gröbner: total degree ˝ ‚ . . . . . . Linear algebra in K r x s{ I ù x i “ h i p x n q all products t f i , t P Monomials p d ´ deg p f i qq Gröbner: lexicographical Maximal degree d reached: d max We stop the computation when # Rows ě # Columns Algorithmic goal: generate full rank matrices

Gröbner Bases: summary f 1 “ ¨ ¨ ¨ “ f m “ 0 terms of degree d • Buchberger (1965) Macaulay Matrix in degree d • F 4 (1999) Gaussian Elimination of m 1 ą m 2 ą ¨ ¨ ¨ ą m k • F 5 (2002) Macaulay matrices up to ¨ ˛ t 1 , 1 f 1 • . . . degree d max . . . t 1 , 2 f 1 ˚ ‹ . . . ˚ ‹ . ˚ ‹ . ˚ ‹ M d “ . coeff p t f i , m j q ˚ ‹ ˚ ‹ t 2 , 1 f 2 . . . Gröbner: total degree ˝ ‚ . . . . . . • FGLM (1993) Linear algebra in K r x s{ I ù x i “ h i p x n q all products t f i , t P Monomials p d ´ deg p f i qq Gröbner: lexicographical Maximal degree d reached: d max We stop the computation when # Rows ě # Columns Algorithmic goal: generate full rank matrices

Gröbner Bases: summary f 1 “ ¨ ¨ ¨ “ f m “ 0 terms of degree d Gaussian Elimination of • Buchberger (1965) Macaulay Matrix in degree d Macaulay matrices up to • F 4 (1999) m 1 ą m 2 ą ¨ ¨ ¨ ą m k degree d max • F 5 (2002) ¨ ˛ t 1 , 1 f 1 • . . . . . . « t 1 , 2 f 1 ˚ ‹ ` n ` d max ˘ ω q . . . ˚ ‹ O p . ˚ ‹ n . ˚ ‹ M d “ . coeff p t f i , m j q ˚ ‹ ˚ ‹ t 2 , 1 f 2 . . . Gröbner: total degree ˝ ‚ . . . . Linear algebra in K r x s{ I . . • FGLM (1993) ù x i “ h i p x n q all products t f i , O p # Sols 3 q ˜ t P Monomials p d ´ deg p f i qq Gröbner: lexicographical Maximal degree d reached: d max We stop the computation when # Rows ě # Columns Algorithmic goal: generate full rank matrices

Research Directions Intrinsic Exponential Complexity: # Sols “ D “ ś deg p f i q and NP-hard when K “ F p Hopeless ?

Research Directions Intrinsic Exponential Complexity: # Sols “ D “ ś deg p f i q and NP-hard when K “ F p Hopeless ? Implementations/ Structured Systems Algorithms Linear Algebra Symmetries ˜ O p # Sols ω q Overdetermined Dedicated Linear Algebra Finite fields Multi-core implementations Bilinear eqs [Lachartre, Martani, Eder] Quasi-homogeneous LGPL Multi-homogeneous

Over F p : Katsura 18, # Sols =262144 Research Directions solutions, Size > 200 Gb Over Q : problem submitted by D. Henrion as a numerical challenge. # Sols =40320 Compute 7 univariate polynomials of size 3.2 Gbytes ☞ Bottleneck: real roots isolation (cannot be read by Maple) Intrinsic Exponential Complexity: # Sols “ D “ ś deg p f i q and NP-hard when K “ F p Hopeless ? Implementations/ Structured Systems Algorithms Linear Algebra Symmetries ˜ O p # Sols ω q Overdetermined Dedicated Linear Algebra Finite fields Multi-core implementations Bilinear eqs [Lachartre, Martani, Eder] Quasi-homogeneous LGPL Multi-homogeneous

Structured Systems

Solving Systems with Symmetries G is a finite group. Compute the roots of the system: V L “ t z P L n | f 1 p z q “ ¨ ¨ ¨ “ f m p z q “ 0 u Difficult case: V L is globally invariant by G : if z P V L then σ . z P V L for all σ P G Open Issue: How to compute efficiently V L { G ?

Solving Systems with Symmetries Open Issue: How to compute efficiently V L { G ? Theorem ([F., Svartz 2013]) I “ p f 1 , . . . , f m q a 0-dimensional ideal, invariant under an Abelian Group G “ Z q 1 ˆ ¨ ¨ ¨ ˆ Z q k . Dedicated F 5 algorithm and divide the GB complexity by: | G | 3 Abelian Group and/or Multi-homogeneous : Grading p d 1 , . . . , d k q with d i P Z q i where q i “ 0 or q i “ p k i i Instead of Macaulay p d q Ý Ñ Ť Macaulay p d 1 , . . . , d k q

Overdetermined Systems Theorem ( Bardet, F., Salvy ) For m “ α n semi-regular quadratic equations in Q r x 1 , . . . , x n s : a d max « p α ´ 1 2 ´ α p α ´ 1 qq n 1 d max { n 0 . 8 0 . 6 0 . 4 0 . 2 0 α 1 1 . 5 2 2 . 5 3

Overdetermined Systems Theorem ( Bardet, F., Salvy ) For m “ α n semi-regular quadratic equations in Q r x 1 , . . . , x n s : a d max « p α ´ 1 2 ´ α p α ´ 1 qq n 1 d max { n If m “ n 1 ` β with 0 ă β ă 1 0 . 8 d max « 1 8 n 1 ´ β 0 . 6 ☞ Sub-exp algorithm 0 . 4 0 . 2 0 α 1 1 . 5 2 2 . 5 3

Improve the complexity when solutions are in a finite field Fact: in F p solving m equations ¨ ¨ ¨ , f i p x 1 , . . . , x n q , ¨ ¨ ¨ in n variables õ solve q k systems of m equations / n ´ k variables ð Overdetermined k tradeoff between exhaustive search and Gröbner General Case [Bettale, F .,Perret, Issac, 2012] direct Gröbner basis approach „ 2 1 . 8 n hybrid approach Boolean case over F 2 p K “ F 2 q Theorem ( [Bardet, F.,Salvy, Spaenlehauer, J. Comp.2012] ) Under precise algebraic assumption, a Boolean quadratic polynomial p f 1 , . . . , f α n q can be solved in probabilistic time faster than exh. search: O p 2 p 1 ´ 0 . 208 α q n q when α ď 1 . 82

Key Ingredients Solving sparse linear systems ! D. Wiedemann. Solving sparse linear equations over finite fields. IEEE Transactions on Information Theory , 32(1):54–62, 1986. E. Kaltofen and B. David Saunders. On Wiedemann’s method of solving sparse linear systems. AAECC , p. 29–38, 1991. G. Villard. Further analysis of Coppersmith’s block Wiedemann algorithm for the solution of sparse linear systems. ISSAC’97, p. 32–39. ACM, 1997. M. Giesbrecht, A. Lobo, and B. D. Saunders. Certifying inconsistency of sparse linear systems. ISSAC’98, p. 113–119, 1998.

Solving α n equations in n variables: 2 c n c : exponent of the complexity Exhaustive search 1 0.79 Dedicated Algorithm Gröbner Bases 0.31 α 0 1 1.82 3 4 5

Bilinear systems Particular case of multi-homogeneous systems: BiLinear ÿ f p h q p x 0 , . . . , x n x , y 0 , . . . , y n y q “ a i , j x i y j . Minrank Input: M 1 , . . . , M k k n ˆ n matrices in K n 2 and r ă n integer Find if any λ 1 , . . . , λ k P ¯ K such that: λ 1 M 1 ` . . . ` λ k M k ´ I n has rank r NP hard ! J.O. Shallit, G.S. Frandsen, and J.F. Buss. The Computational Complexity of some Problems of Linear Algebra . BRICS series report, Aarhus, Denmark, RS-96-33. Can be used to break cryptosystems: HFE, Minrank, . . . Can be used to simplify quadratic system of equations