On The Complexity Of Computing Grbner Bases For Weighted Homogeneous - PowerPoint PPT Presentation

On The Complexity Of Computing Gröbner Bases For Weighted Homogeneous Systems Jean-Charles Faugère 1 Mohab Safey El Din 1 , 2 Thibaut Verron 1 1 Université Pierre et Marie Curie, Paris 6, France INRIA Paris-Rocquencourt, Équipe P OL S YS Laboratoire d’Informatique de Paris 6, UMR CNRS 7606 2 Institut Universitaire de France Journées Nationales de Calcul Formel, 6 novembre 2014

Context Polynomial System Solving ◮ Input: polynomial system f 1 , . . . , f m ∈ K [ X 1 , . . . , X n ] ◮ Output: exact solution Important and difficult ◮ Many applications ◮ Cryptography, mechanics... ◮ Difficult problem ◮ Decision problem is NP-hard ◮ Many tools ◮ Triangular sets [Aubry, Lazard and Moreno Maza 1999] ◮ Resultants [Cattani and Dickenstein 2005] ◮ Geometric resolution [Giusti, Lecerf and Salvy 2001] ◮ Gröbner bases [Buchberger 1965]

Context Polynomial System Solving ◮ Input: polynomial system f 1 , . . . , f m ∈ K [ X 1 , . . . , X n ] ◮ Output: exact solution Computing Gröbner bases (Buchberger, F 4 , F 5 ...) 1. Select a set of pairs of polynomials from a queue 2. Reduce these polynomials 3. Add the new polynomials to the basis, add new pairs to the queue 4. Repeat 1-3 until the queue is empty

Context Polynomial System Solving Importance of structure ◮ Systems from applications are not generic! ◮ Input: polynomial system f 1 , . . . , f m ∈ K [ X 1 , . . . , X n ] ◮ Design dedicated strategies ◮ Output: exact solution ◮ Complexity studies with generic properties Computing Gröbner bases Examples of structures (Buchberger, F 4 , F 5 ...) ◮ Homogeneous systems 1. Select a set of pairs of ◮ Multi-homogeneous systems (Dickenstein, polynomials from a queue Emiris, Faugère/Safey/Spaenlehauer...) 2. Reduce these polynomials ◮ Systems with group symmetries (Colin, 3. Add the new polynomials to Gattermann, Faugère/Rahmany, the basis, add new pairs to Faugère/Svartz...) the queue ◮ Weighted homogeneous systems 4. Repeat 1-3 until the queue is ◮ Sparse systems (Sturmfels, empty Faugère/Spaenlehauer/Svartz...)

Problem statement: an example (1) Discrete Logarithm Problem on Edwards elliptic curves (Faugère, Gaudry, Huot, Renault 2013) 7871 53362 26257 25203 19817 9843 11204               18574 50900 128 23117 29737 3752 25459               e 16 e 8 e 7 e 6 e 2 e 5 e 3 e 4 e 4 e 3 e 5 0 =  14294   36407   3037   28918   52187   27006   58263  + ˜ 1 + ˜ 1 ˜ e 2 + ˜ 1 ˜ 2 + ˜ 1 ˜ 2 + ˜ 1 ˜ 2 + ˜ 1 ˜   5             2  32775   58813   38424   29298   36574   64195   17964                20289 20802 41456 56353 46683 63059 57146 46217 63811 40524 4522 27518           5478 50777 6881 1728 32176           e 3 + 2067 smaller monomials e 2 e 6 e 7 e 8 e 7 e 6  45631   48809   1238   18652   31159  + ˜ 1 ˜ 2 + e 1 ˜ ˜ 2 + ˜ 2 + ˜ 1 ˜ e 3 + ˜ 1 ˜ e 2 ˜            13171   1858   8056   54885   28424            42548 55751 54831 8241 5276 Description of the system Goal: compute a Gröbner basis ◮ Normal strategy (total degree) ◮ Ideal invariant under the group → difficult ( Z / 2 Z ) n − 1 ⋊ S n , → non regular rewritten with the invariants: ◮ Weighted degree strategy � e i := e i ( x 2 1 , . . . , x 2 n ) ( 1 ≤ i ≤ n − 1 ) ˜ Weight (˜ e i ) = 2 · Weight ( e i ) → easier e n ( x 1 , . . . , x n ) → regular ◮ n equations of degree 2 n − 1 in F q [˜ e 1 , . . . , ˜ e n − 1 , e n ] ◮ 1 DLP = thousands of such systems

Problem statement: an example (1) Discrete Logarithm Problem on Edwards elliptic curves (Faugère, Gaudry, Huot, Renault 2013) 7871 53362 26257 25203 19817 9843 11204               18574 50900 128 23117 29737 3752 25459               e 16 e 8 e 7 e 6 e 2 e 5 e 3 e 4 e 4 e 3 e 5 0 =  14294   36407   3037   28918   52187   27006   58263  + ˜ 1 + ˜ 1 ˜ e 2 + ˜ 1 ˜ 2 + ˜ 1 ˜ 2 + ˜ 1 ˜ 2 + ˜ 1 ˜   5             2  32775   58813   38424   29298   36574   64195   17964                20289 20802 41456 56353 46683 63059 57146 46217 63811 40524 4522 27518           5478 50777 6881 1728 32176           e 3 + 2067 smaller monomials e 2 e 6 e 7 e 8 e 7 e 6  45631   48809   1238   18652   31159  + ˜ 1 ˜ 2 + e 1 ˜ ˜ 2 + ˜ 2 + ˜ 1 ˜ e 3 + ˜ 1 ˜ e 2 ˜            13171   1858   8056   54885   28424            42548 55751 54831 8241 5276 Description of the system Goal: compute a Gröbner basis ◮ Normal strategy (total degree) ◮ Ideal invariant under the group → difficult ( Z / 2 Z ) n − 1 ⋊ S n , → non regular rewritten with the invariants: ◮ Weighted degree strategy � e i := e i ( x 2 1 , . . . , x 2 n ) ( 1 ≤ i ≤ n − 1 ) ˜ Weight (˜ e i ) = 2 · Weight ( e i ) → easier e n ( x 1 , . . . , x n ) → regular ◮ n equations of degree 2 n − 1 in F q [˜ e 1 , . . . , ˜ e n − 1 , e n ] ◮ 1 DLP = thousands of such systems

Problem statement: an example (2) Discrete Logarithm Problem on Edwards elliptic curves (Faugère, Gaudry, Huot, Renault 2013) Algorithm F 5 , step by step Degree 30 Normal 20 10 5 1 10 15 20 25 30 35 Step ◮ 5 equations of degree ( 16 , . . . , 16 ) in 5 variables with W = ( 2 , . . . , 2 , 1 ) ◮ 65 536 solutions ◮ Without weights: 2 h (37 steps) ◮ With weights: 15 min (29 steps)

Problem statement: an example (3) Discrete Logarithm Problem on Edwards elliptic curves (Faugère, Gaudry, Huot, Renault 2013) 7871 53362 26257 25203 19817 9843 11204               18574 50900 128 23117 29737 3752 25459               e 16 e 8 e 7 e 6 e 2 e 5 e 3 e 4 e 4 e 3 e 5 0 =  14294   36407   3037   28918   52187   27006   58263  + ˜ 1 + ˜ 1 ˜ e 2 + ˜ 1 ˜ 2 + ˜ 1 ˜ 2 + ˜ 1 ˜ 2 + ˜ 1 ˜   5             2  32775   58813   38424   29298   36574   64195   17964                20289 20802 41456 56353 46683 63059 57146 46217 63811 40524 4522 27518           5478 50777 6881 1728 32176           e 3 + 2067 smaller monomials e 2 e 6 e 7 e 8 e 7 e 6  45631   48809   1238   18652   31159  + ˜ 1 ˜ 2 + e 1 ˜ ˜ 2 + ˜ 2 + ˜ 1 ˜ e 3 + ˜ 1 ˜ e 2 ˜            13171   1858   8056   54885   28424            42548 55751 54831 8241 5276 Description of the system Goal: compute a Gröbner basis ◮ Normal strategy (total degree) ◮ Ideal invariant under the group → difficult ( Z / 2 Z ) n − 1 ⋊ S n , → non regular rewritten with the invariants: ◮ Weighted degree strategy � e i := e i ( x 2 1 , . . . , x 2 n ) ( 1 ≤ i ≤ n − 1 ) ˜ Weight (˜ e i ) = 2 · Weight ( e i ) → easier e n ( x 1 , . . . , x n ) → regular ◮ n equations of degree 2 n − 1 in F q [˜ e 1 , . . . , ˜ e n − 1 , e n ] ◮ 1 DLP = thousands of such systems

Problem statement: an example (4) Discrete Logarithm Problem on Edwards elliptic curves (Faugère, Gaudry, Huot, Renault 2013) Algorithm F 5 , step by step Degree W -degree / 2 30 Normal Weighted 20 10 5 1 10 15 20 25 30 35 Step ◮ 5 equations of degree ( 16 , . . . , 16 ) in 5 variables with W = ( 2 , . . . , 2 , 1 ) ◮ 65 536 solutions ◮ Without weights: 2 h (37 steps) ◮ With weights: 15 min (29 steps)

Problem statement: another example Ideal of relations between 50 monomials of degree 2 in 25 variables Algorithm F 4 , step by step Degree Normal 20 Weighted 10 5 0 15 25 35 Step ◮ 50 equations of degree 2 in 75 variables ◮ GR EV L EX ordering (e.g. for a 2-step strategy) ◮ Without weights: 3 . 9 h (34 steps reaching degree 22) ◮ With weights: 0 . 1 s (5 steps reaching degree 6)

Problem statement: another example Ideal of relations between 50 monomials of degree 2 in 25 variables Algorithm F 4 , step by step Degree Normal 20 Weighted 10 0 5 15 25 35 Step Problem ◮ Strategy for this structure? ◮ Complexity bounds? Relevant generic properties?