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

On The Complexity Of Computing Gröbner Bases For Quasi-Homogeneous Systems Jean-Charles Faugère 1 Mohab Safey El Din 1 , 2 Thibaut Verron 1 , 3 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 3 École Normale Supérieure, Paris, France June 29, 2013

Motivation Discrete Logarithm Problem (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 ˜ 1 + 3037 ˜ 1 ˜ e 2 + 28918 ˜ 1 ˜ 2 + 52187 ˜ 1 ˜ 2 + 27006 ˜ 1 ˜ 2 + 58263 ˜ 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           ˜ 1 ˜ e 1 ˜ ˜ ˜ ˜ 1 ˜ ˜ 1 ˜ e 2 ˜ + 45631 2 + 48809 2 + 1238 2 + 18652 e 3 + 31159                               13171 1858 8056 54885 28424                     42548 55751 54831 8241 5276

Motivation Discrete Logarithm Problem (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 ˜ 1 + 3037 ˜ 1 ˜ e 2 + 28918 ˜ 1 ˜ 2 + 52187 ˜ 1 ˜ 2 + 27006 ˜ 1 ˜ 2 + 58263 ˜ 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           ˜ 1 ˜ e 1 ˜ ˜ ˜ ˜ 1 ˜ ˜ 1 ˜ e 2 ˜ + 45631 2 + 48809 2 + 1238 2 + 18652 e 3 + 31159                               13171 1858 8056 54885 28424                     42548 55751 54831 8241 5276 Description of the system Goal: compute a Gröbner basis ◮ Total degree grading ◮ Ideal invariant under the group → difficult (intractable with Magma) ( Z / 2 Z ) n − 1 ⋊ S n , → non regular rewritten with the invariants: ◮ Weighted degree grading � 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 ◮ Two questions: in F q [˜ e 1 , . . . , ˜ e n − 1 , e n ] ◮ Algorithms for this structure? ◮ 1 DLP = thousands of such systems ◮ Complexity estimates?

Motivation Discrete Logarithm Problem (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 ˜ 1 + 3037 ˜ 1 ˜ e 2 + 28918 ˜ 1 ˜ 2 + 52187 ˜ 1 ˜ 2 + 27006 ˜ 1 ˜ 2 + 58263 ˜ 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           ˜ 1 ˜ e 1 ˜ ˜ ˜ ˜ 1 ˜ ˜ 1 ˜ e 2 ˜ + 45631 2 + 48809 2 + 1238 2 + 18652 e 3 + 31159                               13171 1858 8056 54885 28424                     42548 55751 54831 8241 5276 Description of the system Goal: compute a Gröbner basis ◮ Total degree grading ◮ Ideal invariant under the group → difficult (intractable with Magma) ( Z / 2 Z ) n − 1 ⋊ S n , → non regular rewritten with the invariants: ◮ Weighted degree grading � 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 ◮ Two questions: in F q [˜ e 1 , . . . , ˜ e n − 1 , e n ] ◮ Algorithms for this structure? ◮ 1 DLP = thousands of such systems ◮ Complexity estimates?

Motivation Discrete Logarithm Problem (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 ˜ 1 + 3037 ˜ 1 ˜ e 2 + 28918 ˜ 1 ˜ 2 + 52187 ˜ 1 ˜ 2 + 27006 ˜ 1 ˜ 2 + 58263 ˜ 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           ˜ 1 ˜ e 1 ˜ ˜ ˜ ˜ 1 ˜ ˜ 1 ˜ e 2 ˜ + 45631 2 + 48809 2 + 1238 2 + 18652 e 3 + 31159                               13171 1858 8056 54885 28424                     42548 55751 54831 8241 5276 Description of the system Goal: compute a Gröbner basis ◮ Total degree grading ◮ Ideal invariant under the group → difficult (intractable with Magma) ( Z / 2 Z ) n − 1 ⋊ S n , → non regular rewritten with the invariants: ◮ Weighted degree grading � 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 ◮ Two questions: in F q [˜ e 1 , . . . , ˜ e n − 1 , e n ] ◮ Algorithms for this structure? ◮ 1 DLP = thousands of such systems ◮ Complexity estimates?