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 Thibaut Verron 2 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 Toulouse Universités, INP-ENSEEIHT-IRIT, CNRS, Équipe APO Séminaire Géométrie et Algèbre Effectives , 2 juin 2017

Polynomial system solving Applications: ◮ Numerical: give approximations of ◮ Cryptography the solutions ◮ Physics, industry ◮ Newton’s method ◮ Homotopy continuation method ◮ Mathematics. . . ◮ Symbolic: give exact solutions ◮ Gröbner bases Polynomial equations ◮ Resultant method ◮ Triangular sets f 1 ( X ) = ··· = f m ( X ) = 0 ◮ Geometric resolution Solutions, e.g. find all the solutions if finite (dimension 0)

Computing Gröbner bases for generic systems: the normal strategy Degree 10 Regular Gröbner basis algorithms (e.g. F 5 ) ◮ Compute a basis by iteratively 8 building and reducing matrices of polynomials of same degree 6 ◮ Normal strategy: perform lowest-degree reductions first Degree falls 4 ◮ Degree = indicator of progress Step 0 2 4 6 8 10 12 Regular sequences = ⇒ algorithmic regularity! Degree fall? ◮ F 5 -criterion: no reduction to zero in F 5 ( ⇐ ◮ Definition: reduction resulting in a lower degree polynomial ⇒ all matrices have full-rank) for regular sequences ◮ Example: X · ( Y − 1 ) − Y · ( X − 1 ) = XY − YX + Y − X ◮ Degree falls ⇐ ⇒ Reduction to zero of the highest degree components ◮ Consequence: “next d ” < d + 1 � Regularity in the affine sense = regularity of the highest degree components

Computing Gröbner bases for generic systems: the normal strategy Degree 10 Regular Gröbner basis algorithms (e.g. F 5 ) Irregular ◮ Compute a basis by iteratively 8 building and reducing matrices of polynomials of same degree 6 ◮ Normal strategy: perform lowest-degree reductions first Degree falls 4 ◮ Degree = indicator of progress Step 0 2 4 6 8 10 12 Regular sequences = ⇒ algorithmic regularity! Degree fall? ◮ F 5 -criterion: no reduction to zero in F 5 ( ⇐ ◮ Definition: reduction resulting in a lower degree polynomial ⇒ all matrices have full-rank) for regular sequences ◮ Example: X · ( Y − 1 ) − Y · ( X − 1 ) = XY − YX + Y − X ◮ Degree falls ⇐ ⇒ Reduction to zero of the highest degree components ◮ Consequence: “next d ” < d + 1 � Regularity in the affine sense = regularity of the highest degree components

Computing Gröbner bases for generic systems: the normal strategy Degree 10 Regular Gröbner basis algorithms (e.g. F 5 ) Irregular ◮ Compute a basis by iteratively 8 building and reducing matrices of polynomials of same degree 6 ◮ Normal strategy: perform lowest-degree reductions first Degree falls 4 ◮ Degree = indicator of progress Step 0 2 4 6 8 10 12 Regular sequences = ⇒ algorithmic regularity! Degree fall? ◮ F 5 -criterion: no reduction to zero in F 5 ( ⇐ ◮ Definition: reduction resulting in a lower degree polynomial ⇒ all matrices have full-rank) for regular sequences ◮ Example: X · ( Y − 1 ) − Y · ( X − 1 ) = XY − YX + Y − X ◮ Degree falls ⇐ ⇒ Reduction to zero of the highest degree components ◮ Consequence: “next d ” < d + 1 � Regularity in the affine sense = regularity of the highest degree components

Computing Gröbner bases for generic systems: the normal strategy Degree 10 Regular Gröbner basis algorithms (e.g. F 5 ) Irregular ◮ Compute a basis by iteratively 8 building and reducing matrices of polynomials of same degree 6 ◮ Normal strategy: perform lowest-degree reductions first Degree falls 4 ◮ Degree = indicator of progress Step 0 2 4 6 8 10 12 Regular sequences = ⇒ algorithmic regularity! Degree fall? ◮ F 5 -criterion: no reduction to zero in F 5 ( ⇐ ◮ Definition: reduction resulting in a lower degree polynomial ⇒ all matrices have full-rank) for regular sequences ◮ Example: X · ( Y − 1 ) − Y · ( X − 1 ) = XY − YX + Y − X ◮ Degree falls ⇐ ⇒ Reduction to zero of the highest degree components ◮ Consequence: “next d ” < d + 1 � Regularity in the affine sense = regularity of the highest degree components

Computing Gröbner bases for generic systems: the normal strategy Degree 10 Regular Gröbner basis algorithms (e.g. F 5 ) Irregular ◮ Compute a basis by iteratively 8 building and reducing matrices of polynomials of same degree 6 ◮ Normal strategy: perform lowest-degree reductions first Degree falls 4 ◮ Degree = indicator of progress Step 0 2 4 6 8 10 12 Regular sequences = ⇒ algorithmic regularity! Degree fall? ◮ F 5 -criterion: no reduction to zero in F 5 ( ⇐ ◮ Definition: reduction resulting in a lower degree polynomial ⇒ all matrices have full-rank) for regular sequences ◮ Example: X · ( Y − 1 ) − Y · ( X − 1 ) = XY − YX + Y − X ◮ Degree falls ⇐ ⇒ Reduction to zero of the highest degree components ◮ Consequence: “next d ” < d + 1 � Regularity in the affine sense = regularity of the highest degree components This notion depends on the homogeneous structure!

Strategy and complexity for generic homogeneous systems Homogeneous, generic, with total degree ( d 1 ,..., d n ) (zero-dimensional) F ( X 1 ,..., X n ) Buchberger [Buchberger 1976] F 4 [Faugère 1999] F 5 [Faugère 2002] . . . GR EV L EX basis [Faugère, Gianni, Lazard and Mora 1993] FGLM L EX basis

Strategy and complexity for generic homogeneous systems Homogeneous, generic, with total degree ( d 1 ,..., d n ) (zero-dimensional) F ( X 1 ,..., X n )  Highest degree ∼ # of reduction steps    n    ∑ ( d i − 1 )+ 1 = d reg ≤  F 5 i = 1   � n + d − 1 �   Size of the matrix at degree d =    d GR EV L EX basis Number of solutions = ∏ n i = 1 d i (Bézout bound) FGLM � 3   � 3 � n � n + d reg − 1 L EX basis ∏ O + n d i   d reg i = 1

The weighted homogeneous structure: an example (1) Discrete Logarithm Problem on Edwards elliptic curves (Faugère, Gaudry, Huot, Renault 2013) 41518 49874 45709 46659 32367 37627             33900 32136 10698 59796 23164 25182              X 16  X 8  X 7  X 6 1 X 2  X 5 1 X 3  X 4 1 X 4 0 = 8840 34252 45336 38267 64111 59951 5 + 1 + 1 X 2 + 2 + 2 + 2 +                         22855 24932 26076 39647 63692 60422       29081 11782 55993 27683 29095 11080 27200 64271 49159 59456 17060 55016             38476 43542 11328 49518 60912 15550              X 3 1 X 5  X 2 1 X 6  X 1 X 7  X 8  X 7  X 6 28698 57950 33520 46071 64907 19633 2 + 2 + 2 + 2 + 1 X 3 + 1 X 2 X 3 +                         5708 52276 65039 49716 61073 28147       47718 9739 27178 33760 37208 25442 31264 38258 19475 4467         26817 44188 52270 31828         35757  X 5 1 X 2 46688  X 4 1 X 3 9282  X 3 1 X 4 34222  X 2 1 X 5 2 X 3 + 2063 smaller monomials 2 X 3 + 2 X 3 + 2 X 3 +                 43106 55434 51171 30753     44133 64632 17150 37662 Goal: compute a Gröbner basis Alt. strategy: use weights = substitute X i ← X w i for W = ( w 1 ,..., w 5 ) Normal strategy (total degree): i What weights? ◮ Non generic ◮ W = ( 1 , 1 , 1 , 1 , 1 ) : nothing changed ◮ Non regular in the affine sense ◮ W = ( 2 , 2 , 1 , 1 , 1 ) : better... ◮ Non regular computation ◮ W = ( 2 , 2 , 2 , 2 , 1 ) : regular!