On the Existence of Semi-Regular Sequences Sergio Molina 1 joint - PowerPoint PPT Presentation

On the Existence of Semi-Regular Sequences Sergio Molina 1 joint work with T. J. Hodges 1 J. Schlather 1 Department of Mathematics University of Cincinnati DIMACS, January 2015 Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 1 / 20

Background Important Problem: Finding solutions to systems of polynomial equations of the form p 1 ( x 1 , . . . , x n ) = β 1 , . . . , p m ( x 1 , . . . , x n ) = β m . (1) Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 2 / 20

Background Important Problem: Finding solutions to systems of polynomial equations of the form p 1 ( x 1 , . . . , x n ) = β 1 , . . . , p m ( x 1 , . . . , x n ) = β m . (1) MPKC systems: Multivariate Public Key Cryptographic systems. Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 2 / 20

Background Important Problem: Finding solutions to systems of polynomial equations of the form p 1 ( x 1 , . . . , x n ) = β 1 , . . . , p m ( x 1 , . . . , x n ) = β m . (1) MPKC systems: Multivariate Public Key Cryptographic systems. The security of MPKC systems relies on the difficulty of solving a system (1) of quadratic equations over a finite field. Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 2 / 20

Background Main types of algorithms used to solve such systems of equations are: Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 3 / 20

Background Main types of algorithms used to solve such systems of equations are: Gr¨ obner basis algorithm [Buchberger] and its variants F 4 and F 5 [Faug` ere]. Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 3 / 20

Background Main types of algorithms used to solve such systems of equations are: Gr¨ obner basis algorithm [Buchberger] and its variants F 4 and F 5 [Faug` ere]. The XL algorithms including FXL [Courtois et al.] and mutantXL [Buchmann et al.]. Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 3 / 20

Background To assess complexity of the F 4 and F 5 algorithms for solution of polynomial equations the concept of “semi-regular” sequences over F 2 was introduced by Bardet, Faug` ere, Salvy and Yang. Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 4 / 20

Background To assess complexity of the F 4 and F 5 algorithms for solution of polynomial equations the concept of “semi-regular” sequences over F 2 was introduced by Bardet, Faug` ere, Salvy and Yang. Roughly speaking, semi-regular sequences over F 2 are sequences of homogeneous elements of the algebra B ( n ) = F 2 [ X 1 , ..., X n ] / ( X 2 1 , ..., X 2 n ) which have as few relations between them as possible. Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 4 / 20

Background To assess complexity of the F 4 and F 5 algorithms for solution of polynomial equations the concept of “semi-regular” sequences over F 2 was introduced by Bardet, Faug` ere, Salvy and Yang. Roughly speaking, semi-regular sequences over F 2 are sequences of homogeneous elements of the algebra B ( n ) = F 2 [ X 1 , ..., X n ] / ( X 2 1 , ..., X 2 n ) which have as few relations between them as possible. Experimental evidence has shown that randomly generated sequences tend to be semi-regular. Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 4 / 20

Definitions Let B d ⊂ B ( n ) be the set of homogeneous polynomials of degree d . Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 5 / 20

Definitions Let B d ⊂ B ( n ) be the set of homogeneous polynomials of degree d . Definition 1 Let B ( n ) = F 2 [ X 1 , ..., X n ] / ( X 2 n ) . If λ 1 , ..., λ m ∈ B ( n ) is a sequence 1 , ..., X 2 of homogeneous elements of positive degrees d 1 , ..., d m and I = ( λ 1 , ..., λ m ) then Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 5 / 20

Definitions Let B d ⊂ B ( n ) be the set of homogeneous polynomials of degree d . Definition 1 Let B ( n ) = F 2 [ X 1 , ..., X n ] / ( X 2 n ) . If λ 1 , ..., λ m ∈ B ( n ) is a sequence 1 , ..., X 2 of homogeneous elements of positive degrees d 1 , ..., d m and I = ( λ 1 , ..., λ m ) then Ind( I ) = min { d ≥ 0 | I ∩ B d = B d } Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 5 / 20

Definitions Let B d ⊂ B ( n ) be the set of homogeneous polynomials of degree d . Definition 1 Let B ( n ) = F 2 [ X 1 , ..., X n ] / ( X 2 n ) . If λ 1 , ..., λ m ∈ B ( n ) is a sequence 1 , ..., X 2 of homogeneous elements of positive degrees d 1 , ..., d m and I = ( λ 1 , ..., λ m ) then Ind( I ) = min { d ≥ 0 | I ∩ B d = B d } The sequence λ 1 , . . . , λ m is semi-regular over F 2 if for all i = 1 , 2 , . . . , m , if µ is homogeneous and µλ i ∈ ( λ 1 , . . . , λ i − 1 ) and deg( µ ) + deg( λ i ) < Ind( I ) then µ ∈ ( λ 1 , . . . , λ i ). Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 5 / 20

Characterization with Hilbert Series Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 6 / 20

Characterization with Hilbert Series The truncation of a series � a i z i is defined to be: �� a i z i � � b i z i = where b i = a i if a j > 0 for all j ≤ i , and b i = 0 otherwise. Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 6 / 20

Characterization with Hilbert Series The truncation of a series � a i z i is defined to be: �� a i z i � � b i z i = where b i = a i if a j > 0 for all j ≤ i , and b i = 0 otherwise. For instance [1 + 10 z + z 2 + 20 z 3 − z 4 + z 6 + · · · ] = 1 + 10 z + z 2 + 20 z 3 Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 6 / 20

Characterization with Hilbert Series Theorem 2 (Bardet, Faug` ere, Salvy, Yang) Let λ 1 , ..., λ m ∈ B ( n ) be a sequence of homogeneous elements of positive degrees d 1 , ..., d m and I = ( λ 1 , ..., λ m ) . Then, the sequence λ 1 , ..., λ m is semi-regular if and only if (1 + z ) n � � Hilb B ( n ) / I ( z ) = � m i =1 (1 + z d i ) Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 7 / 20

Characterization with Hilbert Series Theorem 2 (Bardet, Faug` ere, Salvy, Yang) Let λ 1 , ..., λ m ∈ B ( n ) be a sequence of homogeneous elements of positive degrees d 1 , ..., d m and I = ( λ 1 , ..., λ m ) . Then, the sequence λ 1 , ..., λ m is semi-regular if and only if (1 + z ) n � � Hilb B ( n ) / I ( z ) = � m i =1 (1 + z d i ) Let λ 1 , ..., λ m ∈ B ( n ) be a sequence of homogeneous elements and let I = ( λ 1 , ..., λ m ). If the sequence is semi-regular then Ind( λ 1 , ..., λ m ) = 1 + deg( Hilb B ( n ) / I ( z )) Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 7 / 20

Example Consider the element λ = x 1 x 2 + x 3 x 4 + x 5 x 6 in B (6) and let I = ( λ ). Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 8 / 20

Example Consider the element λ = x 1 x 2 + x 3 x 4 + x 5 x 6 in B (6) and let I = ( λ ). HS B (6) / I ( z ) = 1 + 6 z + 14 z 2 + 14 z 3 + z 4 and (1 + z ) 6 = 1 + 6 z + 14 z 2 + 14 z 3 + z 4 − 8 z 5 + · · · 1 + z 2 Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 8 / 20

Example Consider the element λ = x 1 x 2 + x 3 x 4 + x 5 x 6 in B (6) and let I = ( λ ). HS B (6) / I ( z ) = 1 + 6 z + 14 z 2 + 14 z 3 + z 4 and (1 + z ) 6 = 1 + 6 z + 14 z 2 + 14 z 3 + z 4 − 8 z 5 + · · · 1 + z 2 � (1+ z ) 6 � = 1 + 6 z + 14 z 2 + 14 z 3 + z 4 = HS B (6) / I ( z ). 1+ z 2 Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 8 / 20

Example Consider the element λ = x 1 x 2 + x 3 x 4 + x 5 x 6 in B (6) and let I = ( λ ). HS B (6) / I ( z ) = 1 + 6 z + 14 z 2 + 14 z 3 + z 4 and (1 + z ) 6 = 1 + 6 z + 14 z 2 + 14 z 3 + z 4 − 8 z 5 + · · · 1 + z 2 � (1+ z ) 6 � = 1 + 6 z + 14 z 2 + 14 z 3 + z 4 = HS B (6) / I ( z ). 1+ z 2 λ is semi-regular and Ind( λ ) = 5. Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 8 / 20

Existence of Semi-Regular Sequences Sequences that are trivially semi-regular: Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 9 / 20

Existence of Semi-Regular Sequences Sequences that are trivially semi-regular: Sequences of linear elements that are linearly independent are semi-regular. Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 9 / 20

Existence of Semi-Regular Sequences Sequences that are trivially semi-regular: Sequences of linear elements that are linearly independent are semi-regular. Sequences of homogeneous polynomials of degree n − 1 in B ( n ) that are linearly independent are semi-regular. Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 9 / 20

Existence of Semi-Regular Sequences Sequences that are trivially semi-regular: Sequences of linear elements that are linearly independent are semi-regular. Sequences of homogeneous polynomials of degree n − 1 in B ( n ) that are linearly independent are semi-regular. x 1 x 2 · · · x n ∈ B ( n ) is semi-regular. Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 9 / 20

Existence of Semi-Regular Sequences Sequences that are trivially semi-regular: Sequences of linear elements that are linearly independent are semi-regular. Sequences of homogeneous polynomials of degree n − 1 in B ( n ) that are linearly independent are semi-regular. x 1 x 2 · · · x n ∈ B ( n ) is semi-regular. Any a basis of B d the space of homogeneous polynomials of degree d , is semi-regular. Sergio Molina (UC) Semi-Regular Sequences DIMACS 2015 9 / 20