Drinfeld Modules, Hasse Invariants and Factoring Polynomials over - PowerPoint PPT Presentation

Drinfeld Modules, Hasse Invariants and Factoring Polynomials over Finite Fields Anand Kumar Narayanan Laboratoire d’informatique de Paris 6 GTBAC Telecom Paristech 8 March 2018

Polynomial Factorization over Finite Fields Decompose a given monic sqaure-free f ( x ) ∈ F q [ x ] of degree n into its monic irreducible factors. � f ( x ) = p i ( x ) i Gauss->Legendre->Berlekamp->Cantor/Zassenhaus->Camion->von zur Gather/Shoup->Kaltofen/Shoup->Kedlaya-Umans Kaltofen-Shoup algorithm with Kedlaya-Umans fast modular composition takes expected time n 3 / 2 + o ( 1 ) ( log q ) 1 + o ( 1 ) + n 1 + o ( 1 ) ( log q ) 2 + o ( 1 ) . Drinfeld modules and Polynomial Factorization ◮ Panchishkin and Potemine (1989), van der Heiden (2005). This Talk: ◮ Factor Degree Estimation using Euler-Poincare Characteristic of Drinfeld modules. ◮ Rank-2 Drinfeld module analogue of Kaltofen-Lobo’s blackbox Berlekamp algorithm. ◮ Drinfeld modules with complex multiplication, Hasse invariants/Deligne’s congruence.

Polynomial Factorization over Finite Fields Decompose a given monic sqaure-free f ( x ) ∈ F q [ x ] of degree n into its monic irreducible factors. � f ( x ) = p i ( x ) i Gauss->Legendre->Berlekamp->Cantor/Zassenhaus->Camion->von zur Gather/Shoup->Kaltofen/Shoup->Kedlaya-Umans Kaltofen-Shoup algorithm with Kedlaya-Umans fast modular composition takes expected time n 3 / 2 + o ( 1 ) ( log q ) 1 + o ( 1 ) + n 1 + o ( 1 ) ( log q ) 2 + o ( 1 ) . Drinfeld modules and Polynomial Factorization ◮ Panchishkin and Potemine (1989), van der Heiden (2005). This Talk: ◮ Factor Degree Estimation using Euler-Poincare Characteristic of Drinfeld modules. ◮ Rank-2 Drinfeld module analogue of Kaltofen-Lobo’s blackbox Berlekamp algorithm. ◮ Drinfeld modules with complex multiplication, Hasse invariants/Deligne’s congruence.

Degree Estimation using Euler Characteristic of Drinfeld Modules Decompose a given monic f ( x ) ∈ F q [ x ] of degree n into its monic irreducible factors. � f ( x ) = p i ( x ) i Finding an irreducible factor degree with runtime exponent < 3 / 2 ⇓ factorization with exponent < 3 / 2 . An algorithm to find the smallest irreducible factor degree using Euler-Poincare characteristics of random Drinfeld modules.

Rank-2 Drinfeld Modules Let F q [ x ] � σ � denote the skew polynomial ring with the commutation rule σ u ( x ) = u ( x ) q σ, ∀ u ( x ) ∈ F q [ x ] . A rank-2 Drinfeld module over F q ( x ) is (the F q [ x ] module structure on the additive group scheme over F q ( x ) given by) a ring homomorphism φ : F q [ x ] − → F q ( x ) � σ � → x + g φ ( x ) σ + � φ ( x ) σ 2 x �− for some g φ ( x ) ∈ F q ( x ) and non zero � φ ( x ) ∈ F q [ x ] .

Rank-2 Drinfeld Modules Let F q [ x ] � σ � denote the skew polynomial ring with the commutation rule σ u ( x ) = u ( x ) q σ, ∀ u ( x ) ∈ F q [ x ] . A rank-2 Drinfeld module over F q ( x ) is (the F q [ x ] module structure on the additive group scheme over F q ( x ) given by) a ring homomorphism φ : F q [ x ] − → F q ( x ) � σ � → x + g φ ( x ) σ + � φ ( x ) σ 2 x �− for some g φ ( x ) ∈ F q ( x ) and non zero � φ ( x ) ∈ F q [ x ] . 2 deg ( b ) � φ b , i ( x ) σ i For b ( x ) ∈ F q [ x ] , b ( x ) �− → b ( x ) + . i = 1 � �� � Call φ b

Rank-2 Drinfeld Modules Let F q [ x ] � σ � denote the skew polynomial ring with the commutation rule σ u ( x ) = u ( x ) q σ, ∀ u ( x ) ∈ F q [ x ] . A rank-2 Drinfeld module over F q ( x ) is (the F q [ x ] module structure on the additive group scheme over F q ( x ) given by) a ring homomorphism φ : F q [ x ] − → F q ( x ) � σ � → x + g φ ( x ) σ + � φ ( x ) σ 2 x �− for some g φ ( x ) ∈ F q ( x ) and non zero � φ ( x ) ∈ F q [ x ] . 2 deg ( b ) � φ b , i ( x ) σ i For b ( x ) ∈ F q [ x ] , b ( x ) �− → b ( x ) + . i = 1 � �� � Call φ b Let M be an F q [ x ] algebra, say M = F q [ x ] / ( f ( x )) . Retain the addition in M but define a new F q [ x ] action: 2 deg ( b ) � φ b , i ( x ) a ( x ) q i b ( x ) ⋆ a ( x ) := φ b ( a ) = b ( x ) a ( x ) + i = 1 Let φ ( M ) denote the new F q [ x ] module structure thus endowed to M .

Rank-2 Drinfeld Modules Let F q [ x ] � σ � denote the skew polynomial ring with the commutation rule σ u ( x ) = u ( x ) q σ, ∀ u ( x ) ∈ F q [ x ] . A rank-2 Drinfeld module over F q ( x ) is (the F q [ x ] module structure on the additive group scheme over F q ( x ) given by) a ring homomorphism φ : F q [ x ] − → F q ( x ) � σ � → x + g φ ( x ) σ + � φ ( x ) σ 2 x �− for some g φ ( x ) ∈ F q ( x ) and non zero � φ ( x ) ∈ F q [ x ] . 2 deg ( b ) � φ b , i ( x ) σ i For b ( x ) ∈ F q [ x ] , b ( x ) �− → b ( x ) + . i = 1 � �� � Call φ b Let M be an F q [ x ] algebra, say M = F q [ x ] / ( f ( x )) . Retain the addition in M but define a new F q [ x ] action: 2 deg ( b ) � φ b , i ( x ) a ( x ) q i b ( x ) ⋆ a ( x ) := φ b ( a ) = b ( x ) a ( x ) + i = 1 Let φ ( M ) denote the new F q [ x ] module structure thus endowed to M .

Euler-Poincare Characteristic of Finite F q [ x ] Modules An F q [ x ] measure of cardinality: For a finite F q [ x ] module A , χ ( A ) ∈ F q [ x ] is the monic polynomial s.t. ◮ If A ∼ = F q [ x ] / ( p ( x )) for a monic irreducible p ( x ) , then χ ( A ) = p ( x ) . ◮ If 0 → A 1 → A → A 2 → 0 is exact, then χ ( A ) = χ ( A 1 ) χ ( A 2 ) . For a finite Z module G , # G ∈ Z is the positive integer s.t. ◮ If G ∼ = Z / ( p ) for a positive prime p , then # G = p . ◮ If 0 → G 1 → G → G 2 → 0 is exact, then # G = # G 1 # G 2 . Drinfeld module analogue of Hasse bound (Gekeler) For a monic irreducible p ( x ) ∈ F q [ x ] χ φ, p ( x ) := χ ( φ ( F q [ x ] / ( p ( x )))) = p ( x ) + t φ, p ( x ) � �� � ≤ deg ( p ) / 2 #( E ( Z / p Z )) = p + 1 − t E , p ���� − 2 √ p ≤ ≤ 2 √ p χ φ, p ( x ) = p ( x ) + terms of degree at most deg ( p ) / 2 .

Euler-Poincare Characteristic of Finite F q [ x ] Modules An F q [ x ] measure of cardinality: For a finite F q [ x ] module A , χ ( A ) ∈ F q [ x ] is the monic polynomial s.t. ◮ If A ∼ = F q [ x ] / ( p ( x )) for a monic irreducible p ( x ) , then χ ( A ) = p ( x ) . ◮ If 0 → A 1 → A → A 2 → 0 is exact, then χ ( A ) = χ ( A 1 ) χ ( A 2 ) . For a finite Z module G , # G ∈ Z is the positive integer s.t. ◮ If G ∼ = Z / ( p ) for a positive prime p , then # G = p . ◮ If 0 → G 1 → G → G 2 → 0 is exact, then # G = # G 1 # G 2 . Drinfeld module analogue of Hasse bound (Gekeler) For a monic irreducible p ( x ) ∈ F q [ x ] χ φ, p ( x ) := χ ( φ ( F q [ x ] / ( p ( x )))) = p ( x ) + t φ, p ( x ) � �� � ≤ deg ( p ) / 2 #( E ( Z / p Z )) = p + 1 − t E , p ���� − 2 √ p ≤ ≤ 2 √ p χ φ, p ( x ) = p ( x ) + terms of degree at most deg ( p ) / 2 .

Factor Degree Estimation � � f ( x ) = p i ( x ) ⇒ φ ( F q [ x ] / ( f ( x ))) = φ ( F q [ x ] / ( p i ( x ))) i i � � ⇒ χ φ, f ( x ) = χ φ, p i = ( p i ( x ) + t φ, p i ( x )) i i Since ∀ i , deg ( t φ, p i ( x )) ≤ deg ( p i ) / 2 , χ φ, f ( x ) = f ( x ) + terms of smaller degree . If s f denotes the degree of the smallest degree factor of f ( x ) , � � χ φ, f ( x ) − f ( x ) = ( t φ, p j ( x ) p i ( x )) + terms of degree < ( deg ( f ) − ⌈ s f / 2 ⌉ ) j : deg ( p j )= s f i � = j ⇒ ⌈ s f / 2 ⌉ ≤ deg ( f ) − deg ( χ φ, f − f )

Factor Degree Estimation � � f ( x ) = p i ( x ) ⇒ φ ( F q [ x ] / ( f ( x ))) = φ ( F q [ x ] / ( p i ( x ))) i i � � ⇒ χ φ, f ( x ) = χ φ, p i = ( p i ( x ) + t φ, p i ( x )) i i Since ∀ i , deg ( t φ, p i ( x )) ≤ deg ( p i ) / 2 , χ φ, f ( x ) = f ( x ) + terms of smaller degree . If s f denotes the degree of the smallest degree factor of f ( x ) , � � χ φ, f ( x ) − f ( x ) = ( t φ, p j ( x ) p i ( x )) + terms of degree < ( deg ( f ) − ⌈ s f / 2 ⌉ ) j : deg ( p j )= s f i � = j ⇒ ⌈ s f / 2 ⌉ ≤ deg ( f ) − deg ( χ φ, f − f ) � � Theorem : Prob φ ⌈ s f / 2 ⌉ = deg ( f ) − deg ( χ φ, f − f ) ≥ 1 / 4 .

Factor Degree Estimation � � f ( x ) = p i ( x ) ⇒ φ ( F q [ x ] / ( f ( x ))) = φ ( F q [ x ] / ( p i ( x ))) i i � � ⇒ χ φ, f ( x ) = χ φ, p i = ( p i ( x ) + t φ, p i ( x )) i i Since ∀ i , deg ( t φ, p i ( x )) ≤ deg ( p i ) / 2 , χ φ, f ( x ) = f ( x ) + terms of smaller degree . If s f denotes the degree of the smallest degree factor of f ( x ) , � � χ φ, f ( x ) − f ( x ) = ( t φ, p j ( x ) p i ( x )) + terms of degree < ( deg ( f ) − ⌈ s f / 2 ⌉ ) j : deg ( p j )= s f i � = j ⇒ ⌈ s f / 2 ⌉ ≤ deg ( f ) − deg ( χ φ, f − f ) � � Theorem : Prob φ ⌈ s f / 2 ⌉ = deg ( f ) − deg ( χ φ, f − f ) ≥ 1 / 4 .

Computing Euler-Poincare Characteristics ◮ Compute χ φ, f as the characteristic polynomial of the ( F q -linear) φ x action on F q [ x ] / ( f ( x )) . ◮ Only need a Montecarlo algorithm for χ φ, f ( x ) that succeeds with constant probability ! For a ∈ φ ( F q ( x ) / f ( x )) , Ord ( a ) is the smallest degree monic g ( x ) such that φ g ( a ) = 0 . Theorem: It is likely that χ φ, f equals the order Ord ( a ) of a random a ∈ φ ( F q [ x ] / ( f ( x ))) . Ord ( a ) can be computed with run time exponent 3 / 2 by (a Drinfeld version of) automorphism-projection followed by Berlekamp-Massey assuming the matrix multiplication exponent is 2 .