Algorithms and Statistics for Additive Polynomials Mark Giesbrecht with Joachim von zur Gathen and Konstantin Ziegler Symbolic Computation Group Cheriton School of Computer Science University of Waterloo Waterloo, Ontario, Canada November 28, 2013 1/29
Polynomial Composition and Decomposition Functional Composition Let g , h ∈ F [ x ] , for a field F. Compose g , h as functions f ( x ) = g ( h ( x )) = g ◦ h Generally non-distributive operation (not always, as we’ll see!): g ( h 1 ( x ) + h 2 ( x )) � g ( h 1 ( x )) + g ( h 2 ( x )) 2/29
Polynomial Composition and Decomposition Functional Composition Let g , h ∈ F [ x ] , for a field F. Compose g , h as functions f ( x ) = g ( h ( x )) = g ◦ h Generally non-distributive operation (not always, as we’ll see!): g ( h 1 ( x ) + h 2 ( x )) � g ( h 1 ( x )) + g ( h 2 ( x )) Decomposition Given f ∈ F [ x ] , can it be decomposed? Do there exist g , h ∈ F [ x ] such that f = g ◦ h ? f = x 4 − 2 x 3 + 8 x 2 − 7 x + 5 f = g ◦ h g = x 2 + 3 x − 5 h = x 2 − x − 2 2/29
Tame and Wild Decomposition Let F be a field of characteristic p and f ∈ F [ x ] monic of degree d . Normalize f , g , h to monic and original : h ( 0 ) = 0 f is tame if p ∤ d f is wild if p | d Traditionally this describes the ramification of F ( x ) over F ( f ( x )) . 3/29
Tame and Wild Decomposition Let F be a field of characteristic p and f ∈ F [ x ] monic of degree d . Normalize f , g , h to monic and original : h ( 0 ) = 0 f is tame if p ∤ d f is wild if p | d Traditionally this describes the ramification of F ( x ) over F ( f ( x )) . Tame decomposition Ritt (1922) describes all tame decompositions and “ambiguities”. For a fixed s , there are either 0 or 1 monic h ∈ F [ x ] of degree s with h ( 0 ) = 0 such that f ( x ) = g ( h ( x )) . See von zur Gathen (2013) for complete decompositions. 3/29
Tame and Wild Decomposition Let F be a field of characteristic p and f ∈ F [ x ] monic of degree d . Normalize f , g , h to monic and original : h ( 0 ) = 0 f is tame if p ∤ d f is wild if p | d Traditionally this describes the ramification of F ( x ) over F ( f ( x )) . Wild decomposition Life is much more difficult (G, 1988) For a finite field F of characteristic p , there are f ∈ F [ x ] of degree d with > d λ log d monic, original, h ∈ F [ x ] of degree s ≈ √ s such that f ( x ) = g ( h ( x )) , where λ = ( 6 log p ) − 1 . 3/29
Tame and Wild Decomposition Let F be a field of characteristic p and f ∈ F [ x ] monic of degree d . Normalize f , g , h to monic and original : h ( 0 ) = 0 f is tame if p ∤ d f is wild if p | d Traditionally this describes the ramification of F ( x ) over F ( f ( x )) . Wild decomposition On the bright side, there are at most ( d − 1 ) / ( s − 1 ) indecomposable monic, orginal h ∈ F [ x ] of degree s such that f ( x ) = g ( h ( x )) . (Von zur Gathen, G, Ziegler, 2010) 3/29
Additive Polynomials Additive or linearized polynomials are those such that f ( x + y ) = f ( x ) + f ( y ) Non-linear additive polynomials only exist in F [ x ] if F has prime characteristic p , and have the form f = a 0 x + a 1 x p + a 2 x p 2 + · · · + a n x p n ∈ F [ x ] . 4/29
Additive Polynomials Additive or linearized polynomials are those such that f ( x + y ) = f ( x ) + f ( y ) Non-linear additive polynomials only exist in F [ x ] if F has prime characteristic p , and have the form f = a 0 x + a 1 x p + a 2 x p 2 + · · · + a n x p n ∈ F [ x ] . Example Let F 125 = F 5 [ θ ] / ( θ 3 + θ + 1 ) . f = x 25 + ( 3 θ 2 + 4 θ + 2 ) x 5 + ( 3 θ 2 + 4 θ + 2 ) x is an additive polynomial, and f = ( x 5 + ( θ 2 + θ + 4 ) x ) ◦ ( x 5 + 3 θ x ) = ( x 5 + ( 2 θ 2 + 4 θ + 2 ) x ) ◦ ( x 5 + ( θ 2 + 2 θ ) x ) 4/29
Ore’s Legacy In 1932-4, Oystein Ore wrote four seminal papers for finite fields, differential algebra, and computer algebra O. Ore, Formale Theorie der linearen Differentialgleichungen , J. 1 reine angew. Math., v. 168, pp. 233-252, 1932. O. Ore, Theory of Non-Commutative Polynomials , "Annals of 2 Mathematics", v. 34, no. 22, pp. 480–508, 1933. O. Ore, On a Special Class of Polynomials , Trans. Amer. Math. 3 Soc., v. 35, pp. 559-584, 1933. O. Ore, Contributions to the Theory of Finite Fields , Trans. Amer. 4 Math. Soc., v. 36, pp. 243-274, 1934. [1,2] form the basis for modern computational theory of LODEs (Ore_algebra,OreTools) [3,4] have had great influence on theory of finite fields 5/29
Ore Polynomials in Computational Algebra Additive polynomials are employed in Error correcting codes HFE and other cryptosystems Mathematical constructions in algebraic function fields General fun and parlour tricks. Despite their large (exponential) degrees we will see that we can compute very efficiently with them. 6/29
Ore Polynomials and Additive Polynomials Let q = p e for prime p and integer e . F q the finite field with q elements. Additive polynomials over F q : � � � a i x p i ∈ F q [ x ] F q [ x ; p ] = 0 � i � n Ring under usual polynomial addition ( + ) and functional composition( ◦ ), with x p ◦ ax = a p x p . 7/29
Ore Polynomials and Additive Polynomials Let q = p e for prime p and integer e . F q the finite field with q elements. Additive polynomials over F q : � � � a i x p i ∈ F q [ x ] F q [ x ; p ] = 0 � i � n Ring under usual polynomial addition ( + ) and functional composition( ◦ ), with x p ◦ ax = a p x p . Ore polynomials over F q : � � � a i x i ∈ F q [ x ] F q [ x ; σ p ] = 0 � i � n Ring under usual polynomial addition ( + ) and multiplication xa = σ p ( a ) x σ p ( a ) = a p is the Frobenius automorphism of F q / F p 7/29
Ore Polynomials and Additive Polynomials Let q = p e for prime p and integer e . F q the finite field with q elements. Isomorphic Additive polynomials over F q : � � � a i x p i ∈ F q [ x ] F q [ x ; p ] = 0 � i � n Ring under usual polynomial addition ( + ) and functional composition( ◦ ), with x p ◦ ax = a p x p . Ore polynomials over F q : � � � a i x i ∈ F q [ x ] F q [ x ; σ p ] = 0 � i � n Ring under usual polynomial addition ( + ) and multiplication xa = σ p ( a ) x σ p ( a ) = a p is the Frobenius automorphism of F q / F p 7/29
The Geometry of Additive Polynomials Assume f ∈ F q [ x ; p ] squarefree of degree p n 8/29
The Geometry of Additive Polynomials Assume f ∈ F q [ x ; p ] squarefree of degree p n f squarefree ⇐⇒ f ′ = a 0 � 0 8/29
The Geometry of Additive Polynomials Assume f ∈ F q [ x ; p ] squarefree of degree p n f squarefree ⇐⇒ f ′ = a 0 � 0 Roots V f ⊆ F q of f form F p -vector space of dimension n . 8/29
The Geometry of Additive Polynomials Assume f ∈ F q [ x ; p ] squarefree of degree p n f squarefree ⇐⇒ f ′ = a 0 � 0 Roots V f ⊆ F q of f form F p -vector space of dimension n . If W an F p -subspace of V f , and h ∈ F q [ x ] has roots exactly W then h ∈ F q [ x ; p ] and ∃ g ∈ F q [ x ; p ] such that f = g ◦ h . 8/29
The Geometry of Additive Polynomials Assume f ∈ F q [ x ; p ] squarefree of degree p n f squarefree ⇐⇒ f ′ = a 0 � 0 Roots V f ⊆ F q of f form F p -vector space of dimension n . If W an F p -subspace of V f , and h ∈ F q [ x ] has roots exactly W then h ∈ F q [ x ; p ] and ∃ g ∈ F q [ x ; p ] such that f = g ◦ h . Decomposing additive polynomials ≡ finding subspaces of V f 8/29
The Geometry of Additive Polynomials Assume f ∈ F q [ x ; p ] squarefree of degree p n f squarefree ⇐⇒ f ′ = a 0 � 0 Roots V f ⊆ F q of f form F p -vector space of dimension n . If W an F p -subspace of V f , and h ∈ F q [ x ] has roots exactly W then h ∈ F q [ x ; p ] and ∃ g ∈ F q [ x ; p ] such that f = g ◦ h . Decomposing additive polynomials ≡ finding subspaces of V f Let σ q ( a ) = a q , the q -Frobenius automorphism. If W is also σ q -invariant, then h ∈ F q [ x ; p ] Decomposing additive polynomial over F q [ x ] ≡ finding σ q -invariant subspace of V f 8/29
The Geometry of Additive Polynomials (2) Example Again let F 125 = F 5 [ θ ] / ( θ 3 + θ + 1 ) , and f = x 25 + ( 3 θ 2 + 4 θ + 2 ) x 5 + ( 3 θ 2 + 4 θ + 2 ) x Then x 4 + ( θ 2 + 3 θ + 4 ) x 2 + ( 3 θ 2 + 4 θ ) x + ( 4 θ 2 + θ ) � � µ = RootOf x 4 + ( 4 θ 2 + 2 θ + 1 ) x 2 + ( 4 θ 2 + 2 θ ) x + ( 4 θ 2 + θ ) � � ν = RootOf V f = { αµ + βν : α , β ∈ F p } ⊆ F 5 12 � 3 � 3 σ q = (after some ugly calculations) 2 3 Probably not the best way to work with additive polynomials... 9/29
Right Composition Factors as Eigenvectors of σ q Given f ∈ F q [ x ; p ] , find � � h = x p + ax ∈ F q [ x ; p ] : ∃ g ∈ F q [ x ; p ] with f = g ◦ h # The number of right composition factors of f degree p 10/29
Right Composition Factors as Eigenvectors of σ q Given f ∈ F q [ x ; p ] , find � � h = x p + ax ∈ F q [ x ; p ] : ∃ g ∈ F q [ x ; p ] with f = g ◦ h # The number of right composition factors of f degree p = number of 1-dimensional σ q -invariant subspaces of V f = number of eigenvectors of σ q Remember, σ q : V f → V f is a F p -linear map σ q acts like an n × n matrix over F p 10/29
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