Universal elliptic Gauss sums and applications Christian Berghoff - PowerPoint PPT Presentation

Introduction Universal elliptic Gauss sums Universal elliptic Gauss sums and applications Christian Berghoff Rheinische Friedrich-Wilhelms-Universit¨ at Bonn November 19 th , 2015

Introduction Universal elliptic Gauss sums Table of Contents Introduction 1 Universal elliptic Gauss sums 2 Modular functions Definition of universal elliptic Gauss sums

Introduction Universal elliptic Gauss sums Classical Gauss sum Let q � = 2 be a prime, χ : ( Z / q Z ) ∗ → µ n , n | q − 1, ξ an n -th root of unity and ζ a q -th root of unity, � g � = F ∗ q . A (cyclotomic) Gauss sum is defined as q − 1 q − 1 χ ( g i ) ζ g i = ξ mi ζ g i � � i =1 i =1

Introduction Universal elliptic Gauss sums Elliptic Curves Recall: Elliptic curve E over finite field F p ( p � = 2 , 3): Y 2 = X 3 + AX + B . Identify E with set of points ( X , Y ) ∈ F p × F p satisfying the equation together with O . We wish to determine # E ( F p ) = # { ( X , Y ) ∈ F p × F p | ( X , Y ) lies on E } ∪ O . Important problem related to ECC.

Introduction Universal elliptic Gauss sums Definitions and Facts ℓ -torsion : E [ ℓ ] = { P ∈ E | [ ℓ ] P = O} . Later on, ℓ will be prime, ℓ � = p . In this case = Z ℓ Z × Z E [ ℓ ] ∼ ℓ Z . Frobenius endomorphism : ( X , Y ) �→ ( X p , Y p ) φ p : E → E , By restriction, φ p acts as endomorphism of E [ ℓ ]. division polynomials of E : Certain sequence of polynomials, so that ( X , Y ) ∈ E [ ℓ ] ⇔ ψ ℓ ( X ) = 0 holds.

Introduction Universal elliptic Gauss sums Bounds for # E ( F p ) Theorem (Hasse bound (1933)) Let E be an elliptic curve over F p . Then p + 1 − 2 √ p ≤ # E ( F p ) ≤ p + 1 + 2 √ p . Hence # E ( F p ) = p + 1 − t, where t ∈ Z and | t | ≤ 2 √ p. Theorem The Frobenius endomorphism satisfies the quadratic equation χ ( φ p ) := φ 2 p − t φ p + p = 0 . � Schoof’s algorithm

Introduction Universal elliptic Gauss sums Further considerations Consider action of φ p on E [ ℓ ]. Consider roots of χ ℓ ( φ p ) = φ 2 p − t φ p + p mod ℓ . Two roots in F ℓ → ℓ is an Elkies prime . 1 No root in F ℓ → ℓ is an Atkin prime . 2 In the first case χ ℓ ( X ) has a linear factor over F ℓ [ X ] → ψ ℓ ( X ) has factor f ℓ ( X ) = � ( ℓ − 1) / 2 ( X − ( aP ) x ) where a =1 ϕ p ( P ) = λ P . � Elkies procedure with improved run-time

Introduction Universal elliptic Gauss sums Elliptic Gauss sum Let χ be a Dirichlet character of order n | ℓ − 1, then we define an elliptic Gauss sum (Mihailescu) as ℓ − 1 � v = x , n ≡ 1 (2) , � τ e ( χ ) = χ ( a )( aP ) v , v = y , n ≡ 0 (2) . a =1 Lemma The elliptic Gauss sum has the following properties: τ e ( χ ) n ∈ F p [ ζ n ] 1 ϕ p ( τ e ( χ )) = χ − p ( λ ) τ e ( χ p ) 2

Introduction Universal elliptic Gauss sums Modular functions Modular functions I Definition Upper half-plane H := { τ ∈ C : ℑ ( τ ) > 0 } . 1 Γ = SL 2 ( Z ) acts on H via 2 � a � τ �→ a τ + b b γ = : H → H , c τ + d . c d

Introduction Universal elliptic Gauss sums Modular functions Modular functions II Definition Let f ( τ ) be a meromorphic function on H , k ∈ Z . We call f ( τ ) a modular function of weight k for Γ ′ ⊆ SL 2 ( Z ) (where we require 0 1 )) ∈ Γ ′ for some N ∈ N ) if it satisfies the following conditions ( 1 N � a b f ( γτ ) = ( c τ + d ) k f ( τ ) � ∈ Γ ′ . ∀ γ = 1 c d In particular, this implies there is a Laurent series for f ( τ ) in terms of q N = exp( 2 π i τ N ). n ∈ Z a n q n In the Laurent series for f ( γτ ) = � N we have a n = 0 for 2 n < n 0 , n 0 ∈ Z ∀ γ ∈ SL 2 ( Z ). In applications we focus on � a b Γ ′ = Γ 0 ( ℓ ) = � � � γ = ∈ SL 2 ( Z ) : c ≡ 0( ℓ ) , ℓ prime . c d

Introduction Universal elliptic Gauss sums Modular functions Examples 1 1 � ′ E 2 k ( τ ) = ( m + n τ ) 2 k for k > 1 , ζ (2 k ) n , m ∈ Z ∆( τ ) = E 4 ( τ ) 3 − E 6 ( τ ) 2 , 1728 j ( τ ) = E 4 ( τ ) 3 ∆( τ ) , ∞ 1 � (1 − q n ) , η ( q ) = q 24 n =1 2 s m ℓ ( τ ) = ℓ s η ( ℓτ ) � s : s ( ℓ − 1) � , s = min ∈ N , η ( τ ) 12 s ∈ N j ( ℓτ ) .

Introduction Universal elliptic Gauss sums Modular functions Facts on modular functions Lemma Modular functions of weight 0 form a field A 0 (Γ ′ ) . Theorem With notation as on the last slide, we have A 0 (Γ) = C ( j ) , 1 A 0 (Γ 0 ( ℓ )) = C ( j , f ) for f ∈ A 0 (Γ 0 ( ℓ )) \ C ( j ) . 2 So, given g ∈ A 0 (Γ 0 ( ℓ )), there exist P 1 , P 2 ∈ C [ X , Y ] s. t. g = P 1 ( f , j ) P 2 ( f , j ) We now focus on f = m ℓ ( τ ).

Introduction Universal elliptic Gauss sums Modular functions Facts on modular functions II Lemma (B) Let g ∈ A 0 (Γ 0 ( ℓ )) be holomorphic. Then g admits a representation of the form Q ( m ℓ , j ) g ( τ ) = ∂ Y ( m ℓ , j ) , ∂ G ℓ m k ℓ for some k ≥ 0 and a polynomial Q ( X , Y ) ∈ C [ X , Y ] , where � v = s ( ℓ − 1) � deg Y ( Q ) < deg Y ( G ℓ ) = min : v ∈ N 12 s ∈ N and G ℓ ( X , j ) is the minimal polynomial of m ℓ over C ( j ) .

Introduction Universal elliptic Gauss sums Definition of universal elliptic Gauss sums Tate curve Proposition (Tate) Let E 4 , E 6 be as before. Then the quantities ∞ ∞ x ( w , q ) = 1 w mq nm ( w m + w − m ) − 2 mq nm , � � 12 + (1 − w ) 2 + n =1 m =1 ∞ ∞ w + w 2 2(1 − w ) 3 + 1 m ( m + 1) q nm ( w m − w − m ) � � � y ( w , q ) = 2 2 n =1 m =1 + q n ( m +1) ( w m +1 − w − ( m +1) ) � satisfy y ( w , q ) 2 = x ( w , q ) 3 − E 4 ( q ) 48 x ( w , q ) + E 6 ( q ) E q : 864 . E q is called the Tate curve which parametrizes isomorphism classes of elliptic curves over C .

Introduction Universal elliptic Gauss sums Definition of universal elliptic Gauss sums Universal elliptic Gauss sums Lemma (B) Let ℓ be a prime, n | ℓ − 1 , χ : F ∗ ℓ �→ µ n a Dirichlet character, ζ an ℓ -th root of unity and let r , e ∆ be appropriately chosen integers. Let in addition V = x for odd and V = y for even n and define � � χ ( λ ) V ( ζ λ , q ) , x ( ζ λ , q ) . G ℓ, n ( q ) = p 1 ( q ) = λ ∈ F ∗ λ ∈ F ∗ ℓ ℓ Then τ ℓ, n ( q ) := G ℓ, n ( q ) n p 1 ( q ) r , ∆( q ) e ∆ is a modular function of weight 0 for Γ 0 ( ℓ ) , holomorphic on H and has coefficients in Q [ ζ n ] . We call it a universal elliptic Gauss sum . Proof. Study behaviour of Weierstraß ℘ -function under action of SL 2 ( Z ) and use connection between x ( w , q ) , y ( w , q ) and ℘ ( z , τ ) , ℘ ′ ( z , τ ).

Introduction Universal elliptic Gauss sums Definition of universal elliptic Gauss sums An algorithm for computing By general lemma we find Q ( m ℓ , j ) τ ℓ, n ( q ) = ∂ Y ( m ℓ , j ) . ∂ G ℓ m k ℓ So use the following algorithm: Compute τ ℓ, n ( q ) ∂ G ℓ ∂ Y ( m ℓ , j ) =: s up to precision prec( ℓ, n ) , Q := 0. 1 Determine o = ord( s ) and ( i , k ) : iv − k = o and k < v . 2 Compute s := s − cm i ℓ j k , Q := Q + cX i Y k 3 Repeat 2 and 3 until s = 0. 4

Introduction Universal elliptic Gauss sums Definition of universal elliptic Gauss sums Required precision Lemma (B.) We can take prec( ℓ, n ) = ( v + e ∆ ) ℓ . Run-time: Compute τ ℓ, n ( q ): ˜ O ( ℓ nv ) ˜ O ( ℓ 2 v 2 ) Determine Q :

Introduction Universal elliptic Gauss sums Definition of universal elliptic Gauss sums Application Recall Schoof’s algorithm (1985) Compute # E ( F p ) = p + 1 − t , | t | ≤ 2 √ p � Determine t mod ℓ for small primes ℓ by finding t s. t. ϕ 2 p − t ϕ p + p ≡ 0 mod ℓ , then use CRT First polynomial algorithm (in log p ) If ℓ is Elkies prime: Use polynomials of lower degree ⇒ power saving in run-time If ℓ is Atkin prime: Generic approach of equal run-time + sophisticated BSGS � SEA combines Elkies (mostly) + Atkin procedures

Introduction Universal elliptic Gauss sums Definition of universal elliptic Gauss sums Elkies procedure Need to find λ s. t. ϕ p ( P ) = λ P for ℓ -torsion point P . � Compute in F p [ X ] / ( f ℓ ( X )), extension of degree O ( ℓ ). Lemma (Mihailescu, 2006) Let ℓ be a prime, χ be a character with ord( χ ) = n || ℓ − 1 . Let τ e ( χ ) be the elliptic Gauss sum. Then ϕ p ( τ e ( χ )) = χ − p ( λ ) τ e ( χ p ) Writing p = nq + m, one obtains ( τ e ( χ )) n ) q · τ e ( χ ) m τ e ( χ m ) = χ − m ( λ ) Both factors lie in F p [ ζ n ], computations can be done in extension of degree O ( n ) and no searching for λ is required.

Introduction Universal elliptic Gauss sums Definition of universal elliptic Gauss sums Compute the factors Use universal elliptic Gauss sums: We know τ ℓ, n ( q ) = G ℓ, n ( q ) n p 1 ( q ) r = R ( j ( q ) , m ℓ ( q )) . ∆( q ) e ∆ Substitute q = exp(2 π i τ ( E )) ⇒ τ ℓ, n ( E ) = R ( j ( E ) , m ℓ ( E )) for curve E in question. Hence, compute j ( E ) , ∆( E ) , p 1 ( E ) and obtain m ℓ ( E ) as root of G ℓ ( X , j ( E )). � Compute τ e ( χ ) n for our E Similar approach for Jacobi sums � determine λ mod n for all n || ℓ − 1. CRT gives index of λ in ( Z /ℓ Z ) ∗ � t = λ + p /λ mod ℓ and t .

Introduction Universal elliptic Gauss sums Definition of universal elliptic Gauss sums Further research Replace m ℓ by other modular functions to improve run-time 1 Analyse coefficient size 2 ? 3