Weierstra Equations 2 d 4 16 ( 1 t 2 / 4 ) Singular - PowerPoint PPT Presentation

Elliptic curves over F q Introduction History length of ellipses why Elliptic curves? Fields Weierstraß Equations Singular points The Discriminant E LLIPTIC CURVES OVER FINITE FIELDS Elliptic curves / F 2 Elliptic curves / F 3 The sum of points F RANCESCO P APPALARDI Examples Structure of E ( F 2 ) Structure of E ( F 3 ) Further Examples #3 - F IRST S TEPS . S EPTEMBER 4 TH 2015 SEAMS School 2015 Number Theory and Applications in Cryptography and Coding Theory University of Science, Ho Chi Minh, Vietnam August 31 - September 08, 2015

Elliptic curves over F q Proto–History (from W IKIPEDIA ) Introduction History length of ellipses why Elliptic curves? Giulio Carlo, Count Fagnano, and Marquis de Fields Toschi (December 6, 1682 – September 26, 1766) Weierstraß Equations Singular points was an Italian mathematician. He was probably the The Discriminant first to direct attention to the theory of elliptic Elliptic curves / F 2 integrals . Fagnano was born in Senigallia. Elliptic curves / F 3 The sum of points He made his higher studies at the Collegio Examples Carlo Fagnano Structure of E ( F 2 ) Clementino in Rome and there won great Structure of E ( F 3 ) distinction, except in mathematics, to which his Further Examples aversion was extreme. Only after his college course he took up the study of mathematics. Later, without help from any teacher, he mastered mathematics from its foundations. Collegio Clementino Some of His Achievements: 0.6 • π = 2 i log 1 − i 0.4 0.2 1 + 1 0.0 • Length of Lemniscate � 0.2 � 0.4 � 0.6 � 1.5 � 1.0 � 0.5 0.0 0.5 1.0 1.5 Lemniscate ( x 2 + y 2 ) 2 = 2 a 2 ( x 2 − y 2 ) a √ π Γ( 5 ℓ = 4 � a 4 ) a 2 dr √ a 4 − r 4 = Γ( 3 4 ) 0

Elliptic curves over F q Length of Ellipses Introduction History Applying this formula to E : E : x 2 4 + y 2 length of ellipses 16 = 1 why Elliptic curves? Fields 4 � Weierstraß Equations � 2 � � d � � 4 16 ( 1 − t 2 / 4 ) Singular points � The Discriminant ℓ ( E ) = 4 � 1 + dt dt Elliptic curves / F 2 2 0 Elliptic curves / F 3 The sum of points 1 � � 1 + 3 x 2 Examples = 4 1 − x 2 dx x = t / 2 Structure of E ( F 2 ) Structure of E ( F 3 ) 0 0 Further Examples If y is the integrand, then we have the identity: � 2 y 2 ( 1 − x 2 ) = 1 + 3 x 2 Apply the invertible change of variables: � 4 � � 2 � 1 0 1 2 x = 1 − 2 / t The length of the arc of a plane curve y = f ( x ) , f : [ a , b ] → R is: u y = t − 1 � b Arrive to � 1 + ( f ′ ( t )) 2 dt ℓ = u 2 = t 3 − 4 t 2 + 6 t − 3 a

Elliptic curves over F q What are Elliptic Curves? Reasons to study them Introduction History length of ellipses why Elliptic curves? Fields Weierstraß Equations Singular points The Discriminant Elliptic Curves Elliptic curves / F 2 Elliptic curves / F 3 are curves and finite groups at the same time 1 The sum of points Examples are non singular projective curves of genus 1 2 Structure of E ( F 2 ) Structure of E ( F 3 ) have important applications in Algorithmic Number Theory and Cryptography 3 Further Examples are the topic of the Birch and Swinnerton-Dyer conjecture (one of the seven Millennium Prize Problems) 4 have a group law that is a consequence of the fact that they intersect every line in exactly three points (in 5 the projective plane over C and counted with multiplicity) represent a mathematical world in itself ... Each of them does!! 6

Elliptic curves over F q Notations Introduction History length of ellipses Fields of characteristics 0 why Elliptic curves? Fields Q is the field of rational numbers Weierstraß Equations 1 Singular points R and C are the fields of real and complex numbers The Discriminant 2 Elliptic curves / F 2 K ⊂ C , dim Q K < ∞ is a number field 3 Elliptic curves / F 3 √ Q [ d ] , d ∈ Q The sum of points • Q [ α ] , f ( α ) = 0, f ∈ Q [ X ] irreducible Examples • Structure of E ( F 2 ) Structure of E ( F 3 ) Further Examples Finite fields F p = { 0 , 1 , . . . , p − 1 } is the prime field; 1 F q is a finite field with q = p n elements 2 F q = F p [ ξ ] , f ( ξ ) = 0, f ∈ F p [ X ] irreducible, ∂ f = n 3 F 4 = F 2 [ ξ ] , ξ 2 = 1 + ξ 4 F 8 = F 2 [ α ] , α 3 = α + 1 but also F 8 = F 2 [ β ] , β 3 = β 2 + 1, ( β = α 2 + 1) 5 F 101 101 = F 101 [ ω ] , ω 101 = ω + 1 6

Elliptic curves over F q Notations Introduction History length of ellipses why Elliptic curves? Fields Algebraic Closure of F q Weierstraß Equations Singular points • C ⊃ Q satisfies that Fundamental Theorem of Algebra ! (i.e. ∀ f ∈ Q [ x ] , ∂ f > 1 , ∃ α ∈ C , f ( α ) = 0 ) The Discriminant Elliptic curves / F 2 • We need a field that plays the role, for F q , that C plays for Q . It will be F q , called algebraic closure of F q Elliptic curves / F 3 The sum of points Examples ∀ n ∈ N , we fix an F q n 1 Structure of E ( F 2 ) We also require that F q n ⊆ F q m if n | m 2 Structure of E ( F 3 ) � Further Examples We let F q = F q n 3 n ∈ N • Fact: F q is algebraically closed (i.e. ∀ f ∈ F q [ x ] , ∂ f > 1 , ∃ α ∈ F q , f ( α ) = 0 ) If F ( x , y ) ∈ Q [ x , y ] a point of the curve F = 0, means ( x 0 , y 0 ) ∈ C 2 s.t. F ( x 0 , y 0 ) = 0. 2 If F ( x , y ) ∈ F q [ x , y ] a point of the curve F = 0, means ( x 0 , y 0 ) ∈ F q s.t. F ( x 0 , y 0 ) = 0.

Elliptic curves over F q The (general) Weierstraß Equation Introduction An elliptic curve E over a F q (finite field) is given by an equation History length of ellipses E : y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 why Elliptic curves? Fields Weierstraß Equations where a 1 , a 3 , a 2 , a 4 , a 6 ∈ F q Singular points The Discriminant Elliptic curves / F 2 Elliptic curves / F 3 The sum of points Examples Structure of E ( F 2 ) Structure of E ( F 3 ) Further Examples The equation should not be singular

Elliptic curves over F q Tangent line to a plane curve Introduction History length of ellipses Given f ( x , y ) ∈ F q [ x , y ] and a point ( x 0 , y 0 ) such that f ( x 0 , y 0 ) = 0, the tangent line is: why Elliptic curves? Fields ∂ x ( x 0 , y 0 )( x − x 0 ) + ∂ f ∂ f Weierstraß Equations ∂ y ( x 0 , y 0 )( y − y 0 ) = 0 Singular points The Discriminant If Elliptic curves / F 2 Elliptic curves / F 3 ∂ x ( x 0 , y 0 ) = ∂ f ∂ f ∂ y ( x 0 , y 0 ) = 0 , The sum of points Examples Structure of E ( F 2 ) such a tangent line cannot be computed and we say that ( x 0 , y 0 ) is singular Structure of E ( F 3 ) Further Examples Definition A non singular curve is a curve without any singular point Example The tangent line to x 2 + y 2 = 1 over F 7 at ( 2 , 2 ) is x + y = 4

Elliptic curves over F q Singular points The classical definition Introduction History length of ellipses why Elliptic curves? Fields Definition Weierstraß Equations Singular points A singular point ( x 0 , y 0 ) on a curve f ( x , y ) = 0 is a point such that The Discriminant Elliptic curves / F 2 Elliptic curves / F 3 � ∂ f ∂ x ( x 0 , y 0 ) = 0 The sum of points ∂ f ∂ y ( x 0 , y 0 ) = 0 Examples Structure of E ( F 2 ) Structure of E ( F 3 ) Further Examples So, at a singular point there is no (unique) tangent line!! In the special case of Weierstraß equations: E : y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 we have � � a 1 y = 3 x 2 + 2 a 2 x + a 4 ∂ x = 0 − → ∂ y = 0 2 y + a 1 x + a 3 = 0 We can express this condition in terms of the coefficients a 1 , a 2 , a 3 , a 4 , a 5 .

Elliptic curves over F q The Discriminant of an Equation The condition of absence of singular points in terms of a 1 , a 2 , a 3 , a 4 , a 6 Introduction History With a bit of Mathematica length of ellipses why Elliptic curves? Ell:=-a_6-a_4x-a_2xˆ2-xˆ3+a_3y+a_1xy+yˆ2; Fields SS := Solve[{D[Ell,x]==0,D[Ell,y]==0},{y,x}]; Weierstraß Equations Simplify[ReplaceAll[Ell,SS[[1]]]*ReplaceAll[Ell,SS[[2]]]] Singular points The Discriminant we obtain Elliptic curves / F 2 Elliptic curves / F 3 The sum of points 1 Examples ∆ ′ � − a 5 1 a 3 a 4 − 8 a 3 1 a 2 a 3 a 4 − 16 a 1 a 2 2 a 3 a 4 + 36 a 2 1 a 2 E := 3 a 4 Structure of E ( F 2 ) 2 4 3 3 Structure of E ( F 3 ) Further Examples − a 4 1 a 2 4 − 8 a 2 1 a 2 a 2 4 − 16 a 2 2 a 2 4 + 96 a 1 a 3 a 2 4 + 64 a 3 4 + a 6 1 a 6 + 12 a 4 1 a 2 a 6 + 48 a 2 1 a 2 2 a 6 + 64 a 3 2 a 6 − 36 a 3 1 a 3 a 6 − 144 a 1 a 2 a 3 a 6 − 72 a 2 1 a 4 a 6 − 288 a 2 a 4 a 6 + 432 a 2 � 6 Definition The discriminant of a Weierstraß equation over F q , q = p n , p ≥ 5 is ∆ E := 3 3 ∆ ′ E

Elliptic curves over F q The discriminant of E / F 2 α Introduction History length of ellipses E : y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 , a i ∈ F 2 α why Elliptic curves? Fields If p = 2, the singularity condition becomes: Weierstraß Equations Singular points The Discriminant � � a 1 y = x 2 + a 4 Elliptic curves / F 2 ∂ x = 0 − → Elliptic curves / F 3 ∂ y = 0 a 1 x + a 3 = 0 The sum of points Examples Structure of E ( F 2 ) Structure of E ( F 3 ) Classification of Weierstraß equations over F 2 α Further Examples El:=a6+a4x+a2xˆ2+xˆ3+a3y+a1xy+yˆ2; Simplify[ReplaceAll[El, { x → a3/a1,y → ((a3/a1)ˆ2+a4)/a1 } ]] • Case a 1 � = 0: ∆ E := ( a 6 1 a 6 + a 5 1 a 3 a 4 + a 4 1 a 2 a 2 3 + a 4 1 a 2 4 + a 3 1 a 3 3 + a 4 3 ) / a 6 1 we obtain • Case a 1 = 0 and a 3 � = 0: curve non singular ( ∆ E := a 3 ) ( x 0 , y 0 ) , ( x 2 0 = a 4 , y 2 • Case a 1 = 0 and a 3 = 0: curve singular 0 = a 2 a 4 + a 6 ) singular point!