Introduction Basic definitions First properties Field extensions Newton lift Algorithmic p − adic integers Basic operations A point counting algorithm An introduction to the algorithmic of p -adic numbers David Lubicz 1 1 Universté de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France D. Lubicz p -adic numbers
Introduction Basic definitions First properties Field extensions Newton lift Algorithmic p − adic integers Basic operations A point counting algorithm Outline Introduction 1 Basic definitions 2 First properties 3 Field extensions 4 Newton lift 5 Algorithmic p − adic integers 6 Basic operations 7 A point counting algorithm 8 D. Lubicz p -adic numbers
Introduction Basic definitions First properties Field extensions Newton lift Algorithmic p − adic integers Basic operations A point counting algorithm When do we need p -adic numbers? In elliptic curve cryptography, most of time, the important objects to manipulate are finite fields F q . Sometimes, we would like to use formulas coming from the classical theory of elliptic curves over C but they have no meaning in characteristic p because for instance they imply the evaluation of 1 / p . D. Lubicz p -adic numbers
Introduction Basic definitions First properties Field extensions Newton lift Algorithmic p − adic integers Basic operations A point counting algorithm Cryptographic applications Main cryptographic applications of p -adic numbers : point counting algorithms; CM-methods; isogeny computations. D. Lubicz p -adic numbers
Introduction Basic definitions First properties Field extensions Newton lift Algorithmic p − adic integers Basic operations A point counting algorithm What are the p -adic numbers? A dictionary : Function fields Number theory C [ X ] Z C ( X ) Q a monomial ( X − α ) p prime finite extension of C ( X ) finite extension of Q Laurent series about α p -adic numbers D. Lubicz p -adic numbers
Introduction Basic definitions First properties Field extensions Newton lift Algorithmic p − adic integers Basic operations A point counting algorithm Construction of p -adic numbers I Let p be a prime, let A n = Z / p n Z . We have a natural morphism φ : A n → A n − 1 provided by the reduction modulo p n − 1 . The sequence . . . A n → A n − 1 → . . . → A 2 → A 1 is an inverse system. Definition The ring of p -adic numbers is by definition Z p = lim ← ( A n , φ n ) . D. Lubicz p -adic numbers
Introduction Basic definitions First properties Field extensions Newton lift Algorithmic p − adic integers Basic operations A point counting algorithm Construction of p -adic numbers II An element of a = Z p can be represented as a sequence of elements a = ( a 1 , a 2 , . . . , a n , . . . ) with a i ∈ Z / p i Z and a i mod p i − 1 = a i − 1 . The ring structure is the one inherited from that of Z / p i Z . The neutral element is ( 1 , . . . , 1 , . . . ) . There exists natural projections p i : Z p → Z / p i Z , a �→ a i = a mod p i . D. Lubicz p -adic numbers
Introduction Basic definitions First properties Field extensions Newton lift Algorithmic p − adic integers Basic operations A point counting algorithm First properties I Proposition Let x ∈ Z p , x is invertible if and only if x mod p is invertible. Let x ∈ Z p , there exists a unique ( u , n ) where u is an invertible element of Z p and n a positive integer such that x = p n u . The integer n is called the valuation of x and denoted by v ( x ) . D. Lubicz p -adic numbers
Introduction Basic definitions First properties Field extensions Newton lift Algorithmic p − adic integers Basic operations A point counting algorithm First properties II Z p is a characteristic 0 ring; Z p is integral; Z p has a unique maximal ideal O p = { x ∈ Z p | v ( x ) > 0 } ; There is a canonical isomorphism Z p / O p ≃ F p . D. Lubicz p -adic numbers
Introduction Basic definitions First properties Field extensions Newton lift Algorithmic p − adic integers Basic operations A point counting algorithm The field of p -adics Definition The field of p -adic numbers noted Q p is by definition the field of fractions of Z p . The valuation of Z p extend immediately to Q p by letting v ( x / y ) = v ( x ) − v ( y ) for x , y ∈ Z p ; Q p comes with a norm called the p -adic norm given by | x | Q p = p − v ( x ) . D. Lubicz p -adic numbers
Introduction Basic definitions First properties Field extensions Newton lift Algorithmic p − adic integers Basic operations A point counting algorithm Representation as a series I Definition An element π ∈ Z p is called a uniformizing element if v ( π ) = 1. Let p 1 be the canonical projection from Z p to F p . A map ω : F p → Z p is a system of representatives of F p if for all � � x ∈ F p we have p 1 ω ( x ) = x . Definition An element x ∈ Z p is called a lift of an element x 0 ∈ F p if p 1 ( x ) = x 0 . Consequently, for all x ∈ F p , ω ( x ) is a lift of x . D. Lubicz p -adic numbers
Introduction Basic definitions First properties Field extensions Newton lift Algorithmic p − adic integers Basic operations A point counting algorithm Representation as a series II Let π be a uniformizing element of Z p , ω a system of representatives of F p in Z p and x ∈ Z p . Let n = v ( x ) , then x /π n is an invertible element of Z p and there exists a unique x n ∈ F p x − π n ω ( x n ) � � such that v = n + 1. Iterating this process, we obtain that Proposition There exists a unique sequence ( x i ) i � 0 of elements of F p such that ∞ � ω ( x i ) π i . x = i = 0 D. Lubicz p -adic numbers
Introduction Basic definitions First properties Field extensions Newton lift Algorithmic p − adic integers Basic operations A point counting algorithm Field extensions I Let K be a finite extension of Q p defined by an irreducible polynomial m ∈ Q p [ X ] . There exists a unique norm | · | K on K extending the p -adic norm on Q p . R = { x ∈ K | | x | K ≤ 1 } is the valuation ring of K . M = { x ∈ R | | x | K < 1 } is be the unique maximal ideal of R . D. Lubicz p -adic numbers
Introduction Basic definitions First properties Field extensions Newton lift Algorithmic p − adic integers Basic operations A point counting algorithm Field extension II Definition Keeping the notation from above : The field F q = R / M is an algebraic extension of F p , the degree of which is called the inertia degree of K and is denoted by f . The absolute ramification index of K is the integer � � e = v K ψ ( p ) , where ψ : Z → K is the canonical embedding of Z into K . D. Lubicz p -adic numbers
Introduction Basic definitions First properties Field extensions Newton lift Algorithmic p − adic integers Basic operations A point counting algorithm Unramified extensions I We have the Theorem Let d be the degree of K / Q p , then d = ef. Definition Let K / Q p be a finite extension. Then K is called absolutely unramified if e = 1. An absolutely unramified extension of degree d is denoted by Q q with q = p d and its valuation ring by Z q . D. Lubicz p -adic numbers
Introduction Basic definitions First properties Field extensions Newton lift Algorithmic p − adic integers Basic operations A point counting algorithm Unramified extensions II Proposition Let K be a finite extension of Q p defined by an irreducible polynomial m ∈ Q p [ X ] . Denote by P 1 the reduction morphism R [ X ] → F q [ X ] induced by p 1 and let m be the irreducible polynomial defined by P 1 ( m ) . The extension K / Q p is absolutely unramified if and only if deg m = deg m. Let d = deg m and F q = F p d the finite field defined by m, then we have p 1 ( R ) = F q . D. Lubicz p -adic numbers
Introduction Basic definitions First properties Field extensions Newton lift Algorithmic p − adic integers Basic operations A point counting algorithm Unramified extensions III The classification of unramified extension is given by their degree. Proposition Let K 1 and K 2 be two unramified extensions of Q p defined respectively by m 1 and m 2 then K 1 ≃ K 2 if and only if deg m 1 = deg m 2 . D. Lubicz p -adic numbers
Introduction Basic definitions First properties Field extensions Newton lift Algorithmic p − adic integers Basic operations A point counting algorithm Unramified extensions IV The Galois properties of unramified extensions of Q p is the same as that of finite fields. Proposition An unramified extension K of Q p is Galois and its Galois group is cyclic generated by an element Σ that reduces to the Frobenius morphism on the residue field. We call this automorphism the Frobenius substitution on K. D. Lubicz p -adic numbers
Introduction Basic definitions First properties Field extensions Newton lift Algorithmic p − adic integers Basic operations A point counting algorithm Lefschetz principle I The field Q p and its unramified extensions enjoy several important properties: Their Galois groups reflect the structure of finite field extensions; Their are big enough to be characteristic 0 fields... ...but small enough so that there exists an field morphism K → C for any K finite extension of Q p . Warning : Q p / Q is NOT an algebraic extension. D. Lubicz p -adic numbers
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