Reminder on elliptic curves Endomorphism ring Volcano of ℓ -isogeny and Frobenius endomorphism ℓ -adic tower Structure of Volcano of ℓ -isogeny applied to Couveignes’s algorithm Luca De Feo, Cyril Hugounenq, Jerome Plut, Eric Schost Université Versailles Saint Quentin en Yvelines, Paris-Saclay March 15, 2016 1/ 28 Luca De Feo, Cyril Hugounenq, Jerome Plut, Eric Schost Structure of Volcano of ℓ -isogeny applied to Couveignes’s algorithm
Reminder on elliptic curves Endomorphism ring Volcano of ℓ -isogeny and Frobenius endomorphism ℓ -adic tower Summary Reminder on elliptic curves, 1 Endomorphism ring of elliptic curves following Kohel in 1996 [5], 2 Volcanoes of ℓ -isogenies and Frobenius endomorphism, 3 Working on ℓ -adic tower. 4 2/ 28 Luca De Feo, Cyril Hugounenq, Jerome Plut, Eric Schost Structure of Volcano of ℓ -isogeny applied to Couveignes’s algorithm
Reminder on elliptic curves Endomorphism ring Volcano of ℓ -isogeny and Frobenius endomorphism ℓ -adic tower Reminder on elliptic curves F q a finite field of characteristic p . Definition E an elliptic curve defined over F q , we denote by : E ( F q ) the set of rational points of E over F q During all this presentation we will consider only elliptic curves on the finite field F q , ℓ is a prime different from p 3/ 28 Luca De Feo, Cyril Hugounenq, Jerome Plut, Eric Schost Structure of Volcano of ℓ -isogeny applied to Couveignes’s algorithm
Reminder on elliptic curves Endomorphism ring Volcano of ℓ -isogeny and Frobenius endomorphism ℓ -adic tower Definition ( m torsion points) m ∈ N , we denote by E [ m ] = { P ∈ E , mP = 0 E } E ( F q )[ m ] = { P ∈ E ( F q ) , mP = 0 E } 4/ 28 Luca De Feo, Cyril Hugounenq, Jerome Plut, Eric Schost Structure of Volcano of ℓ -isogeny applied to Couveignes’s algorithm
Reminder on elliptic curves Endomorphism ring Volcano of ℓ -isogeny and Frobenius endomorphism ℓ -adic tower Reminder on isogenies Definition (isogeny) E and E ′ two ellitpic curves, φ : E → E ′ a surjective morphism such that φ (0 E ) = 0 E ′ , then φ is an isogeny. An isogeny is a group morphism. We say that E and E ′ are isogenous if there exist an isogeny φ between the two curves. Proposition E and E ′ two ellitpic curves, φ : E → E ′ an isogeny, if φ is separable , then we have: deg φ = | ker( φ ) | 5/ 28 Luca De Feo, Cyril Hugounenq, Jerome Plut, Eric Schost Structure of Volcano of ℓ -isogeny applied to Couveignes’s algorithm
Reminder on elliptic curves Endomorphism ring Volcano of ℓ -isogeny and Frobenius endomorphism ℓ -adic tower Definition E and E ′ two elliptic curves and ℓ a prime number, φ : E → E ′ a non constant isogeny. We say that φ is an ℓ -isogeny if we have deg φ = ℓ Theorem (Tate) E and E ′ two elliptic curves and φ : E → E ′ an isogeny. Then | E ( F q ) | = | E ′ ( F q ) | 6/ 28 Luca De Feo, Cyril Hugounenq, Jerome Plut, Eric Schost Structure of Volcano of ℓ -isogeny applied to Couveignes’s algorithm
Reminder on elliptic curves Endomorphism ring Volcano of ℓ -isogeny and Frobenius endomorphism ℓ -adic tower Theorem E, E ′ two elliptic curves. There is a bijection between finite subgroups of E ′ and separable isogenies : ( φ : E → E ′ ) �→ ker φ ( E → E / C ) �→ C Remark E an elliptic curve defined over F q , let ℓ be a prime different from p , then we define an ℓ -isogeny by a primitive ℓ -torsion point: P φ : E → E / � P � 7/ 28 Luca De Feo, Cyril Hugounenq, Jerome Plut, Eric Schost Structure of Volcano of ℓ -isogeny applied to Couveignes’s algorithm
Reminder on elliptic curves Endomorphism ring Volcano of ℓ -isogeny and Frobenius endomorphism ℓ -adic tower Isogeny computation Couveignes’s algorithm [1] in O ( r 2 ) Require: E,E’ two r -isogenous curves on F p n Ensure: φ : E → E ′ of degree r Main steps of Couveignes’s algorithm: determine p k primitive torsion points on E and E ′ with p k > 4 r , 1 since E [ p k ] is cyclic, the algorithm just has to interpolate p k torsion 2 points on p k torsion points according to the group law, test if the interpolation is good, 3 if the test is good, then return the isogeny. 4 Mainly used in S.E.A. for counting points 8/ 28 Luca De Feo, Cyril Hugounenq, Jerome Plut, Eric Schost Structure of Volcano of ℓ -isogeny applied to Couveignes’s algorithm
Reminder on elliptic curves Endomorphism ring Volcano of ℓ -isogeny and Frobenius endomorphism ℓ -adic tower Isogeny computation Other existing algorithms .[BMSS] et [CCR] work only for r ≪ p in O ( M ( r ) log( r )) 1 p -adic algorithms [Satoh] with p fixed are exponential in log( p ) 2 .[LS08] works for every p in O ( r 2 ) 3 9/ 28 Luca De Feo, Cyril Hugounenq, Jerome Plut, Eric Schost Structure of Volcano of ℓ -isogeny applied to Couveignes’s algorithm
Reminder on elliptic curves Endomorphism ring Volcano of ℓ -isogeny and Frobenius endomorphism ℓ -adic tower Definition (Endomorphism ring) End ( E ) = { isogenies φ : E → E } is a ring with the addition law and composition law. Remark We have Z ⊂ End ( E ) 10/ 28 Luca De Feo, Cyril Hugounenq, Jerome Plut, Eric Schost Structure of Volcano of ℓ -isogeny applied to Couveignes’s algorithm
Reminder on elliptic curves Endomorphism ring Volcano of ℓ -isogeny and Frobenius endomorphism ℓ -adic tower Definition (Frobenius Endomorphism) E an elliptic curve defined over F q . The function π : ( x , y ) �→ ( x q , y q ) is called Frobenius endomorphism. It belongs to End ( E ). Remark E an elliptic curve defined over F q , then we always have Z [ π ] ⊂ End ( E ) . 11/ 28 Luca De Feo, Cyril Hugounenq, Jerome Plut, Eric Schost Structure of Volcano of ℓ -isogeny applied to Couveignes’s algorithm
Reminder on elliptic curves Endomorphism ring Volcano of ℓ -isogeny and Frobenius endomorphism ℓ -adic tower Proposition E an elliptic curve defined over F q is ordinary if it satisfies any of the two equivalent conditions: E [ p r ] = Z / p r Z 1 End ( E ) is isomorphic to an order in a quadratic imaginary extension 2 of Q . From now we will only work with ordinary elliptic curves. Definition An order in a quadratic imaginary number field K is a subring of K 1 a Z -modulus of rank 2 2 12/ 28 Luca De Feo, Cyril Hugounenq, Jerome Plut, Eric Schost Structure of Volcano of ℓ -isogeny applied to Couveignes’s algorithm
Reminder on elliptic curves Endomorphism ring Volcano of ℓ -isogeny and Frobenius endomorphism ℓ -adic tower Definition We denote by O K the algebraic integers of K . We can associate to any elliptic curve E his endomorphism ring: O ≃ End ( E ) We will denote O (resp. O ′ ) the End ( E ) (resp. End ( E ′ )) up to isomorphism. Remark For an ordinary elliptic curve we have: Z [ π ] ⊂ O ⊂ O K 13/ 28 Luca De Feo, Cyril Hugounenq, Jerome Plut, Eric Schost Structure of Volcano of ℓ -isogeny applied to Couveignes’s algorithm
Reminder on elliptic curves Endomorphism ring Volcano of ℓ -isogeny and Frobenius endomorphism ℓ -adic tower Lemma (Kohel 1996) E and E ′ two elliptic curves defined over F q , φ : E → E ′ an ℓ -isogeny, with ℓ � = p . Then ℓ = [ O : O ′ ] we say then that 1 φ is a descending isogeny, ℓ = [ O ′ : O ] we say then that 2 φ is an ascending isogeny, O = O ′ we say then that φ 3 is an horizontal isogeny. 14/ 28 Luca De Feo, Cyril Hugounenq, Jerome Plut, Eric Schost Structure of Volcano of ℓ -isogeny applied to Couveignes’s algorithm
Reminder on elliptic curves Endomorphism ring Volcano of ℓ -isogeny and Frobenius endomorphism ℓ -adic tower • • • • • • • • • • • • • • • • • • • • ( d K /ℓ ) = − 1 ( d K /ℓ ) = 0 ( d K /ℓ ) = +1 Figure: The three shapes of volcanoes of 2-isogenies 15/ 28 Luca De Feo, Cyril Hugounenq, Jerome Plut, Eric Schost Structure of Volcano of ℓ -isogeny applied to Couveignes’s algorithm
Reminder on elliptic curves Endomorphism ring Volcano of ℓ -isogeny and Frobenius endomorphism ℓ -adic tower Remark In the rest of this talk we consider only volcanoes with cyclic crater (i.e. ( d K /ℓ ) = +1), so that ℓ is an Elkies prime for these curves. This implies that the Frobenius automorphism on T ℓ ( E ), which we write π | T ℓ ( E ), has two distinct eigenvalues λ � = µ . The depth of the volcano of F q -rational ℓ -isogenies is h = v ℓ ( λ − µ ). Proposition Let E be a curve on a volcano of ℓ isogeny with cyclic crater. Then there exists a unique a ∈ { 0 , ℓ, . . . , ℓ h − 1 } such that π | T ℓ ( E ) is conjugate, � λ a over Z ℓ , to the matrix � . 0 µ Moreover a = 0 if E lies on the crater. 16/ 28 Luca De Feo, Cyril Hugounenq, Jerome Plut, Eric Schost Structure of Volcano of ℓ -isogeny applied to Couveignes’s algorithm
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