Computing finite fields’ embeddings

Ludovic Brieulle, Luca De Feo, Javad Doliskani, Jean-Pierre Flori, Éric Schost

ANSSI, UVSQ, University of Western Ontario

September 28, 2015

USING CATS

BDFDFS cats (Miaou) Computing FF embeddings September 28, 2015 1 / 10

Computing finite fields’ embeddings

Ludovic Brieulle, Luca De Feo, Javad Doliskani, Jean-Pierre Flori, Éric Schost

ANSSI, UVSQ, University of Western Ontario

September 28, 2015

USING CATS

BDFDFS cats (Miaou) Computing FF embeddings September 28, 2015 1 / 10

Random (keyboard) cat

Fqr ⊆ Fq[X ]/(f (X ))

Fqr ⊆ Fq[X ]/(g(X ))

∼BDFDFS cats (Miaou) Computing FF embeddings September 28, 2015 2 / 10

Random (keyboard) cat

Fqr ⊆ Fq[X ]/(f (X ))

Fqr ⊆ Fq[X ]/(g(X ))

∼

Different minimal polynomials . . .

BDFDFS cats (Miaou) Computing FF embeddings September 28, 2015 2 / 10

Cycl(otom)ic cat

Fix a minimal polynomialwith easy to compute roots

x l = 1

BDFDFS cats (Miaou) Computing FF embeddings September 28, 2015 3 / 10

Cycl(otom)ic cat

Fix a minimal polynomialwith easy to compute roots

x l = 1

Cats are not Galois conjugates!

BDFDFS cats (Miaou) Computing FF embeddings September 28, 2015 3 / 10

Gauss period cats

Sum the cats!< qr , H >= F×l ,

∑h∈H xh

BDFDFS cats (Miaou) Computing FF embeddings September 28, 2015 4 / 10

Gauss period cats

Sum the cats!< qr , H >= F×l ,

∑h∈H xh

(Use Gauss periods)

BDFDFS cats (Miaou) Computing FF embeddings September 28, 2015 5 / 10

Elliptic period cats

What about ellipticperiod cat?< λr , H >= F×l /{±1},

∑h∈H([h]Pr )x

(Mentioned in MMS)

Works well in practice,but no proof so far

BDFDFS cats (Miaou) Computing FF embeddings September 28, 2015 6 / 10

Elliptic period cats

What about ellipticperiod cat?< λr , H >= F×l /{±1},

∑h∈H([h]Pr )x

(Mentioned in MMS)

Works well in practice,but no proof so far

BDFDFS cats (Miaou) Computing FF embeddings September 28, 2015 6 / 10

The (rainbow) diagram

Kl = K [X ]/(fl(X )) = K0[Θ] = K0[X ]/(fl ,λ(X ))

K

K0 ' K [X ]/(Φl(j(E ), X ))

K ′

Fq

Fqr

Fqr × · · · × Fqr

BDFDFS cats (Miaou) Computing FF embeddings September 28, 2015 7 / 10

The (rainbow) diagram

Kl = K [X ]/(fl(X )) = K0[Θ] = K0[X ]/(fl ,λ(X ))

K

K0 ' K [X ]/(Φl(j(E ), X ))

K ′

Fq

Fqr

Fqr × · · · × Fqr

Suggestions welcome!!!BDFDFS cats (Miaou) Computing FF embeddings September 28, 2015 8 / 10

Other cat species

BDFDFS cats (Miaou) Computing FF embeddings September 28, 2015 9 / 10

Github repository

More fun athttps://github.com/defeo/ffisom/

BDFDFS cats (Miaou) Computing FF embeddings September 28, 2015 10 / 10

https://github.com/defeo/ffisom/

anm0: anm1: anm2: anm3: anm4: anm5: anm6: