ludovic brieulle, luca de feo, javad doliskani, jean-pierre flori, Éric...
TRANSCRIPT
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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
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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
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Random (keyboard) cat
Fqr ⊆ Fq[X ]/(f (X ))
Fqr ⊆ Fq[X ]/(g(X ))
∼BDFDFS cats (Miaou) Computing FF embeddings September 28, 2015 2 / 10
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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
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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
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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
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Gauss period cats
Sum the cats!< qr , H >= F×l ,
∑h∈H xh
BDFDFS cats (Miaou) Computing FF embeddings September 28, 2015 4 / 10
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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
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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
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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
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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
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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
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Other cat species
BDFDFS cats (Miaou) Computing FF embeddings September 28, 2015 9 / 10
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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: