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Deirdre Connolly @durumcrustulum Supersingular Isogeny Diffie-Hellman

Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

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Page 1: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Deirdre Connolly@durumcrustulum

Supersingular Isogeny Diffie-Hellman

Page 2: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 3: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Shor’s Algorithm

Page 4: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Diffie-Hellman Key Exchange

Page 5: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Diffie-Hellman Key Exchange

Page 6: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Diffie-Hellman Key Exchange

Page 7: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Diffie-Hellman Key Exchange

Page 8: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Diffie-Hellman Key Exchange

Page 9: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Diffie-Hellman Key Exchange

Page 10: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Diffie-Hellman Key Exchange

Page 11: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Originally FFDH: key = gab mod p

Exchange integers

Diffie-Hellman Key Exchange

Page 12: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Originally FFDH: key = gab mod p

Exchange integers

Then ECDH: key = abP mod p

Exchange points on curve

Diffie-Hellman Key Exchange

Page 13: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 14: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 15: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 16: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 17: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 18: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 19: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 20: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 21: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 22: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Originally FFDH: key = gab mod p

Exchange integers

Diffie-Hellman Key Exchange

Page 23: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Originally FFDH: key = gab mod p

Exchange integers

Then ECDH: key = abP mod p

Exchange points on curve

Diffie-Hellman Key Exchange

Page 24: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Originally FFDH: key = gab mod p

Exchange integers

Then ECDH: key = abP mod p

Exchange points on curve

Diffie-Hellman Key Exchange

Page 25: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Originally FFDH: key = gab mod p

Exchange integers

Then ECDH: key = abP mod p

Exchange points on curve

Diffie-Hellman Key Exchange

Page 26: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Originally FFDH: key = gab mod p

Exchange integers

Then ECDH: key = abP mod p

Exchange points on curve

Now SIDH:

key = ϕ′a(ϕb(E)) = ϕ′B(ϕA(E))

Exchange whole curves

Diffie-Hellman Key Exchange

Page 27: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 28: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 29: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 30: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 31: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 32: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 33: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 34: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 35: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 36: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 37: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 38: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 39: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 40: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 41: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 42: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 43: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 44: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Grover’s Algorithm

Page 45: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Best known attacks

Classical:

Quantum:

Page 46: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice Bob

Page 47: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice Bob

Page 48: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice Bob

Page 49: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice Bob

Page 50: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice Bob

Page 51: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice Bob

Page 52: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice Bob

Page 53: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice Bob

Page 54: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice

Page 55: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice

Page 56: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice

Page 57: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice

Page 58: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice

Page 59: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice

Page 60: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice

Page 61: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice

Page 62: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice

Page 63: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice

Page 64: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice

Page 65: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice

Page 66: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice

Page 67: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice

Page 68: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice

Page 69: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice

Page 70: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice

Page 71: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice

Page 72: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice

Page 73: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice

Page 74: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice

Page 75: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Alice

Page 76: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 77: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 78: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 79: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 80: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 81: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 82: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

2.9X!

Page 83: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 84: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 85: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 86: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 87: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 88: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Towards quantum-resistant cryptosystems from supersingular elliptic curve isogeniesLuca De Feo and David Jao and Jérôme Plût, 2011

Efficient algorithms for supersingular isogeny Diffie-HellmanCraig Costello and Patrick Longa and Michael Naehrig, 2016

On the Security of Supersingular Isogeny CryptosystemsSteven D. Galbraith and Christophe Petit and Barak Shani and Yan Bo Ti, 2016

https://eprint.iacr.org/2011/506.pdf
https://eprint.iacr.org/2011/506.pdf
https://eprint.iacr.org/2011/506.pdf
https://eprint.iacr.org/2011/506.pdf
https://eprint.iacr.org/2016/413.pdf
https://eprint.iacr.org/2016/413.pdf
https://eprint.iacr.org/2016/859.pdf
https://eprint.iacr.org/2016/859.pdf
Page 89: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =
Page 90: Supersingular Isogeny Diffie-Hellman · Originally FFDH: key = gab mod p Exchange integers Then ECDH: key = abP mod p Exchange points on curve Now SIDH: key = ϕ′ a (ϕ b (E)) =

Deirdre Connolly@durumcrustulum

Supersingular Isogeny Diffie-Hellman


dzackgarza.com...2.4 Homomorphisms 2.4 Homomorphisms Definition: LetG,G0begroups,thenϕ: G→G0isahomomorphismifϕ(ab) = ϕ(a)ϕ(b). Examplesofhomomorphisms: • exp(:)(R,+) →(R>0

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