CSIDH : An Efficient Post-Quantum Commutative Group Action Wouter Castryck 1 Tanja Lange 2 Chloe Martindale 2 Lorenz Panny 2 Joost Renes 3 1 KU Leuven 2 TU Eindhoven 3 Radboud Universiteit Brisbane, 6 December 2018
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Why CSIDH? ◮ Drop-in post-quantum replacement for (EC)DH. https://csidh.isogeny.org 1/15
Why CSIDH? ◮ Drop-in post-quantum replacement for (EC)DH. ◮ Non-interactive key exchange (full public-key validation); previously only slow solutions post-quantumly. https://csidh.isogeny.org 1/15
Why CSIDH? ◮ Drop-in post-quantum replacement for (EC)DH. ◮ Non-interactive key exchange (full public-key validation); previously only slow solutions post-quantumly. ◮ Small keys: 64 bytes at conjectured AES-128 security level https://csidh.isogeny.org 1/15
Why CSIDH? ◮ Drop-in post-quantum replacement for (EC)DH. ◮ Non-interactive key exchange (full public-key validation); previously only slow solutions post-quantumly. ◮ Small keys: 64 bytes at conjectured AES-128 security level ◮ Competitive speed: ∼ 35 ms per operation. (Skylake i5 w / TurboBoost) https://csidh.isogeny.org 1/15
Why CSIDH? ◮ Drop-in post-quantum replacement for (EC)DH. ◮ Non-interactive key exchange (full public-key validation); previously only slow solutions post-quantumly. ◮ Small keys: 64 bytes at conjectured AES-128 security level ◮ Competitive speed: ∼ 35 ms per operation. (Skylake i5 w / TurboBoost) ◮ Clean mathematical structure: a true group action. (No noise, no auxiliary points, no compromises.) https://csidh.isogeny.org 1/15
Why CSIDH? ◮ Drop-in post-quantum replacement for (EC)DH. ◮ Non-interactive key exchange (full public-key validation); previously only slow solutions post-quantumly. ◮ Small keys: 64 bytes at conjectured AES-128 security level ◮ Competitive speed: ∼ 35 ms per operation. (Skylake i5 w / TurboBoost) ◮ Clean mathematical structure: a true group action. (No noise, no auxiliary points, no compromises.) ◮ By the way: not ‘better’ or ‘worse’ than SIDH. It’s simply different and likely to be useful for different applications. https://csidh.isogeny.org 1/15
Ordinary isogeny graphs Nodes: Ordinary elliptic curves defined over k up to ∼ = k . Edges: 3-, 5-, and 7-isogenies defined over k up to ∼ = k . Components look something like this: https://csidh.isogeny.org 2/15
Ordinary isogeny graphs (cycles) Nodes: Ordinary elliptic curves defined over k up to ∼ = k . Edges: 3-, 5-, and 7-isogenies defined over k up to ∼ = k . https://csidh.isogeny.org 2/15
Ordinary isogeny graphs (cycles) Nodes: Ordinary elliptic curves defined over k up to ∼ = k . Edges: 3-, 5-, and 7-isogenies defined over k up to ∼ = k . ??? Easy: Compute a random path, output the final node. Hard problem: Find a path between two given nodes. https://csidh.isogeny.org 2/15
Alice vs. Eve Intuition: Combining edges from different cycles allows taking shortcuts to remote parts of the graph! https://csidh.isogeny.org 3/15
Alice vs. Eve g 0 g 1 · g 1 g 3 · g 2 · g 8 g 11 Intuition: Combining edges from different cycles allows taking shortcuts to remote parts of the graph! cf. Square-&-Multiply: Alice gets an advantage over Eve. https://csidh.isogeny.org 3/15
Point counting De Feo–Kieffer–Smith want an ordinary curve E / F q with many small primes ℓ | E ( F q ) . This seems difficult. https://csidh.isogeny.org 4/15
https://csidh.isogeny.org 5/15
Pictures: https://github.com/CardsAgainstCryptography https://csidh.isogeny.org 5/15
I’ve been experimenting with supersingular curves in this context, because they have all the properties Kieffer was looking for. Are there any security issues with using supersingular curves? Hope I did not overlook anything stupid here! — an anonymous CSIDH coauthor Pictures: https://github.com/CardsAgainstCryptography https://csidh.isogeny.org 5/15
I’ve been experimenting with supersingular curves in this context, because they have all the properties Kieffer was looking for. Are there any security issues with using supersingular curves? Hope I did not overlook anything stupid here! — an anonymous CSIDH coauthor Wouter, you are a genius! — me Pictures: https://github.com/CardsAgainstCryptography https://csidh.isogeny.org 5/15
Supersingular isogeny graphs Nodes: Supersingular elliptic curves defined over k up to ∼ = k . Edges: 3-, 5-, and 7-isogenies defined over k up to ∼ = k . https://csidh.isogeny.org 6/15
Supersingular isogeny graphs Nodes: Supersingular elliptic curves defined over k up to ∼ = k . Edges: 3-, 5-, and 7-isogenies defined over k up to ∼ = k . k = F 419 2 (same as F 419 ) https://csidh.isogeny.org 6/15
Supersingular isogeny graphs Nodes: Supersingular elliptic curves defined over k up to ∼ = k . Edges: 3-, 5-, and 7-isogenies defined over k up to ∼ = k . k = F 419 2 (same as F 419 ) k = F 419 https://csidh.isogeny.org 6/15
Supersingular isogeny graphs Theorem. The F p -rational endomorphism ring of an elliptic curve defined over F p is an imaginary quadratic order. https://csidh.isogeny.org 7/15
Supersingular isogeny graphs Theorem. The F p -rational endomorphism ring of an elliptic curve defined over F p is an imaginary quadratic order. ...even in the supersingular case! https://csidh.isogeny.org 7/15
Supersingular isogeny graphs Theorem. The F p -rational endomorphism ring of an elliptic curve defined over F p is an imaginary quadratic order. ...even in the supersingular case! Theorem/fact/definition. Let p > 3. An elliptic curve E over F p is supersingular if and only if # E ( F p ) = p + 1. https://csidh.isogeny.org 7/15
Supersingular isogeny graphs Theorem. The F p -rational endomorphism ring of an elliptic curve defined over F p is an imaginary quadratic order. ...even in the supersingular case! Theorem/fact/definition. Let p > 3. An elliptic curve E over F p is supersingular if and only if # E ( F p ) = p + 1. = ⇒ We can simply craft a curve with a good number of points. https://csidh.isogeny.org 7/15
Reminder The class group action is defined as follows: ◮ Inputs : An elliptic curve E with endomorphism ring O , an ideal a ⊆ O of prime norm ℓ . ◮ Output : The elliptic curve [ a ] E . 1. Compute the subgroup E [ a ] = � α ∈ a ker α killed by a . → E ′ with kernel E [ a ] . 2. Compute an ℓ -isogeny E − 3. Output E ′ . https://csidh.isogeny.org 8/15
Reminder The class group action is defined as follows: ◮ Inputs : An elliptic curve E with endomorphism ring O , an ideal a ⊆ O of prime norm ℓ . ◮ Output : The elliptic curve [ a ] E . 1. Compute the subgroup E [ a ] = � α ∈ a ker α killed by a . → E ′ with kernel E [ a ] . 2. Compute an ℓ -isogeny E − 3. Output E ′ . Typically E [ a ] is only defined over F q m for m ≈ ℓ . = ⇒ Complexity of computing with E [ a ] is exponentia ℓ ... : ( https://csidh.isogeny.org 8/15
CSIDH in one cslide (terrible pun totally intended) https://csidh.isogeny.org 9/15
CSIDH in one cslide (terrible pun totally intended) ◮ Choose some small odd primes ℓ 1 , ..., ℓ n . 1. ◮ Make sure p = 4 · ℓ 1 · · · ℓ n − 1 is prime. ◮ Let X = { supersingular y 2 = x 3 + Ax 2 + x defined over F p } . https://csidh.isogeny.org 9/15
CSIDH in one cslide (terrible pun totally intended) ◮ Choose some small odd primes ℓ 1 , ..., ℓ n . 1. ◮ Make sure p = 4 · ℓ 1 · · · ℓ n − 1 is prime. ◮ Let X = { supersingular y 2 = x 3 + Ax 2 + x defined over F p } . ◮ All curves in X have F p -endomorphism ring O = Z [ √ p ] . 2. Define the ideals l i = ( ℓ i , π − 1 ) of O . ◮ Let K = { [ l e 1 1 · · · l e 1 n ] | ( e 1 , ..., e n ) is ‘short’ } ⊆ cl ( O ) . https://csidh.isogeny.org 9/15
CSIDH in one cslide (terrible pun totally intended) ◮ Choose some small odd primes ℓ 1 , ..., ℓ n . 1. ◮ Make sure p = 4 · ℓ 1 · · · ℓ n − 1 is prime. ◮ Let X = { supersingular y 2 = x 3 + Ax 2 + x defined over F p } . ◮ All curves in X have F p -endomorphism ring O = Z [ √ p ] . 2. Define the ideals l i = ( ℓ i , π − 1 ) of O . ◮ Let K = { [ l e 1 1 · · · l e 1 n ] | ( e 1 , ..., e n ) is ‘short’ } ⊆ cl ( O ) . 3. magic math happens! ∗ ∗ see next slides https://csidh.isogeny.org 9/15
CSIDH in one cslide (terrible pun totally intended) ◮ Choose some small odd primes ℓ 1 , ..., ℓ n . 1. ◮ Make sure p = 4 · ℓ 1 · · · ℓ n − 1 is prime. ◮ Let X = { supersingular y 2 = x 3 + Ax 2 + x defined over F p } . ◮ All curves in X have F p -endomorphism ring O = Z [ √ p ] . 2. Define the ideals l i = ( ℓ i , π − 1 ) of O . ◮ Let K = { [ l e 1 1 · · · l e 1 n ] | ( e 1 , ..., e n ) is ‘short’ } ⊆ cl ( O ) . 3. magic math happens! ∗ ∗ see next slides 4. ◮ cl ( O ) acts on X and the action of K is very efficient! https://csidh.isogeny.org 9/15
Magic (base field arithmetic) ◮ All the ideals ℓ i O split as l i · l i where l i = ( ℓ i , π − 1 ) . = ⇒ We can use all ℓ i we started with (generally: about 1/2) . https://csidh.isogeny.org 10/15
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