Lewis Carroll and Zero Knowledge Proofs
Elliptic curves are often used in cryptography, and in particular they are used in zero-knowledge proofs (ZKP). Cryptocurrencies such as Zcash use ZKP to protect the privacy of users.
Several of the elliptic curves used in ZKP have whimsical names taken from characters by Lewis Carroll. This post will look at these five elliptic curves:
- Jubjub
- Baby Jubjub
- Bandersnatch
- Tweedledee
- Tweedledum
Charles Dodgson was a mathematician, and perhaps there's some connection from his mathematical work to elliptic curves and ZKP, the connection explored here is with his literary works written under the name Lewis Carroll.
Jabberwocky namesThe first three curves-Jubjub, Baby Jubjub, and Bandersnatch-all get their name from Lewis Carroll's poem Jabberwocky.
Beware the Jabberwock, my son!
The jaws that bite, the claws that catch!
Beware the Jubjub bird, and shun
The frumious Bandersnatch!"
These curves all have a have a twisted Edwards curve form and a Montgomery curve form, just like the relationship between Ed25519 and Curve25519 that I wrote about a few days ago.
As its name suggests, the Baby Jubjub elliptic curve is related to the Jubjub curve but smaller.
Bandersnatch is similar to Jubjub, but arithmetic over this curve can be implemented more efficiently.
Looking Glass namesThe last two curves-Tweedledum and Tweedledee-take their names from Through the Looking Glass.
And as their names suggest, Tweedledum and Tweedledee and very closely related. Both have the equation
y^2 = x^3 + 5
but over different fields. Tweedledum is defined over the integers mod p and has q elements. Tweedledee is defined over the integers mod q and has p elements. (Mind your ps and qs!)
Here
p = 2254 + 4707489545178046908921067385359695873
q = 2254 + 4707489544292117082687961190295928833