Naming elliptic curves used in cryptography

There are an infinite number of elliptic curves, but a small number that are used in cryptography, and these special curves have names.

Apparently there are no hard and fast rules for how the names are chosen, but there are patterns.

The named elliptic curves are over a prime field, i.

e.

a finite field with a prime number of elements p.

The curve names usually contain a number which is the number of bits in the binary representation of p.

Let’s see how that plays out with a list of elliptic curves.

|——————+———–| | Name | bits in p | |——————+———–| | ANSSI FRP256v1 | 256 | | BN(2, 254) | 254 | | brainpoolP256t1 | 256 | | Curve1174 | 251 | | Curve25519 | 255 | | Curve383187 | 383 | | E-222 | 222 | | E-382 | 382 | | E-521 | 521 | | Ed448-Goldilocks | 448 | | M-211 | 221 | | M-383 | 383 | | M-511 | 511 | | NIST P-224 | 224 | | NIST P-256 | 256 | | secp256k1 | 256 | |——————+———–| The first three curves in the list use a prime p that does not have a simple binary representation.

In Curve25519, p = 2255 – 19 and in Curve 383187, p = 2383 – 187.

Here the number of bits in p is part of the name but another number is stuck on.

The only mystery on the list is Curve1174 where p has 251 bits.

The equation for the curve isx² + y² = 1 – 1174 x²y²and so the 1174 in the name comes from a coefficient rather than from the number of bits in p.

Edwards curvesThe equation for Curve1174 doesn’t look like an elliptic curve.

It doesn’t have the familiar (Weierstrass) formy² = x³ + ax + bIt is an example of an Edwards curve, named after Harold Edwards.

So are all the curves above whose names start with “E”.

These curves have the formx² + y² = 1 + d x² y².

where d is not 0 or 1.

So some Edwards curves are named after their d parameter and some are named after the number of bits in p.

It’s not obvious that an Edwards curve can be changed into Weierstrass form, but apparently it’s possible; this paper goes into the details.

The advantage of Edwards curves is that the elliptic curve group addition has a simple, convenient form.

Also, when d is not a square in the underlying field, there are no exceptional points to consider for group addition.

Is d = -1174 a square in the field underlying Curve1174?.For that curve p = 2251 – 9, and we can use the Jacobi symbol code from earlier this week to show that d is not a square.

p = 2**251 – 9 d = p-1174 print(jacobi(d, p)) This prints -1, indicating that d is not a square.

Note that we set d to p – 1174 rather than -1174 because our code assumes the first argument is positive, and -1174 and p – 1174 are equivalent mod p.

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