Naming Elliptic Curves Used in Cryptography

There is an infinite number of elliptic curves, but a small number 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 number of points on the elliptic curve is on the order of p [1].

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 few named elliptic curves.

In Curve25519, p = 2 ²⁵⁵ - 19 and in Curve 383187, p = 2 ³⁸³ - 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 is:

x² + y² = 1 – 1174 x²y²

And so, the 1174 in the name comes from a coefficient rather than from the number of bits in p.

Edwards Curves

The equation for Curve1174 doesn't look like an elliptic curve. It doesn't have the familiar (Weierstrass) form:

y² = x³ + ax + b

It 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 form

x² + y² = 1 + dx² 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 = 2 ²⁵¹ - 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.

[1] It is difficult to compute the exact number of points on an elliptic curve over a prime field. However, the number is roughly p ± 2√ p. More precisely, Hasse's theorem says