# Double-Odd Elliptic Curves

Double-odd elliptic curves are a large subset of elliptic curves: these are curves whose order is equal to 2 modulo 4 (i.e. the order is the double of an odd integer). On top of double-odd elliptic curves, we can define a prime order group suitable for building cryptographic protocols. This is similar to what Ristretto achieves for twisted Edwards curves, albeit with a different method. Some highlights of double-odd elliptic curves are:

• There is no cofactor to deal with. A clean prime order group abstraction is provided, with a canonical encoding for elements which is automatically verified when decoding.

• The encoding is economical: for $$n$$-bit security, group elements are encoded over $$2n+1$$ bits (compared to $$2n+3$$ for Ristretto).

• All group operations are supported by unified and complete formulas amenable to efficient and secure (constant-time) implementations.

• No drastically new cryptographic hypothesis is introduced. Double-odd elliptic curves are fairly mundane; about 1/4th of all elliptic curves are double-odd curves. Such curves are considered secure for cryptography, just like any other random elliptic curve with a large enough subgroup of prime order.

• The formulas are fast and provide performance at least on par with twisted Edwards curves. In some respects they are faster; e.g. point doublings can be performed, for half of double-odd curves, with six multiplications (down to 1M+5S for some specific curves).

In order to leverage these techniques, two specific double-odd elliptic curves are defined: do255e and do255s. Both provide "128-bit security" (technically, 127 bits); do255e is a GLV curve that can leverage an internal endomorphism for further improved performance, while do255s is a perfectly ordinary curve.

## Resources

The whitepaper contains all the formulas and demonstrations; it also specifies the use of do255e and do255s in higher-level cryptographic functionalities (key pair generation, key exchange, signature, and hash-to-curve). Most of the present site consists in extracts from this whitepaper. Note: the whitepaper is now also published on ePrint.

Several implementations of do255e and do255s exist and are listed in the implementations section.

## Outline

Subsequent pages in this site include the following: