# 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, on which we define the prime-order groups jq255e and jq255s (the original version of this work defined the do255e and do255s; the "jq" groups are algebraically identical, but use a different encoding process). Both provide "128-bit security" (technically, 127 bits); jq255e is a GLV curve that can leverage an internal endomorphism for further improved performance, while jq255s is a perfectly ordinary curve.

(New: 2022-08-13) The original do255e and do255s have been reinterpreted into a subcase of Jacobi quartics, from which we obtain the new groups jq255e and jq255s (same groups as previously but a different encoding format). They have faster formulas and in particular a better encoding and decoding (as efficient as plain Ed25519 point compression). On top of jq255e and jq255s, we define short signatures of length 48 bytes, noticeably shorter than the usual 64-byte signatures of Ed25519 or P-256/ECDSA; this also makes signature verification faster, down to about 93000 cycles on a 64-bit x86 system (Coffee Lake core).

## Resources

The original 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). The additional whitepaper presents the reinterpretation of double-odd curves as Jacobi quartics, leading to the jq255e and jq255s groups, and new specifications of the higher-level cryptographic schemes, in particular short signatures.

Note: both the original whitepaper and the additional whitepapers are also published on ePrint:

Several implementations of jq255e, jq255s, do255e and do255s exist and are listed in the implementations section.

## Outline

Subsequent pages in this site include the following:

• A geometrical introduction to double-odd elliptic curves: a self-contained explanatory text (with pictures) to explain the motivation and the core ideas of this work.

• (New: 2022-08-13) Double-odd Jacobi quartics presents the reinterpretation of double-odd curves as a special case of Jacobi quartics, which leads to faster formulas and an improved decoding process.

• (New: 2022-08-13) The short signatures describes the signature scheme specified on the jq255e and jq255s groups, i.e. the double-odd curves interpreted as Jacobi quartics. This scheme makes signature substantially shorter (48 bytes instead of 64), and also speeds up signature verification.

• The formulas section lists useful formulas for computing over points of double-odd elliptic curves with several convenient systems of coordinates.

• Explicit curve choices are listed: these are curves do255e/jq255e and do255s/jq255s. Selection criteria and curve parameters are provided.

• Some benchmarks on large (x86) and small (ARM Cortex M0+ and M4) architectures are detailed.

• Available open-source implementations are listed.