# Why Ristretto?

## Background

Many cryptographic protocols require an implementation of a group of prime order \( \ell \), usually an elliptic curve group. However, modern elliptic curve implementations with fast, simple formulas don't provide a prime-order group. Instead, they provide a group of order \(h \cdot \ell \) for a small cofactor, usually \( h = 4 \) or \( h = 8\).

In many existing protocols, the complexity of managing this abstraction is pushed up the stack via ad-hoc protocol modifications. But these modifications are a recurring source of vulnerabilities and subtle design complications, and they usually prevent applying the security proofs of the abstract protocol.

On the other hand, prime-order curves provide the correct abstraction, but their formulas are slower and more difficult to implement in constant time. A clean solution to this dilemma is Mike Hamburg's Decaf proposal, which shows how to use a cofactor-\(4\) curve to provide a prime-order group – with no additional cost.

This provides the best of both choices: the correct abstraction required to implement complex protocols, and the simplicity, efficiency, and speed of a non-prime-order curve. However, many systems use Curve25519, which has cofactor \(8\), not cofactor \(4\).

**Ristretto** is a variant of Decaf designed for compatibility with
cofactor-\(8\) curves, such as Curve25519. It is particularly well-suited
for extending systems using Ed25519 signatures with complex zero-knowledge
protocols.

## Pitfalls of a cofactor

Curve cofactors have caused several vulnerabilities in higher-layer protocol implementations. The abstraction mismatch can also have subtle consequences for programs using these cryptographic protocols, as design quirks in the protocol bubble up—possibly even to the UX level.

The malleability in Ed25519 signatures caused a double-spend vulnerability—or, technically, octuple-spend as \( h = 8\)—in the CryptoNote scheme used by the Monero cryptocurrency, where the adversary could add a low-order point to an existing transaction, producing a new, seemingly-valid transaction.

In Tor, Ed25519 public key malleability would mean that every v3 onion service has eight different addresses, causing mismatches with user expectations and potential gotchas for service operators. Fixing this required expensive runtime checks in the v3 onion services protocol, requiring a full scalar multiplication, point compression, and equality check. This check must be called in several places to validate that the onion service's key does not contain a small torsion component.

## Why can't we just multiply by the cofactor?

In some protocols, designers can specify appropriate places to multiply by the cofactor \(h\) to "fix" the abstraction mismatches; in others, it's unfeasible or impossible. In any case, multiplying by the cofactor often means that security proofs are not cleanly applicable.

As touched upon earlier, a curve point consists of an \( h \)-torsion component and an \( \ell \)-torsion component. Multiplying by the cofactor is frequently referred to as "clearing" the low-order component, however doing so affects the \( \ell \)-torsion component, effectively mangling the point. While this works for some cases, it is not a validation method. To validate that a point is in the prime-order subgroup, one can alternately multiply by \( \ell \) and check that the result is the identity. But this is extremely expensive.

Another option is to mandate that all scalars have particular bit patterns, as in X25519 and Ed25519. However, this means that scalars are no longer well-defined \( \mathrm{mod} \ell \), which makes HKD schemes much more complicated. Yet another approach is to choose a torsion-safe representative: an integer which is \( 0 \mathrm{mod} h \) and with a particular value \( \mathrm{mod} \ell \), so that scalar multiplications remove the low-order component. But these representatives are a few bits too large to be used with existing implementations, and in any case aren't a comprehensive solution.

## A comprehensive solution

Rather than bit-twiddling, point mangling, or otherwise kludged-in ad-hoc fixes, Ristretto is a thin layer that provides protocol implementors with the correct abstraction: a prime-order group.