# The Doppio Group

(This is a work in progress).

This site will describe the construction of a group called "Doppio", designed to provide a prime-order group for use inside of a Bulletproof. Doppio uses the Decaf construction on a curve defined over the ristretto255 scalar field and also serves as a worked example of how to use Decaf with elliptic curves embedded in zero-knowledge proof systems.

# Curve Selection

Our goal is to provide a prime-order group defined over the ristretto255 scalar field $$\mathbb F_q$$, where $q = 2^{252} + 27742317777372353535851937790883648493$ is the order of the ristretto255 group, so that group operations can be efficiently performed inside of a Bulletproof (or some other discrete-log based proof system implemented using ristretto255).

We will proceed in two steps: first, we select a (family of) elliptic curve(s) defined over the ristretto255 scalar field, and then we will use Decaf to construct a prime-order group from a non-prime-order curve. Decaf's conceptual plane does not include just one curve, but a family of curves related by isogenies, depicted in the following diagram parameterized by field elements $$a,d$$:

The Jacobi quartic $$\mathcal J_{a^2, a-2d}$$ is conceptually central; it is connected to bitstrings $$\{0,1\}^n$$ by the encoding function. On the top row is the Edwards curve $$\mathcal E_{a,d}$$ and an isogenous curve $$\mathcal E_{-a, d-a}$$. On the bottom row is the Montgomery curve $$\mathcal M_{a, 2-4d/a}$$ with Montgomery parameter $$A = 2-4d/a$$.
The Decaf strategy is to construct a prime-order group using any of the Edwards or Montgomery curves by transporting the encoding defined on the Jacobi quartic along an isogeny.

As noted in the Decaf paper, results of Ahmadi and Granger show that the curves $$\mathcal E_{a,d}$$ and $$\mathcal E_{-a, d-a}$$ are both $$2$$-isogenous to the same Jacobi quartic and thus $$4$$-isogenous to each other. The Montgomery curve is also $$2$$-isogenous to the Jacobi quartic; depending on the ground field and the parameters, the Montgomery curve may be isomorphic to another Edwards curve $$\mathcal E'$$. This connection is used for Ristretto, but is drawn lightly in the diagram because we will not use it here, as we have freedom to select an arbitrary curve, and are not constrained by legacy compatibility.

While Decaf provides a prime-order group with a canonical encoding and built-in validation, it is concievable that a user might want to use one of the underlying curves directly. Montgomery curves allow fast x-coordinate arithmetic; as described in the Costello-Smith survey, this really operates on the Kummer line of the curve, which unifies the curve and its quadratic twist. For this to be secure, both the Montgomery curve and its twist must have nearly-prime order. Unfortunately, as noted on page 15 of that paper, when $$q \equiv 1 \pmod 4$$, as in our case, it's not possible for both the curve and its quadratic twist to have minimal cofactor $$4$$; one can have cofactor $$4$$ but the other must have cofactor $$8$$.

The "base" curve is usually selected to have cofactor $$8$$, so that multiplication by the "base" cofactor also clears the "twist" cofactor. But this notion of twist security is only relevant when using the Kummer line, which unifies both curves and from whose perspective there is no canonically distinguished "base" and "twist". So it is sufficent to specify that Montgomery arithmetic should multiply by $$8$$, the lcm of the cofactors of the curves it unifies; the "SafeCurves" criteria is an essentially arbitrary choice which selects Curve25519.

On the other hand, specifying a cofactor $$4$$ curve for use with Decaf has an important advantage. While Decaf can be extended to cover curves with cofactor $$8$$, as in ristretto255, this comes at a theoretical cost of complexity and a practical cost of an extra sign check. While the cost of a sign check is quite small in the machine cost model for software implementations, it is much more significant in the circuit cost model for proof systems (as noted by Daira Hopwood). Sean Bowe proposed a cofactor-$$8$$ Montgomery curve intended for use with Ristretto inside of a circuit, but for the reasons above it is preferable to discard the arbitrary "SafeCurves" cofactor criterion and to instead select a curve of cofactor $$4$$ with twist cofactor $$8$$.

To select curve parameters, we follow an analogous procedure as in the [selection of Ed448][ed448], setting $$a = 1$$ and searching for $$d$$ nonsquare with minimal absolute value such that $\mathcal E_{a,d} : ax^2 + y^2 = 1 + dx^2y^2$ has cofactor $$4$$ and its quadratic twist has cofactor $$8$$. Using the derive.sage script contained in this repo (modified from the script used to select JubJub), we find two pairs of small parameters:

• $$(a, d) = (1, -63071)$$
• $$(a, d) = (1, 63072)$$

and choose the one with $$d$$ least in absolute value, $$d = -63071$$.

This curve $\mathcal E_{a,d} : x^2 + y^2 = 1 - 63071x^2y^2$ fits into the top-left position in the diagram above; the isogenous curves are $\mathcal E_{-a, d-a}: -x^2 + y^2 = 1 - 63072x^2y^2 \\ \mathcal M_{a,2-4d/a}: v^2 = u(u^2 + 252286u + 1),$ and a wire-compatible Decaf implementation is free to use any of these curves internally, depending on which curve model is optimal in the relevant cost model. For instance, a software implementation may prefer $$\mathcal E_{-a,d-a}$$ to allow use of the fast extended coordinates of Hisil, Wong, Carter, and Dawson, while a circuit implementation may prefer using Daira Hopwood's ordinary Montgomery formulas which are expressible with fewer constraints.

Note also that the curve $$\mathcal E_{-a,d-a}$$ has parameters $$(-1, -63072)$$, and is isomorphic to the second parameter pair found during the search, which explains why the search found two adjacent parameter pairs.