bitcoin-bitcoin-core/doc/ellswift.md

ElligatorSwift for secp256k1 explained

In this document we explain how the ellswift module implementation is related to the construction in the "SwiftEC: Shallue–van de Woestijne Indifferentiable Function To Elliptic Curves" paper by Jorge Chávez-Saab, Francisco Rodríguez-Henríquez, and Mehdi Tibouchi.

• 1. Introduction
• 2. The decoding function
  • 2.1 Decoding for secp256k1
• 3. The encoding function
  • 3.1 Switching to v, w coordinates
  • 3.2 Avoiding computing all inverses
  • 3.3 Finding the inverse
  • 3.4 Dealing with special cases
  • 3.5 Encoding for secp256k1
• 4. Encoding and decoding full (x, y) coordinates
  • 4.1 Full (x, y) coordinates for secp256k1

1. Introduction

The ellswift module effectively introduces a new 64-byte public key format, with the property that (uniformly random) public keys can be encoded as 64-byte arrays which are computationally indistinguishable from uniform byte arrays. The module provides functions to convert public keys from and to this format, as well as convenience functions for key generation and ECDH that operate directly on ellswift-encoded keys.

The encoding consists of the concatenation of two (32-byte big endian) encoded field elements u and t. Together they encode an x-coordinate on the curve x, or (see further) a full point (x, y) on the curve.

Decoding consists of decoding the field elements u and t (values above the field size p are taken modulo p), and then evaluating F_u(t), which for every u and t results in a valid x-coordinate on the curve. The functions F_u will be defined in Section 2.

Encoding a given x coordinate is conceptually done as follows:

• Loop:
  • Pick a uniformly random field element u.
  • Compute the set L = F_u^{-1}(x) of t values for which F_u(t) = x, which may have up to 8 elements.
  • With probability 1 - \dfrac{\\#L}{8}, restart the loop.
  • Select a uniformly random t \in L and return (u, t).

This is the ElligatorSwift algorithm, here given for just x-coordinates. An extension to full (x, y) points will be given in Section 4. The algorithm finds a uniformly random (u, t) among (almost all) those for which F_u(t) = x. Section 3.2 in the paper proves that the number of such encodings for almost all x-coordinates on the curve (all but at most 39) is close to two times the field size (specifically, it lies in the range 2q \pm (22\sqrt{q} + O(1)), where q is the size of the field).

2. The decoding function

First some definitions:

• \mathbb{F} is the finite field of size q, of characteristic 5 or more, and q \equiv 1 \mod 3.
  • For secp256k1, q = 2^{256} - 2^{32} - 977, which satisfies that requirement.
• Let E be the elliptic curve of points (x, y) \in \mathbb{F}^2 for which y^2 = x^3 + ax + b, with a and b public constants, for which \Delta_E = -16(4a^3 + 27b^2) is a square, and at least one of (-b \pm \sqrt{-3 \Delta_E} / 36)/2 is a square. This implies that the order of E is either odd, or a multiple of 4. If a=0, this condition is always fulfilled.
  • For secp256k1, a=0 and b=7.
• Let the function g(x) = x^3 + ax + b, so the E curve equation is also y^2 = g(x).
• Let the function h(x) = 3x^3 + 4a.
• Define V as the set of solutions (x_1, x_2, x_3, z) to z^2 = g(x_1)g(x_2)g(x_3).
• Define S_u as the set of solutions (X, Y) to X^2 + h(u)Y^2 = -g(u) and Y \neq 0.
• P_u is a function from \mathbb{F} to S_u that will be defined below.
• \psi_u is a function from S_u to V that will be defined below.

Note: In the paper:

• F_u corresponds to F_{0,u} there.
• P_u(t) is called P there.
• All S_u sets together correspond to S there.
• All \psi_u functions together (operating on elements of S) correspond to \psi there.

Note that for V, the left hand side of the equation z^2 is square, and thus the right hand must also be square. As multiplying non-squares results in a square in \mathbb{F}, out of the three right-hand side factors an even number must be non-squares. This implies that exactly 1 or exactly 3 out of \\{g(x_1), g(x_2), g(x_3)\\} must be square, and thus that for any (x_1,x_2,x_3,z) \in V, at least one of \\{x_1, x_2, x_3\\} must be a valid x-coordinate on E. There is one exception to this, namely when z=0, but even then one of the three values is a valid x-coordinate.

Define the decoding function F_u(t) as:

• Let (x_1, x_2, x_3, z) = \psi_u(P_u(t)).
• Return the first element x of (x_3, x_2, x_1) which is a valid x-coordinate on E (i.e., g(x) is square).

P_u(t) = (X(u, t), Y(u, t)), where:

\begin{array}{lcl} X(u, t) & = & \left\{\begin{array}{ll} \dfrac{g(u) - t^2}{2t} & a = 0 \ \dfrac{g(u) + h(u)(Y_0(u) + X_0(u)t)^2}{X_0(u)(1 + h(u)t^2)} & a \neq 0 \end{array}\right. \ Y(u, t) & = & \left\{\begin{array}{ll} \dfrac{X(u, t) + t}{u \sqrt{-3}} = \dfrac{g(u) + t^2}{2tu\sqrt{-3}} & a = 0 \ Y_0(u) + t(X(u, t) - X_0(u)) & a \neq 0 \end{array}\right. \end{array}

P_u(t) is defined:

• For a=0, unless:
  • u = 0 or t = 0 (division by zero)
  • g(u) = -t^2 (would give Y=0).
• For a \neq 0, unless:
  • X_0(u) = 0 or h(u)t^2 = -1 (division by zero)
  • Y_0(u) (1 - h(u)t^2) = 2X_0(u)t (would give Y=0).

The functions X_0(u) and Y_0(u) are defined in Appendix A of the paper, and depend on various properties of E.

The function \psi_u is the same for all curves: \psi_u(X, Y) = (x_1, x_2, x_3, z), where:

\begin{array}{lcl} x_1 & = & \dfrac{X}{2Y} - \dfrac{u}{2} && \ x_2 & = & -\dfrac{X}{2Y} - \dfrac{u}{2} && \ x_3 & = & u + 4Y^2 && \ z & = & \dfrac{g(x_3)}{2Y}(u^2 + ux_1 + x_1^2 + a) = \dfrac{-g(u)g(x_3)}{8Y^3} \end{array}

2.1 Decoding for secp256k1

Put together and specialized for a=0 curves, decoding (u, t) to an x-coordinate is:

Define F_u(t) as:

• Let X = \dfrac{u^3 + b - t^2}{2t}.
• Let Y = \dfrac{X + t}{u\sqrt{-3}}.
• Return the first x in (u + 4Y^2, \dfrac{-X}{2Y} - \dfrac{u}{2}, \dfrac{X}{2Y} - \dfrac{u}{2}) for which g(x) is square.

To make sure that every input decodes to a valid x-coordinate, we remap the inputs in case P_u is not defined (when u=0, t=0, or g(u) = -t^2):

Define F_u(t) as:

• Let u'=u if u \neq 0; 1 otherwise (guaranteeing u' \neq 0).
• Let t'=t if t \neq 0; 1 otherwise (guaranteeing t' \neq 0).
• Let t''=t' if g(u') \neq -t'^2; 2t' otherwise (guaranteeing t'' \neq 0 and g(u') \neq -t''^2).
• Let X = \dfrac{u'^3 + b - t''^2}{2t''}.
• Let Y = \dfrac{X + t''}{u'\sqrt{-3}}.
• Return the first x in (u' + 4Y^2, \dfrac{-X}{2Y} - \dfrac{u'}{2}, \dfrac{X}{2Y} - \dfrac{u'}{2}) for which x^3 + b is square.

The choices here are not strictly necessary. Just returning a fixed constant in any of the undefined cases would suffice, but the approach here is simple enough and gives fairly uniform output even in these cases.

Note: in the paper these conditions result in \infty as output, due to the use of projective coordinates there. We wish to avoid the need for callers to deal with this special case.

This is implemented in secp256k1_ellswift_xswiftec_frac_var (which decodes to an x-coordinate represented as a fraction), and in secp256k1_ellswift_xswiftec_var (which outputs the actual x-coordinate).

3. The encoding function

To implement F_u^{-1}(x), the function to find the set of inverses t for which F_u(t) = x, we have to reverse the process:

• Find all the (X, Y) \in S_u that could have given rise to x, through the x_1, x_2, or x_3 formulas in \psi_u.
• Map those (X, Y) solutions to t values using P_u^{-1}(X, Y).
• For each of the found t values, verify that F_u(t) = x.
• Return the remaining t values.

The function P_u^{-1}, which finds t given (X, Y) \in S_u, is significantly simpler than P_u:

P_u^{-1}(X, Y) = \left\{\begin{array}{ll} Yu\sqrt{-3} - X & a = 0 \ \dfrac{Y-Y_0(u)}{X-X_0(u)} & a \neq 0 \land X \neq X_0(u) \ \dfrac{-X_0(u)}{h(u)Y_0(u)} & a \neq 0 \land X = X_0(u) \land Y = Y_0(u) \end{array}\right.

The third step above, verifying that F_u(t) = x, is necessary because for the (X, Y) values found through the x_1 and x_2 expressions, it is possible that decoding through \psi_u(X, Y) yields a valid x_3 on the curve, which would take precedence over the x_1 or x_2 decoding. These (X, Y) solutions must be rejected.

Since we know that exactly one or exactly three out of \\{x_1, x_2, x_3\\} are valid x-coordinates for any t, the case where either x_1 or x_2 is valid and in addition also x_3 is valid must mean that all three are valid. This means that instead of checking whether x_3 is on the curve, it is also possible to check whether the other one out of x_1 and x_2 is on the curve. This is significantly simpler, as it turns out.

Observe that \psi_u guarantees that x_1 + x_2 = -u. So given either x = x_1 or x = x_2, the other one of the two can be computed as -u - x. Thus, when encoding x through the x_1 or x_2 expressions, one can simply check whether g(-u-x) is a square, and if so, not include the corresponding t values in the returned set. As this does not need X, Y, or t, this condition can be determined before those values are computed.

It is not possible that an encoding found through the x_1 expression decodes to a different valid x-coordinate using x_2 (which would take precedence), for the same reason: if both x_1 and x_2 decodings were valid, x_3 would be valid as well, and thus take precedence over both. Because of this, the g(-u-x) being square test for x_1 and x_2 is the only test necessary to guarantee the found t values round-trip back to the input x correctly. This is the reason for choosing the (x_3, x_2, x_1) precedence order in the decoder; any order which does not place x_3 first requires more complicated round-trip checks in the encoder.

3.1 Switching to v, w coordinates

Before working out the formulas for all this, we switch to different variables for S_u. Let v = (X/Y - u)/2, and w = 2Y. Or in the other direction, X = w(u/2 + v) and Y = w/2:

• S_u' becomes the set of (v, w) for which w^2 (u^2 + uv + v^2 + a) = -g(u) and w \neq 0.
• For a=0 curves, P_u^{-1} can be stated for (v,w) as P_u^{'-1}(v, w) = w\left(\frac{\sqrt{-3}-1}{2}u - v\right).
• \psi_u can be stated for (v, w) as \psi_u'(v, w) = (x_1, x_2, x_3, z), where

\begin{array}{lcl} x_1 & = & v \ x_2 & = & -u - v \ x_3 & = & u + w^2 \ z & = & \dfrac{g(x_3)}{w}(u^2 + uv + v^2 + a) = \dfrac{-g(u)g(x_3)}{w^3} \end{array}

We can now write the expressions for finding (v, w) given x explicitly, by solving each of the \\{x_1, x_2, x_3\\} expressions for v or w, and using the S_u' equation to find the other variable:

• Assuming x = x_1, we find v = x and w = \pm\sqrt{-g(u)/(u^2 + uv + v^2 + a)} (two solutions).
• Assuming x = x_2, we find v = -u-x and w = \pm\sqrt{-g(u)/(u^2 + uv + v^2 + a)} (two solutions).
• Assuming x = x_3, we find w = \pm\sqrt{x-u} and v = -u/2 \pm \sqrt{-w^2(4g(u) + w^2h(u))}/(2w^2) (four solutions).

3.2 Avoiding computing all inverses

The ElligatorSwift algorithm as stated in Section 1 requires the computation of L = F_u^{-1}(x) (the set of all t such that (u, t) decode to x) in full. This is unnecessary.

Observe that the procedure of restarting with probability (1 - \frac{\\#L}{8}) and otherwise returning a uniformly random element from L is actually equivalent to always padding L with \bot values up to length 8, picking a uniformly random element from that, restarting whenever \bot is picked:

Define ElligatorSwift(x) as:

• Loop:
  • Pick a uniformly random field element u.
  • Compute the set L = F_u^{-1}(x).
  • Let T be the 8-element vector consisting of the elements of L, plus 8 - \\#L times \\{\bot\\}.
  • Select a uniformly random t \in T.
  • If t \neq \bot, return (u, t); restart loop otherwise.

Now notice that the order of elements in T does not matter, as all we do is pick a uniformly random element in it, so we do not need to have all \bot values at the end. As we have 8 distinct formulas for finding (v, w) (taking the variants due to \pm into account), we can associate every index in T with exactly one of those formulas, making sure that:

• Formulas that yield no solutions (due to division by zero or non-existing square roots) or invalid solutions are made to return \bot.
• For the x_1 and x_2 cases, if g(-u-x) is a square, \bot is returned instead (the round-trip check).
• In case multiple formulas would return the same non- \bot result, all but one of those must be turned into \bot to avoid biasing those.

The last condition above only occurs with negligible probability for cryptographically-sized curves, but is interesting to take into account as it allows exhaustive testing in small groups. See Section 3.4 for an analysis of all the negligible cases.

If we define T = (G_{0,u}(x), G_{1,u}(x), \ldots, G_{7,u}(x)), with each G_{i,u} matching one of the formulas, the loop can be simplified to only compute one of the inverses instead of all of them:

Define ElligatorSwift(x) as:

• Loop:
  • Pick a uniformly random field element u.
  • Pick a uniformly random integer c in [0,8).
  • Let t = G_{c,u}(x).
  • If t \neq \bot, return (u, t); restart loop otherwise.

This is implemented in secp256k1_ellswift_xelligatorswift_var.

3.3 Finding the inverse

To implement G_{c,u}, we map c=0 to the x_1 formula, c=1 to the x_2 formula, and c=2 and c=3 to the x_3 formula. Those are then repeated as c=4 through c=7 for the other sign of w (noting that in each formula, w is a square root of some expression). Ignoring the negligible cases, we get:

Define G_{c,u}(x) as:

• If c \in \\{0, 1, 4, 5\\} (for x_1 and x_2 formulas):
  • If g(-u-x) is square, return \bot (as x_3 would be valid and take precedence).
  • If c \in \\{0, 4\\} (the x_1 formula) let v = x, otherwise let v = -u-x (the x_2 formula)
  • Let s = -g(u)/(u^2 + uv + v^2 + a) (using s = w^2 in what follows).
• Otherwise, when c \in \\{2, 3, 6, 7\\} (for x_3 formulas):
  • Let s = x-u.
  • Let r = \sqrt{-s(4g(u) + sh(u))}.
  • Let v = (r/s - u)/2 if c \in \\{3, 7\\}; (-r/s - u)/2 otherwise.
• Let w = \sqrt{s}.
• Depending on c:
  • If c \in \\{0, 1, 2, 3\\}: return P_u^{'-1}(v, w).
  • If c \in \\{4, 5, 6, 7\\}: return P_u^{'-1}(v, -w).

Whenever a square root of a non-square is taken, \bot is returned; for both square roots this happens with roughly 50% on random inputs. Similarly, when a division by 0 would occur, \bot is returned as well; this will only happen with negligible probability. A division by 0 in the first branch in fact cannot occur at all, because u^2 + uv + v^2 + a = 0 implies g(-u-x) = g(x) which would mean the g(-u-x) is square condition has triggered and \bot would have been returned already.

Note: In the paper, the case variable corresponds roughly to the c above, but only takes on 4 possible values (1 to 4). The conditional negation of w at the end is done randomly, which is equivalent, but makes testing harder. We choose to have the G_{c,u} be deterministic, and capture all choices in c.

Now observe that the c \in \\{1, 5\\} and c \in \\{3, 7\\} conditions effectively perform the same v \rightarrow -u-v transformation. Furthermore, that transformation has no effect on s in the first branch as u^2 + ux + x^2 + a = u^2 + u(-u-x) + (-u-x)^2 + a. Thus we can extract it out and move it down:

Define G_{c,u}(x) as:

• If c \in \\{0, 1, 4, 5\\}:
  • If g(-u-x) is square, return \bot.
  • Let s = -g(u)/(u^2 + ux + x^2 + a).
  • Let v = x.
• Otherwise, when c \in \\{2, 3, 6, 7\\}:
  • Let s = x-u.
  • Let r = \sqrt{-s(4g(u) + sh(u))}.
  • Let v = (r/s - u)/2.
• Let w = \sqrt{s}.
• Depending on c:
  • If c \in \\{0, 2\\}: return P_u^{'-1}(v, w).
  • If c \in \\{1, 3\\}: return P_u^{'-1}(-u-v, w).
  • If c \in \\{4, 6\\}: return P_u^{'-1}(v, -w).
  • If c \in \\{5, 7\\}: return P_u^{'-1}(-u-v, -w).

This shows there will always be exactly 0, 4, or 8 t values for a given (u, x) input. There can be 0, 1, or 2 (v, w) pairs before invoking P_u^{'-1}, and each results in 4 distinct t values.

3.4 Dealing with special cases

As mentioned before there are a few cases to deal with which only happen in a negligibly small subset of inputs. For cryptographically sized fields, if only random inputs are going to be considered, it is unnecessary to deal with these. Still, for completeness we analyse them here. They generally fall into two categories: cases in which the encoder would produce t values that do not decode back to x (or at least cannot guarantee that they do), and cases in which the encoder might produce the same t value for multiple c inputs (thereby biasing that encoding):

• In the branch for x_1 and x_2 (where c \in \\{0, 1, 4, 5\\}):
  • When g(u) = 0, we would have s=w=Y=0, which is not on S_u. This is only possible on even-ordered curves. Excluding this also removes the one condition under which the simplified check for x_3 on the curve fails (namely when g(x_1)=g(x_2)=0 but g(x_3) is not square). This does exclude some valid encodings: when both g(u)=0 and u^2+ux+x^2+a=0 (also implying g(x)=0), the S_u' equation degenerates to 0 = 0, and many valid t values may exist. Yet, these cannot be targeted uniformly by the encoder anyway as there will generally be more than 8.
  • When g(x) = 0, the same t would be produced as in the x_3 branch (where c \in \\{2, 3, 6, 7\\}) which we give precedence as it can deal with g(u)=0. This is again only possible on even-ordered curves.
• In the branch for x_3 (where c \in \\{2, 3, 6, 7\\}):
  • When s=0, a division by zero would occur.
  • When v = -u-v and c \in \\{3, 7\\}, the same t would be returned as in the c \in \\{2, 6\\} cases. It is equivalent to checking whether r=0. This cannot occur in the x_1 or x_2 branches, as it would trigger the g(-u-x) is square condition. A similar concern for w = -w does not exist, as w=0 is already impossible in both branches: in the first it requires g(u)=0 which is already outlawed on even-ordered curves and impossible on others; in the second it would trigger division by zero.
• Curve-specific special cases also exist that need to be rejected, because they result in (u,t) which is invalid to the decoder, or because of division by zero in the encoder:
  • For a=0 curves, when u=0 or when t=0. The latter can only be reached by the encoder when g(u)=0, which requires an even-ordered curve.
  • For a \neq 0 curves, when X_0(u)=0, when h(u)t^2 = -1, or when 2w(u + 2v) = 2X_0(u) while also either w \neq 2Y_0(u) or h(u)=0.

Define a version of G_{c,u}(x) which deals with all these cases:

• If a=0 and u=0, return \bot.
• If a \neq 0 and X_0(u)=0, return \bot.
• If c \in \\{0, 1, 4, 5\\}:
  • If g(u) = 0 or g(x) = 0, return \bot (even curves only).
  • If g(-u-x) is square, return \bot.
  • Let s = -g(u)/(u^2 + ux + x^2 + a) (cannot cause division by zero).
  • Let v = x.
• Otherwise, when c \in \\{2, 3, 6, 7\\}:
  • Let s = x-u.
  • Let r = \sqrt{-s(4g(u) + sh(u))}; return \bot if not square.
  • If c \in \\{3, 7\\} and r=0, return \bot.
  • If s = 0, return \bot.
  • Let v = (r/s - u)/2.
• Let w = \sqrt{s}; return \bot if not square.
• If a \neq 0 and w(u+2v) = 2X_0(u) and either w \neq 2Y_0(u) or h(u) = 0, return \bot.
• Depending on c:
  • If c \in \\{0, 2\\}, let t = P_u^{'-1}(v, w).
  • If c \in \\{1, 3\\}, let t = P_u^{'-1}(-u-v, w).
  • If c \in \\{4, 6\\}, let t = P_u^{'-1}(v, -w).
  • If c \in \\{5, 7\\}, let t = P_u^{'-1}(-u-v, -w).
• If a=0 and t=0, return \bot (even curves only).
• If a \neq 0 and h(u)t^2 = -1, return \bot.
• Return t.

Given any u, using this algorithm over all x and c values, every t value will be reached exactly once, for an x for which F_u(t) = x holds, except for these cases that will not be reached:

• All cases where P_u(t) is not defined:
  • For a=0 curves, when u=0, t=0, or g(u) = -t^2.
  • For a \neq 0 curves, when h(u)t^2 = -1, X_0(u) = 0, or Y_0(u) (1 - h(u) t^2) = 2X_0(u)t.
• When g(u)=0, the potentially many t values that decode to an x satisfying g(x)=0 using the x_2 formula. These were excluded by the g(u)=0 condition in the c \in \\{0, 1, 4, 5\\} branch.

These cases form a negligible subset of all (u, t) for cryptographically sized curves.

3.5 Encoding for secp256k1

Specialized for odd-ordered a=0 curves:

Define G_{c,u}(x) as:

• If u=0, return \bot.
• If c \in \\{0, 1, 4, 5\\}:
  • If (-u-x)^3 + b is square, return \bot
  • Let s = -(u^3 + b)/(u^2 + ux + x^2) (cannot cause division by 0).
  • Let v = x.
• Otherwise, when c \in \\{2, 3, 6, 7\\}:
  • Let s = x-u.
  • Let r = \sqrt{-s(4(u^3 + b) + 3su^2)}; return \bot if not square.
  • If c \in \\{3, 7\\} and r=0, return \bot.
  • If s = 0, return \bot.
  • Let v = (r/s - u)/2.
• Let w = \sqrt{s}; return \bot if not square.
• Depending on c:
  • If c \in \\{0, 2\\}: return w(\frac{\sqrt{-3}-1}{2}u - v).
  • If c \in \\{1, 3\\}: return w(\frac{\sqrt{-3}+1}{2}u + v).
  • If c \in \\{4, 6\\}: return w(\frac{-\sqrt{-3}+1}{2}u + v).
  • If c \in \\{5, 7\\}: return w(\frac{-\sqrt{-3}-1}{2}u - v).

This is implemented in secp256k1_ellswift_xswiftec_inv_var.

And the x-only ElligatorSwift encoding algorithm is still:

Define ElligatorSwift(x) as:

• Loop:
  • Pick a uniformly random field element u.
  • Pick a uniformly random integer c in [0,8).
  • Let t = G_{c,u}(x).
  • If t \neq \bot, return (u, t); restart loop otherwise.

Note that this logic does not take the remapped u=0, t=0, and g(u) = -t^2 cases into account; it just avoids them. While it is not impossible to make the encoder target them, this would increase the maximum number of t values for a given (u, x) combination beyond 8, and thereby slow down the ElligatorSwift loop proportionally, for a negligible gain in uniformity.

4. Encoding and decoding full (x, y) coordinates

So far we have only addressed encoding and decoding x-coordinates, but in some cases an encoding for full points with (x, y) coordinates is desirable. It is possible to encode this information in t as well.

Note that for any (X, Y) \in S_u, (\pm X, \pm Y) are all on S_u. Moreover, all of these are mapped to the same x-coordinate. Negating X or negating Y just results in x_1 and x_2 being swapped, and does not affect x_3. This will not change the outcome x-coordinate as the order of x_1 and x_2 only matters if both were to be valid, and in that case x_3 would be used instead.

Still, these four (X, Y) combinations all correspond to distinct t values, so we can encode the sign of the y-coordinate in the sign of X or the sign of Y. They correspond to the four distinct P_u^{'-1} calls in the definition of G_{u,c}.

Note: In the paper, the sign of the y coordinate is encoded in a separately-coded bit.

To encode the sign of y in the sign of Y:

Define Decode(u, t) for full (x, y) as:

• Let (X, Y) = P_u(t).
• Let x be the first value in (u + 4Y^2, \frac{-X}{2Y} - \frac{u}{2}, \frac{X}{2Y} - \frac{u}{2}) for which g(x) is square.
• Let y = \sqrt{g(x)}.
• If sign(y) = sign(Y), return (x, y); otherwise return (x, -y).

And encoding would be done using a G_{c,u}(x, y) function defined as:

Define G_{c,u}(x, y) as:

• If c \in \\{0, 1\\}:
  • If g(u) = 0 or g(x) = 0, return \bot (even curves only).
  • If g(-u-x) is square, return \bot.
  • Let s = -g(u)/(u^2 + ux + x^2 + a) (cannot cause division by zero).
  • Let v = x.
• Otherwise, when c \in \\{2, 3\\}:
  • Let s = x-u.
  • Let r = \sqrt{-s(4g(u) + sh(u))}; return \bot if not square.
  • If c = 3 and r = 0, return \bot.
  • Let v = (r/s - u)/2.
• Let w = \sqrt{s}; return \bot if not square.
• Let w' = w if sign(w/2) = sign(y); -w otherwise.
• Depending on c:
  • If c \in \\{0, 2\\}: return P_u^{'-1}(v, w').
  • If c \in \\{1, 3\\}: return P_u^{'-1}(-u-v, w').

Note that c now only ranges [0,4), as the sign of w' is decided based on that of y, rather than on c. This change makes some valid encodings unreachable: when y = 0 and sign(Y) \neq sign(0).

In the above logic, sign can be implemented in several ways, such as parity of the integer representation of the input field element (for prime-sized fields) or the quadratic residuosity (for fields where -1 is not square). The choice does not matter, as long as it only takes on two possible values, and for x \neq 0 it holds that sign(x) \neq sign(-x).

4.1 Full (x, y) coordinates for secp256k1

For a=0 curves, there is another option. Note that for those, the P_u(t) function translates negations of t to negations of (both) X and Y. Thus, we can use sign(t) to encode the y-coordinate directly. Combined with the earlier remapping to guarantee all inputs land on the curve, we get as decoder:

Define Decode(u, t) as:

• Let u'=u if u \neq 0; 1 otherwise.
• Let t'=t if t \neq 0; 1 otherwise.
• Let t''=t' if u'^3 + b + t'^2 \neq 0; 2t' otherwise.
• Let X = \dfrac{u'^3 + b - t''^2}{2t''}.
• Let Y = \dfrac{X + t''}{u'\sqrt{-3}}.
• Let x be the first element of (u' + 4Y^2, \frac{-X}{2Y} - \frac{u'}{2}, \frac{X}{2Y} - \frac{u'}{2}) for which g(x) is square.
• Let y = \sqrt{g(x)}.
• Return (x, y) if sign(y) = sign(t); (x, -y) otherwise.

This is implemented in secp256k1_ellswift_swiftec_var. The used sign(x) function is the parity of x when represented as in integer in [0,q).

The corresponding encoder would invoke the x-only one, but negating the output t if sign(t) \neq sign(y).

This is implemented in secp256k1_ellswift_elligatorswift_var.

Note that this is only intended for encoding points where both the x-coordinate and y-coordinate are unpredictable. When encoding x-only points where the y-coordinate is implicitly even (or implicitly square, or implicitly in [0,q/2]), the encoder in Section 3.5 must be used, or a bias is reintroduced that undoes all the benefit of using ElligatorSwift in the first place.