musig: add key aggregation spec draft

src/modules/musig/musig-spec.mediawiki

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+<pre>
+  Title: MuSig Key Aggregation
+  Author:
+  Status: Draft
+  License: BSD-2-Clause
+  Created: 2020-01-19
+</pre>
+
+== Introduction ==
+
+=== Abstract ===
+
+This document describes MuSig Key Aggregation in libsecp256k1-zkp.
+
+=== Copyright ===
+
+This document is licensed under the 2-clause BSD license.
+
+=== Motivation ===
+
+== Description ==
+
+=== Design ===
+
+* A function for sorting public keys allows to aggregate keys independent of the (initial) order.
+* The MuSig coefficient is computed by hashing the key instead of key index because. Otherwise, if the pubkey list gets sorted, the signer needs to translate between key indices pre- and post-sorting.
+* The second unique key in the pubkey list gets the constant MuSig coefficient 1 which saves an exponentiation.
+
+
+=== Specification ===
+
+The following conventions are used, with constants as defined for [https://www.secg.org/sec2-v2.pdf secp256k1]. We note that adapting this specification to other elliptic curves is not straightforward and can result in an insecure scheme<ref>Among other pitfalls, using the specification with a curve whose order is not close to the size of the range of the nonce derivation function is insecure.</ref>.
+* Lowercase variables represent integers or byte arrays.
+** The constant ''p'' refers to the field size, ''0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F''.
+** The constant ''n'' refers to the curve order, ''0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141''.
+* Uppercase variables refer to points on the curve with equation ''y<sup>2</sup> = x<sup>3</sup> + 7'' over the integers modulo ''p''.
+** ''is_infinite(P)'' returns whether or not ''P'' is the point at infinity.
+** ''x(P)'' and ''y(P)'' are integers in the range ''0..p-1'' and refer to the X and Y coordinates of a point ''P'' (assuming it is not infinity).
+** The constant ''G'' refers to the base point, for which ''x(G) = 0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798'' and ''y(G) = 0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8''.
+** Addition of points refers to the usual [https://en.wikipedia.org/wiki/Elliptic_curve#The_group_law elliptic curve group operation].
+** [https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication Multiplication (⋅) of an integer and a point] refers to the repeated application of the group operation.
+* Functions and operations:
+** ''||'' refers to byte array concatenation.
+** The function ''x[i:j]'', where ''x'' is a byte array and ''i, j &ge; 0'', returns a ''(j - i)''-byte array with a copy of the ''i''-th byte (inclusive) to the ''j''-th byte (exclusive) of ''x''.
+** The function ''bytes(x)'', where ''x'' is an integer, returns the 32-byte encoding of ''x'', most significant byte first.
+** The function ''bytes(P)'', where ''P'' is a point, returns ''bytes(x(P))''.
+** The function ''int(x)'', where ''x'' is a 32-byte array, returns the 256-bit unsigned integer whose most significant byte first encoding is ''x''.
+** The function ''has_even_y(P)'', where ''P'' is a point for which ''not is_infinite(P)'', returns ''y(P) mod 2 = 0''.
+** The function ''lift_x(x)'', where ''x'' is an integer in range ''0..p-1'', returns the point ''P'' for which ''x(P) = x''<ref>
+    Given a candidate X coordinate ''x'' in the range ''0..p-1'', there exist either exactly two or exactly zero valid Y coordinates. If no valid Y coordinate exists, then ''x'' is not a valid X coordinate either, i.e., no point ''P'' exists for which ''x(P) = x''. The valid Y coordinates for a given candidate ''x'' are the square roots of ''c = x<sup>3</sup> + 7 mod p'' and they can be computed as ''y = &plusmn;c<sup>(p+1)/4</sup> mod p'' (see [https://en.wikipedia.org/wiki/Quadratic_residue#Prime_or_prime_power_modulus Quadratic residue]) if they exist, which can be checked by squaring and comparing with ''c''.</ref> and ''has_even_y(P)'', or fails if no such point exists. The function ''lift_x(x)'' is equivalent to the following pseudocode:
+*** Let ''c = x<sup>3</sup> + 7 mod p''.
+*** Let ''y = c<sup>(p+1)/4</sup> mod p''.
+*** Fail if ''c &ne; y<sup>2</sup> mod p''.
+*** Return the unique point ''P'' such that ''x(P) = x'' and ''y(P) = y'' if ''y mod 2 = 0'' or ''y(P) = p-y'' otherwise.
+** The function ''hash<sub>tag</sub>(x)'' where ''tag'' is a UTF-8 encoded tag name and ''x'' is a byte array returns the 32-byte hash ''SHA256(SHA256(tag) || SHA256(tag) || x)''.
+
+
+==== Key Sorting ====
+
+Input:
+* The number ''u'' of signatures with ''0 < u < 2^32''
+* The public keys ''pk<sub>1..u</sub>'': ''u'' 32-byte arrays
+
+The algorithm ''KeySort(pk<sub>1..u</sub>)'' is defined as:
+* Return ''sort(pk)''
+** where ''pk<sub>i</sub> < pk<sub>j</sub>'' if for the smallest ''k'' such that ''pk<sub>i</sub>[k] &ne pk<sub>j</sub>[k]'' it holds that ''pk<sub>i</sub>[k] < pk<sub>j</sub>[k]''.
+
+==== Key Aggregation ====
+
+Input:
+* The number ''u'' of signatures with ''0 < u < 2^32''
+* The public keys ''pk<sub>1..u</sub>'': ''u'' 32-byte arrays
+
+The algorithm ''KeyAgg(pk<sub>1..u</sub>)'' is defined as:
+* For ''i = 1 .. u'':
+** Let ''a<sub>i</sub> = ComputeCoefficient(pk<sub>1..u</sub>, i)''.
+** Let ''P<sub>i</sub> = lift_x(int(pk<sub>i</sub>))''; fail if it fails.
+* Let ''S = a<sub>1</sub>⋅P<sub>1</sub> + a<sub>2</sub>⋅P<sub>1</sub> + ... + a<sub>u</sub>⋅P<sub>u</sub>''
+* Fail if ''is_infinite(S)''.
+* Return ''bytes(S)''.
+
+The algorithm ''HashKeys(pk<sub>1..u</sub>)'' is defined as:
+* Return ''hash(pk<sub>1</sub> || pk<sub>2</sub> || ... || pk<sub>u</sub>)''
+
+The algorithm ''IsSecond(pk<sub>1..u</sub>, i)'' is defined as:
+* For ''j = 1 .. u'':
+** If ''pk<sub>j</sub> &ne; pk<sub>1</sub>'':
+*** Return ''true'' if ''pk<sub>j</sub> = pk<sub>i</sub>'', otherwise return ''false''.
+* Return ''false''
+
+The algorithm ''ComputeCoefficient(pk<sub>1..u</sub>, i)'' is defined as:
+* Let ''L = HashKeys(pk<sub>1..u</sub>)''.
+* Return ''int(hash<sub>MuSig coefficient</sub>(L || pk) mod n'' if ''IsSecond(pk<sub>1..u</sub>, i)'' &ne; 2, otherwise return 1.
+
+== Applications ==
+
+== Test Vectors and Reference Code ==
+
+== Footnotes ==
+
+<references />
+
+== Acknowledgements ==