From 23e95568501ba9b9666c78821de742c27956dcc2 Mon Sep 17 00:00:00 2001 From: JamesC Date: Fri, 2 Aug 2019 11:12:50 -0400 Subject: [PATCH 1/5] Added 1.2 Musig Introduction. --- 1.2-Introduction-to-Musig.ipynb | 186 ++++++++ ...tion-to-Schnorr-Solutions-checkpoint.ipynb | 423 ++++++++++++++++++ .../1.2-Introduction-to-Musig-Solutions.ipynb | 212 +++++++++ images/musig_intro_0.jpg | Bin 0 -> 113197 bytes images/musig_intro_1.jpg | Bin 0 -> 75061 bytes images/musig_intro_2.jpg | Bin 0 -> 75305 bytes images/musig_intro_3.jpg | Bin 0 -> 57775 bytes 7 files changed, 821 insertions(+) create mode 100644 1.2-Introduction-to-Musig.ipynb create mode 100644 Solutions/.ipynb_checkpoints/1.1-Introduction-to-Schnorr-Solutions-checkpoint.ipynb create mode 100644 Solutions/1.2-Introduction-to-Musig-Solutions.ipynb create mode 100644 images/musig_intro_0.jpg create mode 100644 images/musig_intro_1.jpg create mode 100644 images/musig_intro_2.jpg create mode 100644 images/musig_intro_3.jpg diff --git a/1.2-Introduction-to-Musig.ipynb b/1.2-Introduction-to-Musig.ipynb new file mode 100644 index 000000000..a87c2f81b --- /dev/null +++ b/1.2-Introduction-to-Musig.ipynb @@ -0,0 +1,186 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "import util\n", + "\n", + "from test_framework.test_framework import BitcoinTestFramework\n", + "from test_framework.key import ECKey, ECPubKey, generate_musig_key, generate_schnorr_nonce\n", + "from test_framework.key import aggregate_schnorr_nonces, aggregate_musig_signatures\n", + "import binascii\n", + "import hashlib" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 1.2 Introduction to n-of-n Musig\n", + "\n", + "* Part 1: Public Key Generation.\n", + "* Part 2: Nonce Generation.\n", + "* Part 3: Signature Generation.\n", + "\n", + "In this chapter, we introduce the interactive musig protocol which allows n-of-n participants to jointly create and spend taproot or tapscript output using aggregated Schnorr signatures. \n", + "\n", + "**The advantage of Musig is both output and joint n-of-n spend are indistinguishable from other pay-to-pubkey transactions on-chain.**\n", + "\n", + "Musig is is an interactive protocol which requires multiple rounds of communication between the wallets. We will proceed with an example execution of the Musig protocol below.\n", + "\n", + "![test](images/musig_intro_0.jpg)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Part 1: Public Key Generation\n", + "\n", + "* Due to linear nature of Schnorr, public keys, nonces and signatures can be aggregated.\n", + "* Due to the [Key cancellation](https://tlu.tarilabs.com/cryptography/digital_signatures/introduction_schnorr_signatures.html#key-cancellation-attack) attack, signers must first commit to individual public keys (Challenge factor).\n", + "* Individual key pairs must be multiplied with respective challenge.\n", + "\n", + "\n", + "![test](images/musig_intro_1.jpg)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### 1.1 Programming Exercise: Compute 3-of-3 musig public key.\n", + "* Use `generate_musig_key([ECPubKey0, ECPubKey1, ...])`.\n", + "* returns `c_map, pk_musig`\n", + "* `c_map` contains `ECPubKey_i, challenge_data_i` key - value pairs.\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "# Compute key pairs.\n", + "\n", + "\n", + "# Compute key Challenges.\n", + "\n", + "\n", + "# Multiply key pairs by challenge factor.\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Part 2: Nonce Generation\n", + "\n", + "* Nonces can be aggregated like public keys and signatures.\n", + "* Security [Proof](https://eprint.iacr.org/2018/068.pdf) requires nonces are randomly generated independently of each other.\n", + "* An initial nonce commitment round is necessary to ensure no nonce is derived as a function of another.\n", + "\n", + "\n", + "![test](images/musig_intro_2.jpg)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### 2.1 Programming Exercise: Compute 3-of-3 musig nonce.\n", + "\n", + "* `generate_schnorr_nonce()` returns a ECkey with with a jacobi symbol of 1.\n", + "* `aggregate_schnorr_nonces()` returns a ECPubKey and a boolean (indicates nonce negation).\n", + "* *Negation of nonce `k_i` isn't necessary, since it is performed by `sign_musig` in next exercise.*" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "# Generate nonces.\n", + "\n", + "\n", + "# Skip nonce point commitment round (we assume nonce commitments are correct).\n", + "\n", + "\n", + "# Aggregate nonces.\n", + "\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Part 3: Signature Generation\n", + "\n", + "* Signatures are generated individually.\n", + "\n", + "![test](images/musig_intro_3.jpg)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### 3.1 Programming exercise: Compute aggregated musig signature.\n", + "\n", + "* `ECKey.sign_musig(nonce_point, negated, nonce_point_aggregated, pk_musig, msg)` returns individual signature.\n", + "* `aggregate_musig_signatures([sig0, sig1, ...])` returns aggregated signature.\n", + "* `ECPubKey.verify(sig, msg)` returns validity of signature.\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "msg = hashlib.sha256(b'transaction').digest()\n", + "\n", + "# Generate individual signatures.\n", + "\n", + "\n", + "# Aggregate signatures.\n", + "\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.6.5" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/Solutions/.ipynb_checkpoints/1.1-Introduction-to-Schnorr-Solutions-checkpoint.ipynb b/Solutions/.ipynb_checkpoints/1.1-Introduction-to-Schnorr-Solutions-checkpoint.ipynb new file mode 100644 index 000000000..8d593ecdb --- /dev/null +++ b/Solutions/.ipynb_checkpoints/1.1-Introduction-to-Schnorr-Solutions-checkpoint.ipynb @@ -0,0 +1,423 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "# Add Path to Bitcoin Core Testframework Library.\n", + "import sys\n", + "# sys.path.insert(0, '/.../bitcoin/test/functional')\n", + "\n", + "# Bitcoin Core Test Framework \n", + "from test_framework.key import SECP256K1_ORDER, SECP256K1_FIELD_SIZE\n", + "from test_framework.key import ECKey, ECPubKey, jacobi_symbol\n", + "from test_framework.key import *\n", + "\n", + "# Python Imports\n", + "import random\n", + "import hashlib\n", + "import pdb" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 1.1 Introduction to Schnorr\n", + "\n", + "* Part 1: Elliptic Curve Math Refresher.\n", + "* Part 2: Schnorr Signatures.\n", + "* Part 3: Deterministic Nonces." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Part 1: Elliptic Curve Math\n", + "\n", + "Elliptic Curve math involves scalars and Points(X,Y). The following provides an overview of all operations of scalars and points over elliptic curves. In Bitcoin, key pair generation and signing is performed over the SECP256k1 curve.\n", + "\n", + "![test](../images/schnorr0.jpg)\n", + "\n", + "### Classes / Methods for Elliptic Curve Math\n", + "\n", + "`int`: All Scalar operations over SECP251k1 can be performed with python integers and the modulo `%` operator:\n", + "* Addition: `a + b % SECP256K1`\n", + "* Subtraction: `-a = SECP256K1 - a`\n", + "* Multiplication: `a * b % % SECP256K1`\n", + "* Division (Fermat's little Theorem): `1/b = b**(SECP256K1-2) % SECP256K1`\n", + "\n", + "`ECKey`: The Bitcoin Core library provides a private key class which can also perform certain scalar operations.\n", + "* Addition: `add`\n", + "* Subtraction: `add` (adding complement of scalar)\n", + "* Multiplication: `mul`\n", + "* Division: `mul` (multiplying with scalar inverse, see Fermat's little theorem) \n", + "\n", + "![test](../images/schnorr1.jpg)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### 1.1 Example: Scalar Addition over SECP256K1 order\n", + "\n", + "Addition can be performed with integer modulo math in python or with the private key class `ECKey`. We can set a `ECKey` object to a certain value, or generate a new private key with the `generate` method.\n", + "\n", + "In the example below, addition is performed with both integers and the `ECKey` class, and evaluated for equality." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "# int() operations\n", + "a = random.randrange(1, SECP256K1_ORDER)\n", + "b = random.randrange(1, SECP256K1_ORDER)\n", + "ab = (a + b) % SECP256K1_ORDER\n", + "\n", + "# ECkey(operation)\n", + "a_key = ECKey()\n", + "a_key.set(a.to_bytes(32,'big'), True) \n", + "ab_key = a_key.add(b.to_bytes(32,'big'))\n", + "\n", + "# Ensure equivalency.\n", + "print(int().from_bytes(ab_key.get_bytes(),'big') == ab)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### 1.2 Example: Scalar Multiplication over SECP256K1 order\n", + "\n", + "In the example below, multiplication is performed with both integers and the `ECKey` class, and evaluated for equality." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "# int() operations\n", + "a = random.randrange(1, SECP256K1_ORDER)\n", + "b = random.randrange(1, SECP256K1_ORDER)\n", + "ab = (a * b) % SECP256K1_ORDER\n", + "\n", + "# ECkey(operation)\n", + "a_key = ECKey()\n", + "a_key.set(a.to_bytes(32,'big'), True) \n", + "ab_key = a_key.mul(b.to_bytes(32,'big'))\n", + "\n", + "# Ensure operations are equivalent.\n", + "print(int().from_bytes(ab_key.get_bytes(),'big') == ab)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### 1.3 _Programming Exercise:_ Distributivity of scalar operations\n", + "\n", + "In this exercise we wish to demonstrate the distributivity property of scalar addition and multiplication, whilst getting familiarized with both integer modulo operations and the private key `ECKey` methods.\n", + "\n", + "Consider: `(a - b) * c = a * c - b * c` over SECP256k1\n", + "* Demonstrate that the equation holds\n", + "* Compute the left side with the Python `int` class.\n", + "* Compute the right side with the Bitcoin Core `ECKey` class.\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "a = random.randrange(1, SECP256K1_ORDER)\n", + "b = random.randrange(1, SECP256K1_ORDER)\n", + "c = random.randrange(1, SECP256K1_ORDER)\n", + "\n", + "# Left: Compute (a - b) * c\n", + "ab = (a + (SECP256K1_ORDER - b)) % SECP256K1_ORDER\n", + "ab_c = (ab * c) % SECP256K1_ORDER\n", + "\n", + "a_key = ECKey()\n", + "b_key = ECKey()\n", + "c_key = ECKey()\n", + "\n", + "# Right: Compute -b * c\n", + "a_key.set(a.to_bytes(32,'big'), True)\n", + "bneg = SECP256K1_ORDER - b\n", + "b_key.set(bneg.to_bytes(32,'big'), True)\n", + "c_key.set(c.to_bytes(32,'big'), True)\n", + "\n", + "# Right: Compute a * c\n", + "ac_key = a_key.mul(c_key.get_bytes())\n", + "bc_key = b_key.mul(c_key.get_bytes())\n", + "\n", + "# Right: Compute a * c - b * c\n", + "ac_bc_key = ac_key.add(bc_key.get_bytes())\n", + "\n", + "# Left/Right: Assert equality\n", + "print(ab_c.to_bytes(32,'big') == ac_bc_key.get_bytes())\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### 1.4 Example: Point Addition over SECP256k1\n", + "\n", + "The public key `ECPubkey` class can be derived from `ECKey` with the `ECKey.get_pubkey` method. \n", + "\n", + "In the following example, we perform point addition with `ECPubkey.add`.\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "a_key = ECKey()\n", + "b_key = ECKey()\n", + "a_key.generate()\n", + "b_key.generate()\n", + "\n", + "A_key = a_key.get_pubkey()\n", + "B_key = b_key.get_pubkey()\n", + "AB_key = A_key.add(B_key)\n", + "\n", + "print(AB_key.get_bytes().hex())" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### 1.5 _Programming Exercise:_ Distributivity over scalar and ec points.\n", + "\n", + "In this exercise we wish to demonstrate the distributivity property of scalar/point operations, whilst getting familiarized with both integer modulo operations and the public key `ECPubKey` methods.\n", + "\n", + "Consider: `(a - b) * P = a * G - b * G`\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "a = random.randrange(1, SECP256K1_ORDER)\n", + "b = random.randrange(1, SECP256K1_ORDER)\n", + "\n", + "# Left: Compute a - b\n", + "a_neg_b = (a + (SECP256K1_ORDER - b)) % SECP256K1_ORDER\n", + "\n", + "# Left: Compute (a - b) * G\n", + "a_neg_b_key = ECKey()\n", + "a_neg_b_key.set(a_neg_b.to_bytes(32,'big'), True) \n", + "Pk_left = a_neg_b_key.get_pubkey()\n", + "\n", + "# Right: Compute aG\n", + "a_key = ECKey()\n", + "a_key.set(a.to_bytes(32,'big'), True) \n", + "Pk_a = a_key.get_pubkey()\n", + "\n", + "# Right: Compute -bG\n", + "# Use int() for -b\n", + "b_neg = SECP256K1_ORDER - b\n", + "b_neg_key = ECKey()\n", + "b_neg_key.set(b_neg.to_bytes(32,'big'), True) \n", + "Pk_neg_b = b_neg_key.get_pubkey()\n", + "\n", + "# Right: Compute aG - bG\n", + "Pk_right = Pk_a.add(Pk_neg_b)\n", + "\n", + "# Left/Right: Assert equality\n", + "print(Pk_left.get_bytes() == Pk_right.get_bytes())\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Part 2: Schnorr Signatures\n", + "\n", + "The Schnorr signature equation is the following:\n", + "\n", + "* `S = R + H(x(R)|P|msg) * P`\n", + "\n", + "Signing involves generating a secret nonce first.\n", + "\n", + "* Generate secret scalar `k`\n", + "\n", + "Then computing s from:\n", + "\n", + "* `s = k - H(x(R)|P|m)`\n", + "\n", + "The resulting signature is:\n", + "\n", + "* `x(R), s`\n", + "\n", + "\n", + "\n", + "![test](../images/schnorr2.jpg)\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Constraint on the private nonce k.\n", + "\n", + "The Schnorr signature provides the verifier with the x-coordinate of the Nonce point R and s value. For a given x-coordinate on the SECP251K1 curve, there are two possible curve points:\n", + "\n", + "* `y^2 = x^3 + 7` (Two y-coordinate values for a given x-coordinate)\n", + " * For x, both P(x,y) and P(x,-y) are valid curve points.\n", + " * One of the y-coordinates is even, and the other one is odd.\n", + "\n", + "The nonce k is therefore constrained by requiring it to be a quadratic residue of p (SECP256k1 curve field size). \n", + "\n", + "* `k` and its complement `SECP26k1_order - k` will have nonce points `R(x, y)` and `R(x, -y)`.\n", + " * Only one will be a quadratic residue of the curve order.\n", + " \n", + "Whether a scalar is a quadratic residue of the secp256k1 field size is determined with the jacobi symbol:\n", + "\n", + "* Bitcoin Core method: `jacobi_symbol(k, SECP256K1_FIELD_SIZE) == 1`\n", + "* If the jacobisymbol is not 1, simply take the inverse of k: `k = SECP26k1_order - k`\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### 2.1 _Programming Exercise:_ Sign a message with Schnorr\n", + "\n", + "* Sign the message with the provided key pair below." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "msg = hashlib.sha256(b'message').digest()\n", + "x = ECKey()\n", + "x.generate()\n", + "P = x.get_pubkey()\n", + "\n", + "# We have to set uncompressed to get the y-coordinate.\n", + "k_int = random.randrange(1, SECP256K1_ORDER)\n", + "k = ECKey()\n", + "k.set(k_int.to_bytes(32,'big'), False) # Uncompressed.\n", + "R = k.get_pubkey()\n", + "\n", + "# Check that nonce is quadratic residue of field size.\n", + "# Method: jacobi_symbol(nonce_int, SECP256K1_FIELD_SIZE)\n", + "if jacobi_symbol(int.from_bytes(R.get_bytes()[33:], \"big\"), SECP256K1_FIELD_SIZE) != 1:\n", + " k_int = SECP256K1_ORDER - k_int\n", + "\n", + "# Generate s = k + H(R_x|P|msg) * x\n", + "H_b = hashlib.sha256(R.get_bytes()[1:33] + P.get_bytes() + msg).digest()\n", + "H_int = int.from_bytes(H_b, 'big')\n", + "x_int = int.from_bytes(x.get_bytes(), 'big')\n", + "s = (k_int + (H_int * x_int) % SECP256K1_ORDER ) % SECP256K1_ORDER\n", + "\n", + "# Generate sig = R_x|s\n", + "sig = R.get_bytes()[1:33] + s.to_bytes(32,'big')\n", + "\n", + "print(P.verify_schnorr(sig, msg))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Part 3: Deterministic Nonces for Schnorr signatures\n", + "\n", + "So far we have used a random secret nonce for creating Schnorr signatures. This has the disadvantage that the the user must rely on the robustness of the random generator for each signing rounds. If the nonce generator is biased or even compromised, the private key can be derived for a given signature and known nonce.\n", + "\n", + "Therefore, for the single signer Schnorr signatures, the [Schnorr BIP](https://github.com/sipa/bips/blob/bip-schnorr/bip-schnorr.mediawiki) proposes a deterministic nonce generation scheme.\n", + "\n", + "* `k = sha256(x|msg)`" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### 3.1 _Programming Exercise:_ Signing Schnorr with deterministic nonce.\n", + "\n", + "* Create a Schnorr signature with the deterministic nonce scheme proposed in the Schnorr BIP.\n", + "* Compare this signature to the private key class method `ECKey.sign_schnorr(msg)`" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "msg = hashlib.sha256(b'message').digest()\n", + "x = ECKey()\n", + "x.generate()\n", + "P = x.get_pubkey()\n", + "\n", + "# We have to set uncompressed to get the y-coordinate.\n", + "k_b = hashlib.sha256(x.get_bytes() + msg).digest()\n", + "k_int = int.from_bytes(k_b, \"big\")\n", + "k = ECKey()\n", + "k.set(k_b, False) \n", + "R = k.get_pubkey()\n", + "\n", + "# Check that nonce is quadratic residue of field size.\n", + "if jacobi_symbol(int.from_bytes(R.get_bytes()[33:], \"big\"), SECP256K1_FIELD_SIZE) != 1:\n", + " k_int = SECP256K1_ORDER - k_int\n", + "\n", + "# Generate s = k + H(R_x|P|msg) * x\n", + "H_b = hashlib.sha256(R.get_bytes()[1:33] + P.get_bytes() + msg).digest()\n", + "H_int = int.from_bytes(H_b, 'big')\n", + "x_int = int.from_bytes(x.get_bytes(), 'big')\n", + "s = (k_int + (H_int * x_int) % SECP256K1_ORDER ) % SECP256K1_ORDER\n", + "\n", + "# Generate sig = R_x|s\n", + "sig = R.get_bytes()[1:33] + s.to_bytes(32,'big')\n", + "sig2 = x.sign_schnorr(msg)\n", + "\n", + "print(P.verify_schnorr(sig, msg))\n", + "print(P.verify_schnorr(sig2, msg))\n", + "print(sig == sig2)" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.6.5" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/Solutions/1.2-Introduction-to-Musig-Solutions.ipynb b/Solutions/1.2-Introduction-to-Musig-Solutions.ipynb new file mode 100644 index 000000000..84168945a --- /dev/null +++ b/Solutions/1.2-Introduction-to-Musig-Solutions.ipynb @@ -0,0 +1,212 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "import sys\n", + "sys.path.insert(0, '../')\n", + "import util\n", + "\n", + "from test_framework.test_framework import BitcoinTestFramework\n", + "from test_framework.key import ECKey, ECPubKey, generate_musig_key, generate_schnorr_nonce\n", + "from test_framework.key import aggregate_schnorr_nonces, aggregate_musig_signatures\n", + "import binascii\n", + "import hashlib" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 1.2 Introduction to n-of-n Musig\n", + "\n", + "* Part 1: Public Key Generation.\n", + "* Part 2: Nonce Generation.\n", + "* Part 3: Signature Generation.\n", + "\n", + "In this chapter, we introduce the interactive musig protocol which allows n-of-n participants to jointly create and spend taproot or tapscript output using aggregated Schnorr signatures. \n", + "\n", + "**The advantage of Musig is both output and joint n-of-n spend are indistinguishable from other pay-to-pubkey transactions on-chain.**\n", + "\n", + "Musig is is an interactive protocol which requires multiple rounds of communication between the wallets. We will proceed with an example execution of the Musig protocol below.\n", + "\n", + "![test](../images/musig_intro_0.jpg)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Part 1: Public Key Generation\n", + "\n", + "* Due to linear nature of Schnorr, public keys, nonces and signatures can be aggregated.\n", + "* Due to the [Key cancellation](https://tlu.tarilabs.com/cryptography/digital_signatures/introduction_schnorr_signatures.html#key-cancellation-attack) attack, signers must first commit to individual public keys (Challenge factor).\n", + "* Individual key pairs must be multiplied with respective challenge.\n", + "\n", + "\n", + "![test](../images/musig_intro_1.jpg)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### 1.1 Programming Exercise: Compute 3-of-3 musig public key.\n", + "* Use `generate_musig_key([ECPubKey0, ECPubKey1, ...])`.\n", + "* returns `c_map, pk_musig`\n", + "* `c_map` contains `ECPubKey_i, challenge_data_i` key - value pairs.\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "# Compute key pairs.\n", + "sec0 = ECKey()\n", + "sec1 = ECKey()\n", + "sec2 = ECKey()\n", + "sec0.generate()\n", + "sec1.generate()\n", + "sec2.generate()\n", + "pk0 = sec0.get_pubkey()\n", + "pk1 = sec1.get_pubkey()\n", + "pk2 = sec2.get_pubkey()\n", + "pk_v = [pk0, pk1, pk2]\n", + "\n", + "# Compute Key Challenges.\n", + "c_map, pk_musig = generate_musig_key(pk_v)\n", + "for pk, c in c_map.items():\n", + " print(\"Compressed Pubkey: \", pk.get_bytes().hex())\n", + " print(\"Challenge: \", c.hex(), '\\n')\n", + " \n", + "# Multiply key pairs by challenge factor.\n", + "sec0_c = sec0.mul(c_map[pk0])\n", + "sec1_c = sec1.mul(c_map[pk1])\n", + "sec2_c = sec2.mul(c_map[pk2])\n", + "pk0_c = pk0.mul(c_map[pk0])\n", + "pk1_c = pk1.mul(c_map[pk1])\n", + "pk2_c = pk2.mul(c_map[pk2])\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Part 2: Nonce Generation\n", + "\n", + "* Nonces can be aggregated like public keys and signatures.\n", + "* Security [Proof](https://eprint.iacr.org/2018/068.pdf) requires nonces are randomly generated independently of each other.\n", + "* An initial nonce commitment round is necessary to ensure no nonce is derived as a function of another.\n", + "\n", + "\n", + "![test](../images/musig_intro_2.jpg)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### 2.1 Programming Exercise: Compute 3-of-3 musig nonce.\n", + "\n", + "* `generate_schnorr_nonce()` returns a ECkey with with a jacobi symbol of 1.\n", + "* `aggregate_schnorr_nonces()` returns a ECPubKey and a boolean (indicates nonce negation).\n", + "* *Negation of nonce `k_i` isn't necessary, since it is performed by `sign_musig` in next exercise.*" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "# Generate nonces.\n", + "k0 = generate_schnorr_nonce()\n", + "k1 = generate_schnorr_nonce()\n", + "k2 = generate_schnorr_nonce()\n", + "R0 = k0.get_pubkey()\n", + "R1 = k1.get_pubkey()\n", + "R2 = k2.get_pubkey()\n", + "\n", + "# Skip nonce point commitment round (we assume nonce commitments are correct).\n", + "\n", + "# Aggregate nonces.\n", + "R_agg, negated = aggregate_schnorr_nonces([R0, R1, R2])\n", + "print(\"R aggregated: \", R_agg.get_bytes().hex())\n", + "print(\"R_agg was negated: \", negated)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Part 3: Signature Generation\n", + "\n", + "* Signatures are generated individually.\n", + "\n", + "![test](images/musig_intro_3.jpg)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### 3.1 Programming exercise: Compute aggregated musig signature.\n", + "\n", + "* `ECKey.sign_musig(nonce_point, negated, nonce_point_aggregated, pk_musig, msg)` returns individual signature.\n", + "* `aggregate_musig_signatures([sig0, sig1, ...])` returns aggregated signature.\n", + "* `ECPubKey.verify(sig, msg)` returns validity of signature.\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "msg = hashlib.sha256(b'transaction').digest()\n", + "\n", + "# Generate individual signatures.\n", + "sig0 = sec0_c.sign_musig(k0, negated, R_agg, pk_musig, msg)\n", + "sig1 = sec1_c.sign_musig(k1, negated, R_agg, pk_musig, msg)\n", + "sig2 = sec2_c.sign_musig(k2, negated, R_agg, pk_musig, msg)\n", + "\n", + "# Aggregate signatures.\n", + "sig_agg = aggregate_musig_signatures([sig0, sig1, sig2])\n", + "print(pk_musig.verify_schnorr(sig_agg, msg))\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + 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zP2=^4@dwwl)^gP?_4*M0LHqc^Pwa8oN?MA7M;<-S>e1uOi4|1((RpERd*3ppWdhBu zGaYva0Rzm&H))Mmf4HpO`UbO>y)mI0?`CBBaisQLyu?*ELu2OS0*}ajb7CSH;(38q zevZ^&>D=}52xD5%R+Cr7f6f@rn&WIV_wxKL67ur@8=4VU%sa!dN)x=0~#ebbKM12!xhLexc%9f&Oa6;rxA9Ax%~%$-F~6FhG? zUbthpWT(@r5^m*JM`F2^mZ#S8XFW4po~Z8>9Cao-zvIxZLW}5(2{#-h@!k1XQ@23pYS1G{X*4(;af%noops@?sfrg6gx&0BL Date: Mon, 2 Sep 2019 18:37:42 +0300 Subject: [PATCH 2/5] Removed checkpoint from 1.1. --- ...tion-to-Schnorr-Solutions-checkpoint.ipynb | 423 ------------------ 1 file changed, 423 deletions(-) delete mode 100644 Solutions/.ipynb_checkpoints/1.1-Introduction-to-Schnorr-Solutions-checkpoint.ipynb diff --git a/Solutions/.ipynb_checkpoints/1.1-Introduction-to-Schnorr-Solutions-checkpoint.ipynb b/Solutions/.ipynb_checkpoints/1.1-Introduction-to-Schnorr-Solutions-checkpoint.ipynb deleted file mode 100644 index 8d593ecdb..000000000 --- a/Solutions/.ipynb_checkpoints/1.1-Introduction-to-Schnorr-Solutions-checkpoint.ipynb +++ /dev/null @@ -1,423 +0,0 @@ -{ - "cells": [ - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "# Add Path to Bitcoin Core Testframework Library.\n", - "import sys\n", - "# sys.path.insert(0, '/.../bitcoin/test/functional')\n", - "\n", - "# Bitcoin Core Test Framework \n", - "from test_framework.key import SECP256K1_ORDER, SECP256K1_FIELD_SIZE\n", - "from test_framework.key import ECKey, ECPubKey, jacobi_symbol\n", - "from test_framework.key import *\n", - "\n", - "# Python Imports\n", - "import random\n", - "import hashlib\n", - "import pdb" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# 1.1 Introduction to Schnorr\n", - "\n", - "* Part 1: Elliptic Curve Math Refresher.\n", - "* Part 2: Schnorr Signatures.\n", - "* Part 3: Deterministic Nonces." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Part 1: Elliptic Curve Math\n", - "\n", - "Elliptic Curve math involves scalars and Points(X,Y). The following provides an overview of all operations of scalars and points over elliptic curves. In Bitcoin, key pair generation and signing is performed over the SECP256k1 curve.\n", - "\n", - "![test](../images/schnorr0.jpg)\n", - "\n", - "### Classes / Methods for Elliptic Curve Math\n", - "\n", - "`int`: All Scalar operations over SECP251k1 can be performed with python integers and the modulo `%` operator:\n", - "* Addition: `a + b % SECP256K1`\n", - "* Subtraction: `-a = SECP256K1 - a`\n", - "* Multiplication: `a * b % % SECP256K1`\n", - "* Division (Fermat's little Theorem): `1/b = b**(SECP256K1-2) % SECP256K1`\n", - "\n", - "`ECKey`: The Bitcoin Core library provides a private key class which can also perform certain scalar operations.\n", - "* Addition: `add`\n", - "* Subtraction: `add` (adding complement of scalar)\n", - "* Multiplication: `mul`\n", - "* Division: `mul` (multiplying with scalar inverse, see Fermat's little theorem) \n", - "\n", - "![test](../images/schnorr1.jpg)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "#### 1.1 Example: Scalar Addition over SECP256K1 order\n", - "\n", - "Addition can be performed with integer modulo math in python or with the private key class `ECKey`. We can set a `ECKey` object to a certain value, or generate a new private key with the `generate` method.\n", - "\n", - "In the example below, addition is performed with both integers and the `ECKey` class, and evaluated for equality." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "# int() operations\n", - "a = random.randrange(1, SECP256K1_ORDER)\n", - "b = random.randrange(1, SECP256K1_ORDER)\n", - "ab = (a + b) % SECP256K1_ORDER\n", - "\n", - "# ECkey(operation)\n", - "a_key = ECKey()\n", - "a_key.set(a.to_bytes(32,'big'), True) \n", - "ab_key = a_key.add(b.to_bytes(32,'big'))\n", - "\n", - "# Ensure equivalency.\n", - "print(int().from_bytes(ab_key.get_bytes(),'big') == ab)\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "#### 1.2 Example: Scalar Multiplication over SECP256K1 order\n", - "\n", - "In the example below, multiplication is performed with both integers and the `ECKey` class, and evaluated for equality." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "# int() operations\n", - "a = random.randrange(1, SECP256K1_ORDER)\n", - "b = random.randrange(1, SECP256K1_ORDER)\n", - "ab = (a * b) % SECP256K1_ORDER\n", - "\n", - "# ECkey(operation)\n", - "a_key = ECKey()\n", - "a_key.set(a.to_bytes(32,'big'), True) \n", - "ab_key = a_key.mul(b.to_bytes(32,'big'))\n", - "\n", - "# Ensure operations are equivalent.\n", - "print(int().from_bytes(ab_key.get_bytes(),'big') == ab)\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "#### 1.3 _Programming Exercise:_ Distributivity of scalar operations\n", - "\n", - "In this exercise we wish to demonstrate the distributivity property of scalar addition and multiplication, whilst getting familiarized with both integer modulo operations and the private key `ECKey` methods.\n", - "\n", - "Consider: `(a - b) * c = a * c - b * c` over SECP256k1\n", - "* Demonstrate that the equation holds\n", - "* Compute the left side with the Python `int` class.\n", - "* Compute the right side with the Bitcoin Core `ECKey` class.\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "a = random.randrange(1, SECP256K1_ORDER)\n", - "b = random.randrange(1, SECP256K1_ORDER)\n", - "c = random.randrange(1, SECP256K1_ORDER)\n", - "\n", - "# Left: Compute (a - b) * c\n", - "ab = (a + (SECP256K1_ORDER - b)) % SECP256K1_ORDER\n", - "ab_c = (ab * c) % SECP256K1_ORDER\n", - "\n", - "a_key = ECKey()\n", - "b_key = ECKey()\n", - "c_key = ECKey()\n", - "\n", - "# Right: Compute -b * c\n", - "a_key.set(a.to_bytes(32,'big'), True)\n", - "bneg = SECP256K1_ORDER - b\n", - "b_key.set(bneg.to_bytes(32,'big'), True)\n", - "c_key.set(c.to_bytes(32,'big'), True)\n", - "\n", - "# Right: Compute a * c\n", - "ac_key = a_key.mul(c_key.get_bytes())\n", - "bc_key = b_key.mul(c_key.get_bytes())\n", - "\n", - "# Right: Compute a * c - b * c\n", - "ac_bc_key = ac_key.add(bc_key.get_bytes())\n", - "\n", - "# Left/Right: Assert equality\n", - "print(ab_c.to_bytes(32,'big') == ac_bc_key.get_bytes())\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "#### 1.4 Example: Point Addition over SECP256k1\n", - "\n", - "The public key `ECPubkey` class can be derived from `ECKey` with the `ECKey.get_pubkey` method. \n", - "\n", - "In the following example, we perform point addition with `ECPubkey.add`.\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "a_key = ECKey()\n", - "b_key = ECKey()\n", - "a_key.generate()\n", - "b_key.generate()\n", - "\n", - "A_key = a_key.get_pubkey()\n", - "B_key = b_key.get_pubkey()\n", - "AB_key = A_key.add(B_key)\n", - "\n", - "print(AB_key.get_bytes().hex())" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "#### 1.5 _Programming Exercise:_ Distributivity over scalar and ec points.\n", - "\n", - "In this exercise we wish to demonstrate the distributivity property of scalar/point operations, whilst getting familiarized with both integer modulo operations and the public key `ECPubKey` methods.\n", - "\n", - "Consider: `(a - b) * P = a * G - b * G`\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "a = random.randrange(1, SECP256K1_ORDER)\n", - "b = random.randrange(1, SECP256K1_ORDER)\n", - "\n", - "# Left: Compute a - b\n", - "a_neg_b = (a + (SECP256K1_ORDER - b)) % SECP256K1_ORDER\n", - "\n", - "# Left: Compute (a - b) * G\n", - "a_neg_b_key = ECKey()\n", - "a_neg_b_key.set(a_neg_b.to_bytes(32,'big'), True) \n", - "Pk_left = a_neg_b_key.get_pubkey()\n", - "\n", - "# Right: Compute aG\n", - "a_key = ECKey()\n", - "a_key.set(a.to_bytes(32,'big'), True) \n", - "Pk_a = a_key.get_pubkey()\n", - "\n", - "# Right: Compute -bG\n", - "# Use int() for -b\n", - "b_neg = SECP256K1_ORDER - b\n", - "b_neg_key = ECKey()\n", - "b_neg_key.set(b_neg.to_bytes(32,'big'), True) \n", - "Pk_neg_b = b_neg_key.get_pubkey()\n", - "\n", - "# Right: Compute aG - bG\n", - "Pk_right = Pk_a.add(Pk_neg_b)\n", - "\n", - "# Left/Right: Assert equality\n", - "print(Pk_left.get_bytes() == Pk_right.get_bytes())\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Part 2: Schnorr Signatures\n", - "\n", - "The Schnorr signature equation is the following:\n", - "\n", - "* `S = R + H(x(R)|P|msg) * P`\n", - "\n", - "Signing involves generating a secret nonce first.\n", - "\n", - "* Generate secret scalar `k`\n", - "\n", - "Then computing s from:\n", - "\n", - "* `s = k - H(x(R)|P|m)`\n", - "\n", - "The resulting signature is:\n", - "\n", - "* `x(R), s`\n", - "\n", - "\n", - "\n", - "![test](../images/schnorr2.jpg)\n", - "\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Constraint on the private nonce k.\n", - "\n", - "The Schnorr signature provides the verifier with the x-coordinate of the Nonce point R and s value. For a given x-coordinate on the SECP251K1 curve, there are two possible curve points:\n", - "\n", - "* `y^2 = x^3 + 7` (Two y-coordinate values for a given x-coordinate)\n", - " * For x, both P(x,y) and P(x,-y) are valid curve points.\n", - " * One of the y-coordinates is even, and the other one is odd.\n", - "\n", - "The nonce k is therefore constrained by requiring it to be a quadratic residue of p (SECP256k1 curve field size). \n", - "\n", - "* `k` and its complement `SECP26k1_order - k` will have nonce points `R(x, y)` and `R(x, -y)`.\n", - " * Only one will be a quadratic residue of the curve order.\n", - " \n", - "Whether a scalar is a quadratic residue of the secp256k1 field size is determined with the jacobi symbol:\n", - "\n", - "* Bitcoin Core method: `jacobi_symbol(k, SECP256K1_FIELD_SIZE) == 1`\n", - "* If the jacobisymbol is not 1, simply take the inverse of k: `k = SECP26k1_order - k`\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "#### 2.1 _Programming Exercise:_ Sign a message with Schnorr\n", - "\n", - "* Sign the message with the provided key pair below." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "msg = hashlib.sha256(b'message').digest()\n", - "x = ECKey()\n", - "x.generate()\n", - "P = x.get_pubkey()\n", - "\n", - "# We have to set uncompressed to get the y-coordinate.\n", - "k_int = random.randrange(1, SECP256K1_ORDER)\n", - "k = ECKey()\n", - "k.set(k_int.to_bytes(32,'big'), False) # Uncompressed.\n", - "R = k.get_pubkey()\n", - "\n", - "# Check that nonce is quadratic residue of field size.\n", - "# Method: jacobi_symbol(nonce_int, SECP256K1_FIELD_SIZE)\n", - "if jacobi_symbol(int.from_bytes(R.get_bytes()[33:], \"big\"), SECP256K1_FIELD_SIZE) != 1:\n", - " k_int = SECP256K1_ORDER - k_int\n", - "\n", - "# Generate s = k + H(R_x|P|msg) * x\n", - "H_b = hashlib.sha256(R.get_bytes()[1:33] + P.get_bytes() + msg).digest()\n", - "H_int = int.from_bytes(H_b, 'big')\n", - "x_int = int.from_bytes(x.get_bytes(), 'big')\n", - "s = (k_int + (H_int * x_int) % SECP256K1_ORDER ) % SECP256K1_ORDER\n", - "\n", - "# Generate sig = R_x|s\n", - "sig = R.get_bytes()[1:33] + s.to_bytes(32,'big')\n", - "\n", - "print(P.verify_schnorr(sig, msg))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Part 3: Deterministic Nonces for Schnorr signatures\n", - "\n", - "So far we have used a random secret nonce for creating Schnorr signatures. This has the disadvantage that the the user must rely on the robustness of the random generator for each signing rounds. If the nonce generator is biased or even compromised, the private key can be derived for a given signature and known nonce.\n", - "\n", - "Therefore, for the single signer Schnorr signatures, the [Schnorr BIP](https://github.com/sipa/bips/blob/bip-schnorr/bip-schnorr.mediawiki) proposes a deterministic nonce generation scheme.\n", - "\n", - "* `k = sha256(x|msg)`" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "#### 3.1 _Programming Exercise:_ Signing Schnorr with deterministic nonce.\n", - "\n", - "* Create a Schnorr signature with the deterministic nonce scheme proposed in the Schnorr BIP.\n", - "* Compare this signature to the private key class method `ECKey.sign_schnorr(msg)`" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "msg = hashlib.sha256(b'message').digest()\n", - "x = ECKey()\n", - "x.generate()\n", - "P = x.get_pubkey()\n", - "\n", - "# We have to set uncompressed to get the y-coordinate.\n", - "k_b = hashlib.sha256(x.get_bytes() + msg).digest()\n", - "k_int = int.from_bytes(k_b, \"big\")\n", - "k = ECKey()\n", - "k.set(k_b, False) \n", - "R = k.get_pubkey()\n", - "\n", - "# Check that nonce is quadratic residue of field size.\n", - "if jacobi_symbol(int.from_bytes(R.get_bytes()[33:], \"big\"), SECP256K1_FIELD_SIZE) != 1:\n", - " k_int = SECP256K1_ORDER - k_int\n", - "\n", - "# Generate s = k + H(R_x|P|msg) * x\n", - "H_b = hashlib.sha256(R.get_bytes()[1:33] + P.get_bytes() + msg).digest()\n", - "H_int = int.from_bytes(H_b, 'big')\n", - "x_int = int.from_bytes(x.get_bytes(), 'big')\n", - "s = (k_int + (H_int * x_int) % SECP256K1_ORDER ) % SECP256K1_ORDER\n", - "\n", - "# Generate sig = R_x|s\n", - "sig = R.get_bytes()[1:33] + s.to_bytes(32,'big')\n", - "sig2 = x.sign_schnorr(msg)\n", - "\n", - "print(P.verify_schnorr(sig, msg))\n", - "print(P.verify_schnorr(sig2, msg))\n", - "print(sig == sig2)" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.6.5" - } - }, - "nbformat": 4, - "nbformat_minor": 2 -} From c5cb9d1a8065a1eb562336267d59b8f7931d9791 Mon Sep 17 00:00:00 2001 From: JamesC Date: Sat, 7 Sep 2019 17:05:25 +0300 Subject: [PATCH 3/5] Incorporated review feedback. Added svg image source files. --- 1.2-Introduction-to-Musig.ipynb | 32 ++-- .../1.2-Introduction-to-Musig-Solutions.ipynb | 63 +++---- images/musig_intro_0.jpg | Bin 113197 -> 113357 bytes images/musig_intro_0.svg | 138 ++++++++++++++ images/musig_intro_1.svg | 102 ++++++++++ images/musig_intro_2.svg | 146 ++++++++++++++ images/musig_intro_3.jpg | Bin 57775 -> 88634 bytes images/musig_intro_3.svg | 178 ++++++++++++++++++ 8 files changed, 611 insertions(+), 48 deletions(-) create mode 100644 images/musig_intro_0.svg create mode 100644 images/musig_intro_1.svg create mode 100644 images/musig_intro_2.svg create mode 100644 images/musig_intro_3.svg diff --git a/1.2-Introduction-to-Musig.ipynb b/1.2-Introduction-to-Musig.ipynb index a87c2f81b..3b5d0e3a8 100644 --- a/1.2-Introduction-to-Musig.ipynb +++ b/1.2-Introduction-to-Musig.ipynb @@ -6,30 +6,28 @@ "metadata": {}, "outputs": [], "source": [ - "import util\n", + "import sys\n", + "import hashlib\n", "\n", - "from test_framework.test_framework import BitcoinTestFramework\n", - "from test_framework.key import ECKey, ECPubKey, generate_musig_key, generate_schnorr_nonce\n", - "from test_framework.key import aggregate_schnorr_nonces, aggregate_musig_signatures\n", - "import binascii\n", - "import hashlib" + "import util\n", + "from test_framework.key import ECKey, ECPubKey, generate_musig_key, generate_schnorr_nonce, aggregate_schnorr_nonces, aggregate_musig_signatures" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "# 1.2 Introduction to n-of-n Musig\n", + "# 1.2 Introduction to n-of-n MuSig\n", "\n", "* Part 1: Public Key Generation.\n", "* Part 2: Nonce Generation.\n", "* Part 3: Signature Generation.\n", "\n", - "In this chapter, we introduce the interactive musig protocol which allows n-of-n participants to jointly create and spend taproot or tapscript output using aggregated Schnorr signatures. \n", + "In this chapter, we introduce the interactive MuSig protocol which allows n-of-n participants to jointly create and spend taproot or tapscript output using aggregated Schnorr signatures. \n", "\n", - "**The advantage of Musig is both output and joint n-of-n spend are indistinguishable from other pay-to-pubkey transactions on-chain.**\n", + "**The advantage of MuSig is both output and joint n-of-n spend are indistinguishable from other pay-to-pubkey transactions on-chain.**\n", "\n", - "Musig is is an interactive protocol which requires multiple rounds of communication between the wallets. We will proceed with an example execution of the Musig protocol below.\n", + "MuSig is is an interactive protocol which requires multiple rounds of communication between the wallets. We will proceed with an example execution of the MuSig protocol below.\n", "\n", "![test](images/musig_intro_0.jpg)\n" ] @@ -45,14 +43,14 @@ "* Individual key pairs must be multiplied with respective challenge.\n", "\n", "\n", - "![test](images/musig_intro_1.jpg)" + "![test](../images/musig_intro_1.jpg)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "### 1.1 Programming Exercise: Compute 3-of-3 musig public key.\n", + "### 1.1 Programming Exercise: Compute 3-of-3 MuSig public key.\n", "* Use `generate_musig_key([ECPubKey0, ECPubKey1, ...])`.\n", "* returns `c_map, pk_musig`\n", "* `c_map` contains `ECPubKey_i, challenge_data_i` key - value pairs.\n" @@ -67,10 +65,11 @@ "# Compute key pairs.\n", "\n", "\n", - "# Compute key Challenges.\n", + "# Compute Key Challenges.\n", "\n", "\n", "# Multiply key pairs by challenge factor.\n", + "\n", "\n" ] }, @@ -92,7 +91,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "### 2.1 Programming Exercise: Compute 3-of-3 musig nonce.\n", + "### 2.1 Programming Exercise: Compute 3-of-3 MuSig nonce.\n", "\n", "* `generate_schnorr_nonce()` returns a ECkey with with a jacobi symbol of 1.\n", "* `aggregate_schnorr_nonces()` returns a ECPubKey and a boolean (indicates nonce negation).\n", @@ -122,7 +121,8 @@ "source": [ "## Part 3: Signature Generation\n", "\n", - "* Signatures are generated individually.\n", + "* Signatures are generated individually, but can be aggregated once they are exchanged amongst all participants in a single roundtrip.\n", + "* Notice that the hash expressions are identical in all signatures, which makes aggregation possible.\n", "\n", "![test](images/musig_intro_3.jpg)" ] @@ -131,7 +131,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "### 3.1 Programming exercise: Compute aggregated musig signature.\n", + "### 3.1 Programming exercise: Compute aggregated MuSig signature.\n", "\n", "* `ECKey.sign_musig(nonce_point, negated, nonce_point_aggregated, pk_musig, msg)` returns individual signature.\n", "* `aggregate_musig_signatures([sig0, sig1, ...])` returns aggregated signature.\n", diff --git a/Solutions/1.2-Introduction-to-Musig-Solutions.ipynb b/Solutions/1.2-Introduction-to-Musig-Solutions.ipynb index 84168945a..e53398c2a 100644 --- a/Solutions/1.2-Introduction-to-Musig-Solutions.ipynb +++ b/Solutions/1.2-Introduction-to-Musig-Solutions.ipynb @@ -7,31 +7,28 @@ "outputs": [], "source": [ "import sys\n", + "import hashlib\n", + "\n", "sys.path.insert(0, '../')\n", "import util\n", - "\n", - "from test_framework.test_framework import BitcoinTestFramework\n", - "from test_framework.key import ECKey, ECPubKey, generate_musig_key, generate_schnorr_nonce\n", - "from test_framework.key import aggregate_schnorr_nonces, aggregate_musig_signatures\n", - "import binascii\n", - "import hashlib" + "from test_framework.key import ECKey, ECPubKey, generate_musig_key, generate_schnorr_nonce, aggregate_schnorr_nonces, aggregate_musig_signatures" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "# 1.2 Introduction to n-of-n Musig\n", + "# 1.2 Introduction to n-of-n MuSig\n", "\n", "* Part 1: Public Key Generation.\n", "* Part 2: Nonce Generation.\n", "* Part 3: Signature Generation.\n", "\n", - "In this chapter, we introduce the interactive musig protocol which allows n-of-n participants to jointly create and spend taproot or tapscript output using aggregated Schnorr signatures. \n", + "In this chapter, we introduce the interactive MuSig protocol which allows n-of-n participants to jointly create and spend taproot or tapscript output using aggregated Schnorr signatures. \n", "\n", - "**The advantage of Musig is both output and joint n-of-n spend are indistinguishable from other pay-to-pubkey transactions on-chain.**\n", + "**The advantage of MuSig is both output and joint n-of-n spend are indistinguishable from other pay-to-pubkey transactions on-chain.**\n", "\n", - "Musig is is an interactive protocol which requires multiple rounds of communication between the wallets. We will proceed with an example execution of the Musig protocol below.\n", + "MuSig is is an interactive protocol which requires multiple rounds of communication between the wallets. We will proceed with an example execution of the MuSig protocol below.\n", "\n", "![test](../images/musig_intro_0.jpg)\n" ] @@ -54,7 +51,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "### 1.1 Programming Exercise: Compute 3-of-3 musig public key.\n", + "### 1.1 Programming Exercise: Compute 3-of-3 MuSig public key.\n", "* Use `generate_musig_key([ECPubKey0, ECPubKey1, ...])`.\n", "* returns `c_map, pk_musig`\n", "* `c_map` contains `ECPubKey_i, challenge_data_i` key - value pairs.\n" @@ -67,27 +64,28 @@ "outputs": [], "source": [ "# Compute key pairs.\n", - "sec0 = ECKey()\n", - "sec1 = ECKey()\n", - "sec2 = ECKey()\n", - "sec0.generate()\n", - "sec1.generate()\n", - "sec2.generate()\n", - "pk0 = sec0.get_pubkey()\n", - "pk1 = sec1.get_pubkey()\n", - "pk2 = sec2.get_pubkey()\n", + "privkey0 = ECKey()\n", + "privkey1 = ECKey()\n", + "privkey2 = ECKey()\n", + "privkey0.generate()\n", + "privkey1.generate()\n", + "privkey2.generate()\n", + "pk0 = privkey0.get_pubkey()\n", + "pk1 = privkey1.get_pubkey()\n", + "pk2 = privkey2.get_pubkey()\n", "pk_v = [pk0, pk1, pk2]\n", "\n", "# Compute Key Challenges.\n", "c_map, pk_musig = generate_musig_key(pk_v)\n", + "print(\"Aggregated Public Key: \", pk_musig.get_bytes().hex())\n", "for pk, c in c_map.items():\n", " print(\"Compressed Pubkey: \", pk.get_bytes().hex())\n", " print(\"Challenge: \", c.hex(), '\\n')\n", - " \n", + "\n", "# Multiply key pairs by challenge factor.\n", - "sec0_c = sec0.mul(c_map[pk0])\n", - "sec1_c = sec1.mul(c_map[pk1])\n", - "sec2_c = sec2.mul(c_map[pk2])\n", + "privkey0_c = privkey0.mul(c_map[pk0])\n", + "privkey1_c = privkey1.mul(c_map[pk1])\n", + "privkey2_c = privkey2.mul(c_map[pk2])\n", "pk0_c = pk0.mul(c_map[pk0])\n", "pk1_c = pk1.mul(c_map[pk1])\n", "pk2_c = pk2.mul(c_map[pk2])\n" @@ -111,7 +109,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "### 2.1 Programming Exercise: Compute 3-of-3 musig nonce.\n", + "### 2.1 Programming Exercise: Compute 3-of-3 MuSig nonce.\n", "\n", "* `generate_schnorr_nonce()` returns a ECkey with with a jacobi symbol of 1.\n", "* `aggregate_schnorr_nonces()` returns a ECPubKey and a boolean (indicates nonce negation).\n", @@ -146,16 +144,17 @@ "source": [ "## Part 3: Signature Generation\n", "\n", - "* Signatures are generated individually.\n", + "* Signatures are generated individually, but can be aggregated once they are exchanged amongst all participants in a single roundtrip.\n", + "* Notice that the hash expressions are identical in all signatures, which makes aggregation possible.\n", "\n", - "![test](images/musig_intro_3.jpg)" + "![test](../images/musig_intro_4.jpg)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "### 3.1 Programming exercise: Compute aggregated musig signature.\n", + "### 3.1 Programming exercise: Compute aggregated MuSig signature.\n", "\n", "* `ECKey.sign_musig(nonce_point, negated, nonce_point_aggregated, pk_musig, msg)` returns individual signature.\n", "* `aggregate_musig_signatures([sig0, sig1, ...])` returns aggregated signature.\n", @@ -171,13 +170,13 @@ "msg = hashlib.sha256(b'transaction').digest()\n", "\n", "# Generate individual signatures.\n", - "sig0 = sec0_c.sign_musig(k0, negated, R_agg, pk_musig, msg)\n", - "sig1 = sec1_c.sign_musig(k1, negated, R_agg, pk_musig, msg)\n", - "sig2 = sec2_c.sign_musig(k2, negated, R_agg, pk_musig, msg)\n", + "sig0 = privkey0_c.sign_musig(k0, negated, R_agg, pk_musig, msg)\n", + "sig1 = privkey1_c.sign_musig(k1, negated, R_agg, pk_musig, msg)\n", + "sig2 = privkey2_c.sign_musig(k2, negated, R_agg, pk_musig, msg)\n", "\n", "# Aggregate signatures.\n", "sig_agg = aggregate_musig_signatures([sig0, sig1, sig2])\n", - "print(pk_musig.verify_schnorr(sig_agg, msg))\n" + "print(\"Signature verifies against MuSig pubkey: \", pk_musig.verify_schnorr(sig_agg, msg))\n" ] }, { diff --git a/images/musig_intro_0.jpg b/images/musig_intro_0.jpg index 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-0,0 +1,138 @@ + + + + + + + + + + + + + + + + + + + + + + MuSig OverviewInteractive Key Generation & Signing + + + + PubkeyGeneration + + + + Individual + + + + Individual + + + + Generate Noncesk0, k1, ...Derive Nonce PointsR0, R1, ... + + + + GenerateP0, P1, ... + + + + P0, P1, ... + + + + Compute key challengesc0, c1, ...Compute musig PubkeyP_agg + + + + H(R0), H(R1), ... + + + + R0, R1, ... + + + + Aggregate Nonce RR_agg, is_negated + + + + Generate Signaturessig0, sig1, ... + + + + sig0, sig1, ... + + + + Coordinator Aggregates Signaturessig_agg + + + + Roundtrip + + + + MuSig Signing + + + + + + + + + + + + + + + + + + + + + + + + 1st Round + + + + 1st + + + + 2nd + + + + 3rd + + + + + + + + + + diff --git a/images/musig_intro_1.svg b/images/musig_intro_1.svg new file mode 100644 index 000000000..d6fcfb6c8 --- /dev/null +++ b/images/musig_intro_1.svg @@ -0,0 +1,102 @@ + + + + + + + + + + + + + + + + + + + Pubkey Generation + + + + x_i, P_i + + + + P_i + + + + Individual + + + + Individual + + + + Roundtrip + + + + Keypair Generation + + + + Public Key Exchange + + + + C_all = H(P_0, P_1, P_2, ..., P_n)C_i = H(C_all | P_i)P_i’ = C_i * P_i, x_i’ = C_i * x_nP_agg = P_0’ + P_1’ + P_2’ + ... + P_n’ + + + + Challenge Factor + + + + Tweaked Keys + + + + Musig Public Key + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/images/musig_intro_2.svg b/images/musig_intro_2.svg new file mode 100644 index 000000000..62287896d --- /dev/null +++ b/images/musig_intro_2.svg @@ -0,0 +1,146 @@ + + + + + + + + + + + + + + + + + + + + + + Nonce Generation + + + + k_i, R_i + + + + H(R_i) + + + + + + + H(R_i) + + + + + + + R_i + + + + Individual + + + + 1st Round + + + + 2nd Round + + + + Individual + + + + Roundtrip + + + + If Nonce points hash correctly:R_agg = R_0 + R_1 + ... + R_nIf jacobi_symbol(R_agg) != 1R_agg = Negate(R_agg)k_i = Negate(k_i) + + + + Store: + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 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zevZ^&>D=}52xD5%R+Cr7f6f@rn&WIV_wxKL67ur@8=4VU%sa!dN)x=0~#ebbKM12!xhLexc%9f&Oa6;rxA9Ax%~%$-F~6FhG? zUbthpWT(@r5^m*JM`F2^mZ#S8XFW4po~Z8>9Cao-zvIxZLW}5(2{#-h@!k1XQ@23pYS1G{X*4(;af%noops@?sfrg6gx&0BL + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + s0 = k0 + H(x(R_agg)|P_agg|m) * x0 + + + + s1 = k1 + H(x(R_agg)|P_agg|m) * x1 + + + + s2 = k2 + H(x(R_agg)|P_agg|m) * x2 + + + + s0 + s1 + s2 = k0 + k1 + k2 + H(x(R_agg)|P_agg|m) * (x0 + x1 + x2) + + + + s_agg + + + + k_agg + + + + = + + + + + + + + + * + + + + + + x_agg + + + + H(x(R_agg)|P_agg|m) + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +Individual Signatures + +Aggregated Signature + + + Linear Aggregation of Schnorr Signatures + + From b0188b4a1610bce515e5b4fe1008432fb26978f6 Mon Sep 17 00:00:00 2001 From: JamesC Date: Sun, 15 Sep 2019 22:19:57 +0200 Subject: [PATCH 4/5] Updated with new musig module. Incorporated feedback and separate musig module methods. --- 1.2-Introduction-to-Musig.ipynb | 53 ++-- .../1.2-Introduction-to-Musig-Solutions.ipynb | 60 +++-- images/musig_intro_0.jpg | Bin 113357 -> 113829 bytes images/musig_intro_0.svg | 187 +++++++++----- images/musig_intro_1.jpg | Bin 75061 -> 72505 bytes images/musig_intro_1.svg | 86 +++++-- images/musig_intro_2.jpg | Bin 75305 -> 76726 bytes images/musig_intro_2.svg | 128 ++++++---- images/musig_intro_3.jpg | Bin 88634 -> 80604 bytes images/musig_intro_3.svg | 236 ++++++++++++------ 10 files changed, 503 insertions(+), 247 deletions(-) diff --git a/1.2-Introduction-to-Musig.ipynb b/1.2-Introduction-to-Musig.ipynb index 3b5d0e3a8..73266bd43 100644 --- a/1.2-Introduction-to-Musig.ipynb +++ b/1.2-Introduction-to-Musig.ipynb @@ -6,11 +6,10 @@ "metadata": {}, "outputs": [], "source": [ - "import sys\n", "import hashlib\n", - "\n", "import util\n", - "from test_framework.key import ECKey, ECPubKey, generate_musig_key, generate_schnorr_nonce, aggregate_schnorr_nonces, aggregate_musig_signatures" + "from test_framework.key import ECKey, ECPubKey, generate_schnorr_nonce\n", + "from test_framework.musig import sign_musig, generate_musig_key, aggregate_schnorr_nonces, aggregate_musig_signatures\n" ] }, { @@ -20,12 +19,15 @@ "# 1.2 Introduction to n-of-n MuSig\n", "\n", "* Part 1: Public Key Generation.\n", - "* Part 2: Nonce Generation.\n", - "* Part 3: Signature Generation.\n", + "* Part 2: Signing.\n", + " * Nonce Aggregation.\n", + " * Signature Aggregation.\n", "\n", "In this chapter, we introduce the interactive MuSig protocol which allows n-of-n participants to jointly create and spend taproot or tapscript output using aggregated Schnorr signatures. \n", "\n", - "**The advantage of MuSig is both output and joint n-of-n spend are indistinguishable from other pay-to-pubkey transactions on-chain.**\n", + "**Advantages of MuSig include:**\n", + "* Privacy\n", + "* Transaction size/fees\n", "\n", "MuSig is is an interactive protocol which requires multiple rounds of communication between the wallets. We will proceed with an example execution of the MuSig protocol below.\n", "\n", @@ -39,11 +41,11 @@ "## Part 1: Public Key Generation\n", "\n", "* Due to linear nature of Schnorr, public keys, nonces and signatures can be aggregated.\n", - "* Due to the [Key cancellation](https://tlu.tarilabs.com/cryptography/digital_signatures/introduction_schnorr_signatures.html#key-cancellation-attack) attack, signers must first commit to individual public keys (Challenge factor).\n", + "* Signers must first commit to individual public keys (challenge factor). Without this, the protocol is vulnerable to the [key cancellation](https://tlu.tarilabs.com/cryptography/digital_signatures/introduction_schnorr_signatures.html#key-cancellation-attack) attack.\n", "* Individual key pairs must be multiplied with respective challenge.\n", "\n", "\n", - "![test](../images/musig_intro_1.jpg)" + "![test](images/musig_intro_1.jpg)" ] }, { @@ -59,11 +61,22 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "scrolled": true + }, "outputs": [], "source": [ "# Compute key pairs.\n", - "\n", + "privkey0 = ECKey()\n", + "privkey1 = ECKey()\n", + "privkey2 = ECKey()\n", + "privkey0.generate()\n", + "privkey1.generate()\n", + "privkey2.generate()\n", + "pk0 = privkey0.get_pubkey()\n", + "pk1 = privkey1.get_pubkey()\n", + "pk2 = privkey2.get_pubkey()\n", + "pk_v = [pk0, pk1, pk2]\n", "\n", "# Compute Key Challenges.\n", "\n", @@ -77,7 +90,11 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "## Part 2: Nonce Generation\n", + "## Part 2: Signing \n", + "\n", + "### 2a - Nonce Generation\n", + "\n", + "In the first step of creating a MuSig signature, each signer generates a nonce. The nonce points are then exchanged and aggregated.\n", "\n", "* Nonces can be aggregated like public keys and signatures.\n", "* Security [Proof](https://eprint.iacr.org/2018/068.pdf) requires nonces are randomly generated independently of each other.\n", @@ -107,11 +124,12 @@ "# Generate nonces.\n", "\n", "\n", - "# Skip nonce point commitment round (we assume nonce commitments are correct).\n", - "\n", + "# Nonce point commitment round.\n", + "R0_digest = hashlib.sha256(R0.get_bytes()).digest()\n", + "R1_digest = hashlib.sha256(R1.get_bytes()).digest()\n", + "R2_digest = hashlib.sha256(R2.get_bytes()).digest()\n", "\n", "# Aggregate nonces.\n", - "\n", "\n" ] }, @@ -119,7 +137,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "## Part 3: Signature Generation\n", + "### 2b - Signature Aggregation\n", "\n", "* Signatures are generated individually, but can be aggregated once they are exchanged amongst all participants in a single roundtrip.\n", "* Notice that the hash expressions are identical in all signatures, which makes aggregation possible.\n", @@ -131,9 +149,9 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "### 3.1 Programming exercise: Compute aggregated MuSig signature.\n", + "### 2.2 Programming exercise: Compute aggregated MuSig signature.\n", "\n", - "* `ECKey.sign_musig(nonce_point, negated, nonce_point_aggregated, pk_musig, msg)` returns individual signature.\n", + "* `sign_musig(ec_key, nonce_point, nonce_point_aggregated, pk_musig, msg)` returns individual signature.\n", "* `aggregate_musig_signatures([sig0, sig1, ...])` returns aggregated signature.\n", "* `ECPubKey.verify(sig, msg)` returns validity of signature.\n" ] @@ -150,7 +168,6 @@ "\n", "\n", "# Aggregate signatures.\n", - "\n", "\n" ] }, diff --git a/Solutions/1.2-Introduction-to-Musig-Solutions.ipynb b/Solutions/1.2-Introduction-to-Musig-Solutions.ipynb index e53398c2a..2c21a34ec 100644 --- a/Solutions/1.2-Introduction-to-Musig-Solutions.ipynb +++ b/Solutions/1.2-Introduction-to-Musig-Solutions.ipynb @@ -11,7 +11,8 @@ "\n", "sys.path.insert(0, '../')\n", "import util\n", - "from test_framework.key import ECKey, ECPubKey, generate_musig_key, generate_schnorr_nonce, aggregate_schnorr_nonces, aggregate_musig_signatures" + "from test_framework.key import ECKey, ECPubKey, generate_schnorr_nonce\n", + "from test_framework.musig import sign_musig, generate_musig_key, aggregate_schnorr_nonces, aggregate_musig_signatures\n" ] }, { @@ -21,12 +22,15 @@ "# 1.2 Introduction to n-of-n MuSig\n", "\n", "* Part 1: Public Key Generation.\n", - "* Part 2: Nonce Generation.\n", - "* Part 3: Signature Generation.\n", + "* Part 2: Signing.\n", + " * Nonce Aggregation.\n", + " * Signature Aggregation.\n", "\n", "In this chapter, we introduce the interactive MuSig protocol which allows n-of-n participants to jointly create and spend taproot or tapscript output using aggregated Schnorr signatures. \n", "\n", - "**The advantage of MuSig is both output and joint n-of-n spend are indistinguishable from other pay-to-pubkey transactions on-chain.**\n", + "**Advantages of MuSig include:**\n", + "* Privacy\n", + "* Transaction size/fees\n", "\n", "MuSig is is an interactive protocol which requires multiple rounds of communication between the wallets. We will proceed with an example execution of the MuSig protocol below.\n", "\n", @@ -40,7 +44,7 @@ "## Part 1: Public Key Generation\n", "\n", "* Due to linear nature of Schnorr, public keys, nonces and signatures can be aggregated.\n", - "* Due to the [Key cancellation](https://tlu.tarilabs.com/cryptography/digital_signatures/introduction_schnorr_signatures.html#key-cancellation-attack) attack, signers must first commit to individual public keys (Challenge factor).\n", + "* Signers must first commit to individual public keys (challenge factor). Without this, the protocol is vulnerable to the [key cancellation](https://tlu.tarilabs.com/cryptography/digital_signatures/introduction_schnorr_signatures.html#key-cancellation-attack) attack.\n", "* Individual key pairs must be multiplied with respective challenge.\n", "\n", "\n", @@ -60,7 +64,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "scrolled": true + }, "outputs": [], "source": [ "# Compute key pairs.\n", @@ -77,10 +83,10 @@ "\n", "# Compute Key Challenges.\n", "c_map, pk_musig = generate_musig_key(pk_v)\n", - "print(\"Aggregated Public Key: \", pk_musig.get_bytes().hex())\n", + "print(\"Aggregated Public Key:\", pk_musig.get_bytes().hex())\n", "for pk, c in c_map.items():\n", - " print(\"Compressed Pubkey: \", pk.get_bytes().hex())\n", - " print(\"Challenge: \", c.hex(), '\\n')\n", + " print(\"Compressed Pubkey:\", pk.get_bytes().hex())\n", + " print(\"Challenge:\", c.hex(), '\\n')\n", "\n", "# Multiply key pairs by challenge factor.\n", "privkey0_c = privkey0.mul(c_map[pk0])\n", @@ -95,7 +101,11 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "## Part 2: Nonce Generation\n", + "## Part 2: Signing \n", + "\n", + "### 2a - Nonce Generation\n", + "\n", + "In the first step of creating a MuSig signature, each signer generates a nonce. The nonce points are then exchanged and aggregated.\n", "\n", "* Nonces can be aggregated like public keys and signatures.\n", "* Security [Proof](https://eprint.iacr.org/2018/068.pdf) requires nonces are randomly generated independently of each other.\n", @@ -130,33 +140,41 @@ "R1 = k1.get_pubkey()\n", "R2 = k2.get_pubkey()\n", "\n", - "# Skip nonce point commitment round (we assume nonce commitments are correct).\n", + "# Nonce point commitment round.\n", + "R0_digest = hashlib.sha256(R0.get_bytes()).digest()\n", + "R1_digest = hashlib.sha256(R1.get_bytes()).digest()\n", + "R2_digest = hashlib.sha256(R2.get_bytes()).digest()\n", "\n", "# Aggregate nonces.\n", "R_agg, negated = aggregate_schnorr_nonces([R0, R1, R2])\n", - "print(\"R aggregated: \", R_agg.get_bytes().hex())\n", - "print(\"R_agg was negated: \", negated)\n" + "if negated:\n", + " k0.negate()\n", + " k1.negate()\n", + " k2.negate()\n", + "\n", + "print(\"R aggregated:\", R_agg.get_bytes().hex())\n", + "print(\"R_agg was negated:\", negated)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "## Part 3: Signature Generation\n", + "### 2b - Signature Aggregation\n", "\n", "* Signatures are generated individually, but can be aggregated once they are exchanged amongst all participants in a single roundtrip.\n", "* Notice that the hash expressions are identical in all signatures, which makes aggregation possible.\n", "\n", - "![test](../images/musig_intro_4.jpg)" + "![test](../images/musig_intro_3.jpg)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "### 3.1 Programming exercise: Compute aggregated MuSig signature.\n", + "### 2.2 Programming exercise: Compute aggregated MuSig signature.\n", "\n", - "* `ECKey.sign_musig(nonce_point, negated, nonce_point_aggregated, pk_musig, msg)` returns individual signature.\n", + "* `sign_musig(ec_key, nonce_point, nonce_point_aggregated, pk_musig, msg)` returns individual signature.\n", "* `aggregate_musig_signatures([sig0, sig1, ...])` returns aggregated signature.\n", "* `ECPubKey.verify(sig, msg)` returns validity of signature.\n" ] @@ -170,13 +188,13 @@ "msg = hashlib.sha256(b'transaction').digest()\n", "\n", "# Generate individual signatures.\n", - "sig0 = privkey0_c.sign_musig(k0, negated, R_agg, pk_musig, msg)\n", - "sig1 = privkey1_c.sign_musig(k1, negated, R_agg, pk_musig, msg)\n", - "sig2 = privkey2_c.sign_musig(k2, negated, R_agg, pk_musig, msg)\n", + "sig0 = sign_musig(privkey0_c, k0, R_agg, pk_musig, msg)\n", + "sig1 = sign_musig(privkey1_c, k1, R_agg, pk_musig, msg)\n", + "sig2 = sign_musig(privkey2_c, k2, R_agg, pk_musig, msg)\n", "\n", "# Aggregate signatures.\n", "sig_agg = aggregate_musig_signatures([sig0, sig1, sig2])\n", - "print(\"Signature verifies against MuSig pubkey: \", pk_musig.verify_schnorr(sig_agg, msg))\n" + "print(\"Signature verifies against MuSig pubkey:\", pk_musig.verify_schnorr(sig_agg, msg))\n" ] }, { diff --git a/images/musig_intro_0.jpg b/images/musig_intro_0.jpg index 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musig pubkey - - Compute key challengesc0, c1, ...Compute musig PubkeyP_agg + + P + agg - - H(R0), H(R1), ... + + k + 0 + , + k + 1 + , ... - - R0, R1, ... + + Generate Nonces - - Aggregate Nonce RR_agg, is_negated + + Derive Nonce Points - - Generate Signaturessig0, sig1, ... + + R + 0, + R + 1, ... - - sig0, sig1, ... + + R + 0, + R + 1, ... - - Coordinator Aggregates Signaturessig_agg + + H( + R + 0 + ), H( + R + 1 + ), ... - - Roundtrip + + Agg + r + egate Nonce R + R + agg + , is_negated + + + + Generate Signatu + r + es + sig + 0 + , + sig + 1 + , ... + + + + sig + 0 + , + sig + 1 + , ... + + + + Coo + r + dinator + Agg + r + egates Signatu + r + es + sig + agg @@ -93,37 +158,17 @@ - - - - - - - - - - - - - - - - - - - 1st Round - - 1st + 1. - 2nd + 2. - 3rd + 3. + + + IndividualComputation + + + + IndividualComputation + + + + ExchangedData + + + + + + + + + + + + + + + + + diff --git a/images/musig_intro_1.jpg b/images/musig_intro_1.jpg index 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zk7?Hrne~t2*)Q46XPf_S*_?n=A3f6}W^^TlpIpQEIoj*bs{`_<)-Zm|{P*Tk5Vb)p z^x$d^MrK%~99{iuF0fC80T_;d56%8H>))$u$0L)3uKxLOhHuUFl1HXl*}zFx2M&wY zBfb?g@}ye49<1t|Qo;5K`j(_i7JHxVK^-=Mq8=P7lqpKViEnLMbYHAVmdCBi*e!PKUN{mM&u zdq?k6=h>dE7v9jTZf_^H>T%K1jC~5%w2mZLbY}@LFA+KUA=dYFc)i|JugId~%n_>v z8y=kt%0B3J{ypC|ogd7)Oq18VmdKhpWrt?L>6ohzR_=VDx=w2G$wL< - + - + - + Pubkey Generation - - x_i, P_i + + x + i + , P + i - - P_i + + P + i @@ -54,20 +58,58 @@ Public Key Exchange - - C_all = H(P_0, P_1, P_2, ..., P_n)C_i = H(C_all | P_i)P_i’ = C_i * P_i, x_i’ = C_i * x_nP_agg = P_0’ + P_1’ + P_2’ + ... + P_n’ + + c + all + = H(P + 0 + , P + 1 + , P + 2 + , ... , P + n + ) + c + i + = H(c + all + | P + i + ) + P + i + ’ = c + i + * P + i + , x + i + ’ = c + i + * x + i + P_agg = P + 0 + ’ + P + 1 + ’ + P + 2 + ’ + ... + P + n + Challenge Factor - - Tweaked Keys + + Tweaked Keys - - Musig Public Key + + Musig Public Key @@ -90,13 +132,13 @@ - - + + - - + + diff --git a/images/musig_intro_2.jpg b/images/musig_intro_2.jpg 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z+-n(1Gx%h(pNHS+N=A@$kcEoH_V2n#*=})zSFv8B-SCS5y)<$QwCS2nZ(&W2cTi)Z s!K6 - + - + - + - + Nonce Generation - - k_i, R_i + + k + n + , R + n - - H(R_i) + + H(R + i + ) - + - - H(R_i) + + H(R + i + ) - + - - R_i + + R + i - - Individual + + Individual @@ -66,81 +74,101 @@ 2nd Round - - Individual + + Individual - - Roundtrip + + Data Exchange - - If Nonce points hash correctly:R_agg = R_0 + R_1 + ... + R_nIf jacobi_symbol(R_agg) != 1R_agg = Negate(R_agg)k_i = Negate(k_i) + + If Nonce points hash correctly: + R + agg + = R + 0 + + R + 1 + + ... + R + n + If jacobi_symbol(R_agg) != 1 + R + agg + = Negate(R + agg + ) + k + i + = Negate(k + i + ) + for all i - - Store: + + Store: - + - + - + - + - + + M202.313,227.364h53.02c1.65,0,3-1.35,3-3V98.222c0-1.65,1.35-3,3-3h56.877c1.65,0,3,1.35,3,3v34.455"/> - + - + - + - + - + - + - + - + - + - + - + - - - + + + diff --git a/images/musig_intro_3.jpg b/images/musig_intro_3.jpg index 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+ k + 1 + + H(x( + R + agg + )| + P + agg + |m) * + x + 1 - - s0 + s1 + s2 = k0 + k1 + k2 + H(x(R_agg)|P_agg|m) * (x0 + x1 + x2) + + s + 2 + = + k + 2 + + H(x( + R + agg + )| + P + agg + |m) * + x + 2 - - s_agg + + s + 0 + + + s + 1 + + + s + 2 + = + k + 0 + + + k + 1 + + + k + 2 + + H(x( + R + agg + )| + P + agg + |m) * ( + x + 0 + + + x + 1 + + + x + 2 + ) - - k_agg + + H(x( + R + agg + )| + P + agg + |m) - - = + + s + agg - - + + + k + agg - - * + + x + agg + + = - - x_agg + + + - - H(x(R_agg)|P_agg|m) + + * - + - + - + - + - + - + - + - + @@ -173,6 +251,6 @@ Aggregated Signature - Linear Aggregation of Schnorr Signatures + Signature Aggregation From c1cf9020e2d05f4613eefa716500dd54987c9828 Mon Sep 17 00:00:00 2001 From: John Newbery Date: Tue, 17 Sep 2019 21:58:36 -0400 Subject: [PATCH 5/5] Chapter 1.2: John's review markups --- 1.2-Introduction-to-Musig.ipynb | 150 ++++++++---- .../1.2-Introduction-to-Musig-Solutions.ipynb | 229 ------------------ .../1.2-Introduction-to-Musig-Solutions.txt | 74 ++++++ 3 files changed, 178 insertions(+), 275 deletions(-) delete mode 100644 Solutions/1.2-Introduction-to-Musig-Solutions.ipynb create mode 100644 Solutions/1.2-Introduction-to-Musig-Solutions.txt diff --git a/1.2-Introduction-to-Musig.ipynb b/1.2-Introduction-to-Musig.ipynb index 73266bd43..15d684392 100644 --- a/1.2-Introduction-to-Musig.ipynb +++ b/1.2-Introduction-to-Musig.ipynb @@ -8,8 +8,8 @@ "source": [ "import hashlib\n", "import util\n", - "from test_framework.key import ECKey, ECPubKey, generate_schnorr_nonce\n", - "from test_framework.musig import sign_musig, generate_musig_key, aggregate_schnorr_nonces, aggregate_musig_signatures\n" + "from test_framework.key import generate_schnorr_nonce, ECKey, ECPubKey\n", + "from test_framework.musig import aggregate_musig_signatures, aggregate_schnorr_nonces, generate_musig_key, sign_musig" ] }, { @@ -23,15 +23,23 @@ " * Nonce Aggregation.\n", " * Signature Aggregation.\n", "\n", - "In this chapter, we introduce the interactive MuSig protocol which allows n-of-n participants to jointly create and spend taproot or tapscript output using aggregated Schnorr signatures. \n", + "In this chapter, we introduce the interactive [MuSig protocol](https://eprint.iacr.org/2018/068.pdf) which allows n-of-n participants to jointly create and spend taproot or tapscript outputs using aggregated schnorr signatures. \n", "\n", - "**Advantages of MuSig include:**\n", - "* Privacy\n", - "* Transaction size/fees\n", + "Using a signature aggregation scheme like MuSig has two significant advantages over using Script's `OP_CHECKMULTISIG` and tapscript's `OP_CHECKSIGADD` opcodes:\n", "\n", - "MuSig is is an interactive protocol which requires multiple rounds of communication between the wallets. We will proceed with an example execution of the MuSig protocol below.\n", + "* **Transaction Size/Fees**: an aggregate MuSig pubkey and signature is indistinguishable from a single-key pubkey and signature, meaning that the transaction size (and required fee) for a multi-key output are the same as for a single-key output.\n", + "* **Privacy and Fungibility**: an aggregate MuSig pubkey and signature is indistinguishable from a single-key pubkey and signature, making it impossible for anyone to use the public block chain data to identify where a multi-key scheme has been used.\n", "\n", - "![test](images/musig_intro_0.jpg)\n" + "The MuSig protocol covers both the initial setup (generating an aggregate pubkey for all participants), and the signing protocol (creating a valid signature for the aggregate pubkey). The signing requires multiple rounds of communication between the individual signers.\n", + "\n", + "Bip-schnorr is linear in the nonce points and public keys, which means that public keys, nonces and signatures can be aggregated. A very naive multiparty signature scheme could be achieved by simply summing the individual pubkeys to generate an aggregate pubkey, each participant signing with a shared nonce, and then summing the signatures. Such a scheme would be vulnerable to both the [key cancellation attack](https://tlu.tarilabs.com/cryptography/digital_signatures/introduction_schnorr_signatures.html#key-cancellation-attack) and private key extraction by exploiting weak or known nonces. Countering these attacks is what adds some complexity to the MuSig protocol." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "![test](images/musig_intro_0.jpg)" ] }, { @@ -40,11 +48,17 @@ "source": [ "## Part 1: Public Key Generation\n", "\n", - "* Due to linear nature of Schnorr, public keys, nonces and signatures can be aggregated.\n", - "* Signers must first commit to individual public keys (challenge factor). Without this, the protocol is vulnerable to the [key cancellation](https://tlu.tarilabs.com/cryptography/digital_signatures/introduction_schnorr_signatures.html#key-cancellation-attack) attack.\n", - "* Individual key pairs must be multiplied with respective challenge.\n", + "To counter the key cancellation attack, each participant's pubkey is _tweaked_ by a _challenge factor,_ which is generated by hashing all the participants' pubkeys together. Doing this ensures that no individual participant (or group of participants) is able to create a pubkey that cancels out other participants' pubkeys.\n", "\n", + "The challenge factor is unique for each participant, but all challenge factors are based on a hash of all participants' pubkeys.\n", "\n", + "No interactive round-trips are required in this step. All participants can provide their pubkey to a central co-ordinator, which can generate the challenge factors and aggregate pubkey." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ "![test](images/musig_intro_1.jpg)" ] }, @@ -52,10 +66,13 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "### 1.1 Programming Exercise: Compute 3-of-3 MuSig public key.\n", - "* Use `generate_musig_key([ECPubKey0, ECPubKey1, ...])`.\n", - "* returns `c_map, pk_musig`\n", - "* `c_map` contains `ECPubKey_i, challenge_data_i` key - value pairs.\n" + "#### 1.2.1 _Programming Exercise:_ Compute 3-of-3 MuSig public key\n", + "\n", + "In this exercise, we'll use the `generate_musig_key()` function to generate challenge factors for each participant and an aggregate MuSig pubkey.\n", + "\n", + "`generate_musig_key()` takes a list of the participants' keys `generate_musig_key([ECPubKey0, ECPubKey1, ...])` and returns a challenge map and the aggregate pubkey:\n", + "* the challenge map contains `ECPubKey_i, challenge_data_i` key - value pairs.\n", + "* The aggregate pubkey is an `ECPubKey` object" ] }, { @@ -66,7 +83,7 @@ }, "outputs": [], "source": [ - "# Compute key pairs.\n", + "# Compute key pairs\n", "privkey0 = ECKey()\n", "privkey1 = ECKey()\n", "privkey2 = ECKey()\n", @@ -78,12 +95,21 @@ "pk2 = privkey2.get_pubkey()\n", "pk_v = [pk0, pk1, pk2]\n", "\n", - "# Compute Key Challenges.\n", + "# Compute Key Challenges\n", + "\n", "\n", + "# Multiply key pairs by challenge factor\n", + "privkey0_c = # TODO: implement\n", + "privkey1_c = # TODO: implement\n", + "privkey2_c = # TODO: implement\n", + "pk0_c = # TODO: implement\n", + "pk1_c = # TODO: implement\n", + "pk2_c = # TODO: implement\n", "\n", - "# Multiply key pairs by challenge factor.\n", "\n", - "\n" + "print(\"Tweaked privkey0 is {}\".format(privkey0_c.secret))\n", + "print(\"Tweaked privkey1 is {}\".format(privkey1_c.secret))\n", + "print(\"Tweaked privkey2 is {}\".format(privkey2_c.secret))" ] }, { @@ -94,25 +120,34 @@ "\n", "### 2a - Nonce Generation\n", "\n", - "In the first step of creating a MuSig signature, each signer generates a nonce. The nonce points are then exchanged and aggregated.\n", + "The first step of creating a MuSig signature requires each signer to generate their own nonce and nonce point. The participants then exchange those nonces points and an aggregate nonce point is derived by summing all the nonce points.\n", "\n", - "* Nonces can be aggregated like public keys and signatures.\n", - "* Security [Proof](https://eprint.iacr.org/2018/068.pdf) requires nonces are randomly generated independently of each other.\n", - "* An initial nonce commitment round is necessary to ensure no nonce is derived as a function of another.\n", + "The security proof for MuSig requires that nonces are randomly generated and are independent of each other. To ensure that no individual participant (or group of participants) can create their nonce as a function of the other nonces or individually control what the aggregate nonce point will be, there is an initial round of exchanging hash commitments to the individual nonce points.\n", "\n", + "Individual participants should only exchange their nonce point when they have received all commitments, and only proceed with signing if all nonce points match their commitments.\n", "\n", - "![test](images/musig_intro_2.jpg)\n" + "Finally, if the aggregate nonce is not a quadratic residue, then it is negated and all individual nonce points are also negated." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "![test](images/musig_intro_2.jpg)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "### 2.1 Programming Exercise: Compute 3-of-3 MuSig nonce.\n", + "#### 1.2.2 _Programming Exercise:_ Compute 3-of-3 MuSig nonce\n", + "\n", + "In this exercise, we'll generate nonces for individual participants, calculate the nonce point commitments, and then generate an aggregate nonce point.\n", "\n", - "* `generate_schnorr_nonce()` returns a ECkey with with a jacobi symbol of 1.\n", - "* `aggregate_schnorr_nonces()` returns a ECPubKey and a boolean (indicates nonce negation).\n", - "* *Negation of nonce `k_i` isn't necessary, since it is performed by `sign_musig` in next exercise.*" + "* Use `generate_schnorr_nonce()` to generate a random nonce. The function returns an `ECkey` object which is a valid schnorr nonce (quadratic residue y-value).\n", + "* Use `aggregate_schnorr_nonces()` to aggregate those individual nonces. The function takes a list of `ECKey` nonces and returns a the aggregate nonce and a negation flag:\n", + " * the aggregate nonce is an `ECPubKey` object\n", + " * the negation flag is a boolean and indicates that the individual nonces should be negated" ] }, { @@ -122,7 +157,9 @@ "outputs": [], "source": [ "# Generate nonces.\n", - "\n", + "k0 = # TODO: implement\n", + "k1 = # TODO: implement\n", + "k2 = # TODO: implement\n", "\n", "# Nonce point commitment round.\n", "R0_digest = hashlib.sha256(R0.get_bytes()).digest()\n", @@ -130,7 +167,10 @@ "R2_digest = hashlib.sha256(R2.get_bytes()).digest()\n", "\n", "# Aggregate nonces.\n", - "\n" + "R_agg = # TODO: implement\n", + "\n", + "print(\"Individual nonces are {}, {}, {}.\".format(k0.secret, k1.secret, k2.secret))\n", + "print(\"Aggregate nonce point is {}\".format(R_agg.get_bytes().hex()))" ] }, { @@ -139,36 +179,40 @@ "source": [ "### 2b - Signature Aggregation\n", "\n", - "* Signatures are generated individually, but can be aggregated once they are exchanged amongst all participants in a single roundtrip.\n", - "* Notice that the hash expressions are identical in all signatures, which makes aggregation possible.\n", + "Once all participants have their individual nonces and the aggregate nonce point, then can all sign individually. \n", "\n", - "![test](images/musig_intro_3.jpg)" + "The individual `s` values are then exchanged and summed together. The aggregate `s` value and aggregate nonce point `R` form a valid bip-schnorr signature for the aggregate pubkey.\n", + "\n", + "Notice that the hash expressions are identical in all signatures, which makes aggregation possible." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "### 2.2 Programming exercise: Compute aggregated MuSig signature.\n", - "\n", - "* `sign_musig(ec_key, nonce_point, nonce_point_aggregated, pk_musig, msg)` returns individual signature.\n", - "* `aggregate_musig_signatures([sig0, sig1, ...])` returns aggregated signature.\n", - "* `ECPubKey.verify(sig, msg)` returns validity of signature.\n" + "![test](images/musig_intro_3.jpg)" ] }, { - "cell_type": "code", - "execution_count": null, + "cell_type": "markdown", "metadata": {}, - "outputs": [], "source": [ - "msg = hashlib.sha256(b'transaction').digest()\n", + "#### 1.2.3 _Programming exercise:_ Compute aggregated MuSig signature\n", + "\n", + "In this exercise, we'll create individual signatures and then aggregate them to create a valid signature.\n", "\n", - "# Generate individual signatures.\n", + "Use the `sign_musig()` function to create individual signatures. `sign_musig()` takes:\n", + " - the individual participant's private key (an `ECKey` object)\n", + " - the invididual participant's nonce (an `ECKey` object)\n", + " - the aggregate nonce point (an `ECPubKey` object)\n", + " - the aggregate pubkey (an `ECPubKey` object)\n", + " - the message (a 32 byte `bytes` object)\n", "\n", + "and returns an individual signature (a 64 byte `bytes` object containing `R(x)` and `s`).\n", "\n", - "# Aggregate signatures.\n", - "\n" + "Use `aggregate_musig_signatures()` to aggregate the individual signatures. `aggregate_musig_signatures()` takes a list of signatures and returns the aggregated signature.\n", + "\n", + "Use `ECPubKey.verify(sig, msg)` to verify that the signature is valid." ] }, { @@ -176,7 +220,21 @@ "execution_count": null, "metadata": {}, "outputs": [], - "source": [] + "source": [ + "msg = hashlib.sha256(b'transaction').digest()\n", + "\n", + "# Generate individual signatures\n", + "sig0 = # TODO: implement\n", + "sig1 = # TODO: implement\n", + "sig2 = # TODO: implement\n", + "\n", + "# Aggregate signatures\n", + "sig_agg = # TODO: implement\n", + "\n", + "# Verify signature\n", + "assert pk_musig.verify_schnorr(sig_agg, msg)\n", + "print(\"Success!\")" + ] } ], "metadata": { diff --git a/Solutions/1.2-Introduction-to-Musig-Solutions.ipynb b/Solutions/1.2-Introduction-to-Musig-Solutions.ipynb deleted file mode 100644 index 2c21a34ec..000000000 --- a/Solutions/1.2-Introduction-to-Musig-Solutions.ipynb +++ /dev/null @@ -1,229 +0,0 @@ -{ - "cells": [ - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "import sys\n", - "import hashlib\n", - "\n", - "sys.path.insert(0, '../')\n", - "import util\n", - "from test_framework.key import ECKey, ECPubKey, generate_schnorr_nonce\n", - "from test_framework.musig import sign_musig, generate_musig_key, aggregate_schnorr_nonces, aggregate_musig_signatures\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# 1.2 Introduction to n-of-n MuSig\n", - "\n", - "* Part 1: Public Key Generation.\n", - "* Part 2: Signing.\n", - " * Nonce Aggregation.\n", - " * Signature Aggregation.\n", - "\n", - "In this chapter, we introduce the interactive MuSig protocol which allows n-of-n participants to jointly create and spend taproot or tapscript output using aggregated Schnorr signatures. \n", - "\n", - "**Advantages of MuSig include:**\n", - "* Privacy\n", - "* Transaction size/fees\n", - "\n", - "MuSig is is an interactive protocol which requires multiple rounds of communication between the wallets. We will proceed with an example execution of the MuSig protocol below.\n", - "\n", - "![test](../images/musig_intro_0.jpg)\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Part 1: Public Key Generation\n", - "\n", - "* Due to linear nature of Schnorr, public keys, nonces and signatures can be aggregated.\n", - "* Signers must first commit to individual public keys (challenge factor). Without this, the protocol is vulnerable to the [key cancellation](https://tlu.tarilabs.com/cryptography/digital_signatures/introduction_schnorr_signatures.html#key-cancellation-attack) attack.\n", - "* Individual key pairs must be multiplied with respective challenge.\n", - "\n", - "\n", - "![test](../images/musig_intro_1.jpg)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 1.1 Programming Exercise: Compute 3-of-3 MuSig public key.\n", - "* Use `generate_musig_key([ECPubKey0, ECPubKey1, ...])`.\n", - "* returns `c_map, pk_musig`\n", - "* `c_map` contains `ECPubKey_i, challenge_data_i` key - value pairs.\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "scrolled": true - }, - "outputs": [], - "source": [ - "# Compute key pairs.\n", - "privkey0 = ECKey()\n", - "privkey1 = ECKey()\n", - "privkey2 = ECKey()\n", - "privkey0.generate()\n", - "privkey1.generate()\n", - "privkey2.generate()\n", - "pk0 = privkey0.get_pubkey()\n", - "pk1 = privkey1.get_pubkey()\n", - "pk2 = privkey2.get_pubkey()\n", - "pk_v = [pk0, pk1, pk2]\n", - "\n", - "# Compute Key Challenges.\n", - "c_map, pk_musig = generate_musig_key(pk_v)\n", - "print(\"Aggregated Public Key:\", pk_musig.get_bytes().hex())\n", - "for pk, c in c_map.items():\n", - " print(\"Compressed Pubkey:\", pk.get_bytes().hex())\n", - " print(\"Challenge:\", c.hex(), '\\n')\n", - "\n", - "# Multiply key pairs by challenge factor.\n", - "privkey0_c = privkey0.mul(c_map[pk0])\n", - "privkey1_c = privkey1.mul(c_map[pk1])\n", - "privkey2_c = privkey2.mul(c_map[pk2])\n", - "pk0_c = pk0.mul(c_map[pk0])\n", - "pk1_c = pk1.mul(c_map[pk1])\n", - "pk2_c = pk2.mul(c_map[pk2])\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Part 2: Signing \n", - "\n", - "### 2a - Nonce Generation\n", - "\n", - "In the first step of creating a MuSig signature, each signer generates a nonce. The nonce points are then exchanged and aggregated.\n", - "\n", - "* Nonces can be aggregated like public keys and signatures.\n", - "* Security [Proof](https://eprint.iacr.org/2018/068.pdf) requires nonces are randomly generated independently of each other.\n", - "* An initial nonce commitment round is necessary to ensure no nonce is derived as a function of another.\n", - "\n", - "\n", - "![test](../images/musig_intro_2.jpg)\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 2.1 Programming Exercise: Compute 3-of-3 MuSig nonce.\n", - "\n", - "* `generate_schnorr_nonce()` returns a ECkey with with a jacobi symbol of 1.\n", - "* `aggregate_schnorr_nonces()` returns a ECPubKey and a boolean (indicates nonce negation).\n", - "* *Negation of nonce `k_i` isn't necessary, since it is performed by `sign_musig` in next exercise.*" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "# Generate nonces.\n", - "k0 = generate_schnorr_nonce()\n", - "k1 = generate_schnorr_nonce()\n", - "k2 = generate_schnorr_nonce()\n", - "R0 = k0.get_pubkey()\n", - "R1 = k1.get_pubkey()\n", - "R2 = k2.get_pubkey()\n", - "\n", - "# Nonce point commitment round.\n", - "R0_digest = hashlib.sha256(R0.get_bytes()).digest()\n", - "R1_digest = hashlib.sha256(R1.get_bytes()).digest()\n", - "R2_digest = hashlib.sha256(R2.get_bytes()).digest()\n", - "\n", - "# Aggregate nonces.\n", - "R_agg, negated = aggregate_schnorr_nonces([R0, R1, R2])\n", - "if negated:\n", - " k0.negate()\n", - " k1.negate()\n", - " k2.negate()\n", - "\n", - "print(\"R aggregated:\", R_agg.get_bytes().hex())\n", - "print(\"R_agg was negated:\", negated)\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 2b - Signature Aggregation\n", - "\n", - "* Signatures are generated individually, but can be aggregated once they are exchanged amongst all participants in a single roundtrip.\n", - "* Notice that the hash expressions are identical in all signatures, which makes aggregation possible.\n", - "\n", - "![test](../images/musig_intro_3.jpg)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 2.2 Programming exercise: Compute aggregated MuSig signature.\n", - "\n", - "* `sign_musig(ec_key, nonce_point, nonce_point_aggregated, pk_musig, msg)` returns individual signature.\n", - "* `aggregate_musig_signatures([sig0, sig1, ...])` returns aggregated signature.\n", - "* `ECPubKey.verify(sig, msg)` returns validity of signature.\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "msg = hashlib.sha256(b'transaction').digest()\n", - "\n", - "# Generate individual signatures.\n", - "sig0 = sign_musig(privkey0_c, k0, R_agg, pk_musig, msg)\n", - "sig1 = sign_musig(privkey1_c, k1, R_agg, pk_musig, msg)\n", - "sig2 = sign_musig(privkey2_c, k2, R_agg, pk_musig, msg)\n", - "\n", - "# Aggregate signatures.\n", - "sig_agg = aggregate_musig_signatures([sig0, sig1, sig2])\n", - "print(\"Signature verifies against MuSig pubkey:\", pk_musig.verify_schnorr(sig_agg, msg))\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.6.5" - } - }, - "nbformat": 4, - "nbformat_minor": 2 -} diff --git a/Solutions/1.2-Introduction-to-Musig-Solutions.txt b/Solutions/1.2-Introduction-to-Musig-Solutions.txt new file mode 100644 index 000000000..03f967a48 --- /dev/null +++ b/Solutions/1.2-Introduction-to-Musig-Solutions.txt @@ -0,0 +1,74 @@ +#### 1.2.1 Compute 3-of-3 MuSig public key + +``` +# Compute key pairs +privkey0 = ECKey() +privkey1 = ECKey() +privkey2 = ECKey() +privkey0.generate() +privkey1.generate() +privkey2.generate() +pk0 = privkey0.get_pubkey() +pk1 = privkey1.get_pubkey() +pk2 = privkey2.get_pubkey() +pk_v = [pk0, pk1, pk2] + +# Compute Key Challenges +c_map, pk_musig = generate_musig_key(pk_v) +print("Aggregated Public Key is {}\n".format(pk_musig.get_bytes().hex())) + +# Multiply key pairs by challenge factor +privkey0_c = privkey0.mul(c_map[pk0]) +privkey1_c = privkey1.mul(c_map[pk1]) +privkey2_c = privkey2.mul(c_map[pk2]) +pk0_c = pk0.mul(c_map[pk0]) +pk1_c = pk1.mul(c_map[pk1]) +pk2_c = pk2.mul(c_map[pk2]) + +print("Tweaked privkey0 is {}".format(privkey0_c.secret)) +print("Tweaked privkey1 is {}".format(privkey1_c.secret)) +print("Tweaked privkey2 is {}".format(privkey2_c.secret)) +``` + +#### 1.2.2 Compute 3-of-3 MuSig nonce + +``` +# Generate nonces. +k0 = generate_schnorr_nonce() +k1 = generate_schnorr_nonce() +k2 = generate_schnorr_nonce() +R0 = k0.get_pubkey() +R1 = k1.get_pubkey() +R2 = k2.get_pubkey() + +# Nonce point commitment round. +R0_digest = hashlib.sha256(R0.get_bytes()).digest() +R1_digest = hashlib.sha256(R1.get_bytes()).digest() +R2_digest = hashlib.sha256(R2.get_bytes()).digest() + +# Aggregate nonces. +R_agg, negated = aggregate_schnorr_nonces([R0, R1, R2]) +if negated: + k0.negate() + k1.negate() + k2.negate() + +print("Individual nonces are {}, {}, {}.".format(k0.secret, k1.secret, k2.secret)) +print("Aggregate nonce point is {}".format(R_agg.get_bytes().hex())) +print("R_agg was negated:", negated) +``` + +#### 1.2.3 Compute aggregated MuSig signature + +``` +msg = hashlib.sha256(b'transaction').digest() + +# Generate individual signatures. +sig0 = sign_musig(privkey0_c, k0, R_agg, pk_musig, msg) +sig1 = sign_musig(privkey1_c, k1, R_agg, pk_musig, msg) +sig2 = sign_musig(privkey2_c, k2, R_agg, pk_musig, msg) + +# Aggregate signatures. +sig_agg = aggregate_musig_signatures([sig0, sig1, sig2]) +print("Signature verifies against MuSig pubkey:", pk_musig.verify_schnorr(sig_agg, msg)) +```