From 50b13d2941bb7195a8f1d4aa4d20b75234b5bfd6 Mon Sep 17 00:00:00 2001
From: Stefan Thomas <justmoon@members.fsf.org>
Date: Sat, 3 Sep 2011 19:36:45 +0100
Subject: [PATCH] More work on split key protocol.
---
demo/split-key.html | 48 +++++++++++++++++---
demo/split-key.js | 104 +++++++++++++++++++++++++++++++++-----------
src/paillier.js | 25 ++++++++---
test/test.js | 12 +++++
4 files changed, 153 insertions(+), 36 deletions(-)
diff --git a/demo/split-key.html b/demo/split-key.html
index b3b9915..82175bd 100755
--- a/demo/split-key.html
+++ b/demo/split-key.html
@@ -81,7 +81,7 @@ jQuery(function ($) {
<label for="q1">Q<sub>1</sub>=</label>
<input id="q1" type="text" readonly="readonly"/>
</div>
- <p>She then encrypts z<sub>1</sub> with her Paillier secret to create α = E<sub>pk</sub>(z<sub>1</sub>)</p>
+ <p>She then encrypts z<sub>1</sub> using Paillier to create α = E<sub>pk</sub>(z<sub>1</sub>)</p>
<div>
<label for="alpha">α=</label>
<input id="alpha" type="text" readonly="readonly"/>
@@ -91,9 +91,34 @@ jQuery(function ($) {
<label for="beta">β=</label>
<input id="beta" type="text" readonly="readonly"/>
</div>
+ <p>And also generates an encrypted blinding factor A = E<sub>pk</sub>(c) for some c ∈ [1, n<sub>P</sub>/n<sub>EC</sub>]</p>
+ <div>
+ <label for="A">A=</label>
+ <input id="A" type="text" readonly="readonly"/>
+ </div>
+ <p>Alice composes the encrypted signature σ<sub>1</sub> = (α ×<sub>pk</sub> e) +<sub>pk</sub> (β ×<sub>pk</sub> r) +<sub>pk</sub> (A ×<sub>pk</sub> n)</p>
+ <div>
+ <label for="sigma_1">σ<sub>1</sub>=</label>
+ <input id="sigma_1" type="text" readonly="readonly"/>
+ </div>
+ <p>She deterministically rerandomizes it to receive σ<sub>1</sub>' = σ<sub>1</sub>HASH(σ<sub>1</sub>)<sup>n</sub> mod n<sup>2</sup></p>
+ <div>
+ <label for="sigma_1n">σ<sub>1</sub>'=</label>
+ <input id="sigma_1n" type="text" readonly="readonly"/>
+ </div>
+ <p>And decrypts σ<sub>1</sub>' to receive s<sub>1</sub></p>
+ <div>
+ <label for="s_1">s<sub>1</sub>=</label>
+ <input id="s_1" type="text" readonly="readonly"/>
+ </div>
+ <p>And v', the randomizing factor in σ<sub>1</sub>'</p>
+ <div>
+ <label for="v_n">v<sub>'</sub>=</label>
+ <input id="v_n" type="text" readonly="readonly"/>
+ </div>
</div>
<div class="messageright"><div class="arrow"></div>
-Q<sub>1</sub>, α, β, message, e, pk
+Q<sub>1</sub>, α, β, message, e, pk, A, s<sub>1</sub>, v'
</div>
<div class="bob">
<p>Bob validates Q<sub>1</sub> by ensuring that
@@ -104,6 +129,13 @@ Q<sub>1</sub>, α, β, message, e, pk
<li>nQ<sub>1</sub> = O</li>
</ol></p>
<p>And verifies the message to be signed</p>
+ <p>He then verifies s<sub>1</sub> as a valid signature</p>
+ <p>Bob also calculates σ<sub>1</sub>' from α, β and A</p>
+ <div>
+ <label for="sigma_1n_b">σ<sub>1</sub>'=</label>
+ <input id="sigma_1n_b" type="text" readonly="readonly"/>
+ </div>
+ <p>And verifies it matches E<sub>pk</sub>(s<sub>1</sub>, v')</p>
<p>He then generates his share k<sub>2</sub> of the private one-time value k</p>
<div>
<label for="k2">k<sub>2</sub>=</label>
@@ -124,12 +156,12 @@ Q<sub>1</sub>, α, β, message, e, pk
<label for="q2">Q<sub>2</sub>=</label>
<input id="q2" type="text" readonly="readonly"/>
</div>
- <p>Bob prepares a random value c to use for blinding<p>
+ <p>Bob prepares a random value B ∈ [1, n<sub>P</sub>/n<sub>EC</sub>] to use for blinding<p>
<div>
- <label for="c">c=</label>
- <input id="c" type="text" readonly="readonly"/>
+ <label for="B">B=</label>
+ <input id="B" type="text" readonly="readonly"/>
</div>
- <p>Finally he calculates σ = (α ×<sub>pk</sub> z<sub>2</sub>e) +<sub>pk</sub> (β ×<sub>pk</sub> z<sub>2</sub>d<sub>2</sub>r) +<sub>pk</sub> E<sub>pk</sub>(cn)</p>
+ <p>Finally he calculates σ = (α ×<sub>pk</sub> z<sub>2</sub>e) +<sub>pk</sub> (β ×<sub>pk</sub> z<sub>2</sub>d<sub>2</sub>r) +<sub>pk</sub> E<sub>pk</sub>(Bn<sub>EC</sub>)</p>
<div>
<label for="sigma">σ=</label>
<input id="sigma" type="text" readonly="readonly"/>
@@ -153,6 +185,10 @@ Q<sub>2</sub>, r, σ
<input id="s" type="text" readonly="readonly"/>
</div>
<p>She verifies the signature using r and the combined public key before publishing.</p>
+ <div>
+ <label for="result"></label>
+ <input id="result" type="text" readonly="readonly"/>
+ </div>
</div>
</body>
</html>
diff --git a/demo/split-key.js b/demo/split-key.js
index 582e1e9..0ded297 100644
--- a/demo/split-key.js
+++ b/demo/split-key.js
@@ -39,7 +39,7 @@ function log() {
postMessage({ "cmd": "log", "args": Array.prototype.slice.apply(arguments) });
};
-self.onmessage = function (event) {
+function start() {
var ecparams = getSECCurveByName("secp256k1");
var rng = new SecureRandom();
@@ -73,6 +73,10 @@ self.onmessage = function (event) {
return this.pub = P.multiply(this.d1).getEncoded();
};
+ Alice.prototype.getPubShare = function () {
+ return G.multiply(this.d1);
+ };
+
Bob.prototype.getPubShare = function () {
return G.multiply(this.d2);
};
@@ -87,24 +91,43 @@ self.onmessage = function (event) {
this.z1 = this.k1.modInverse(n);
ff('z1', this.z1);
- var Q1 = G.multiply(this.k1);
- ff('q1', Q1);
+ var Q_1 = G.multiply(this.k1);
+ ff('q1', Q_1);
var alpha = this.paillier.encrypt(this.z1);
- var beta = this.paillier.encrypt(this.d1.multiply(this.z1).mod(n));
-
ff('alpha', alpha);
+
+ var beta = this.paillier.encrypt(this.d1.multiply(this.z1).mod(n));
ff('beta', beta);
- // TODO: Generate a proof that alpha and beta are safe
+ var r_1 = Q_1.getX().toBigInteger().mod(n);
+ var A = this.paillier.encrypt(Bitcoin.ECDSA.getBigRandom(this.paillier.n.divide(n)));
+ ff('A', A);
+ var s_a = this.paillier.multiply(alpha, this.e);
+ var s_b = this.paillier.multiply(beta, r_1);
+ var sigma_1 = this.paillier.addCrypt(this.paillier.addCrypt(s_a, s_b), this.paillier.multiply(A, n));
+ ff('sigma_1', sigma_1);
+
+ var e = Crypto.SHA256(sigma_1.toByteArrayUnsigned(), {asBytes: true});
+ e = BigInteger.fromByteArrayUnsigned(e);
+ var sigma_1n = this.paillier.rerandomize(sigma_1, e);
+ ff('sigma_1n', sigma_1n);
+
+ var s_1 = this.paillier.decrypt(sigma_1n);
+ ff('s_1', s_1);
+ var v_n = this.paillier.decryptR(sigma_1n, s_1);
+ ff('v_n', v_n);
return {
- message: message,
- e: this.e,
- Q1: Q1,
+ Q_1: Q_1,
+ P_1: this.getPubShare(),
alpha: alpha,
beta: beta,
- paillier: this.paillier.pub
+ message: message,
+ paillier: this.paillier.pub,
+ A: A,
+ s_1: s_1,
+ v_n: v_n
};
};
@@ -113,29 +136,46 @@ self.onmessage = function (event) {
// does what we want.
// Throws exception on error
- pkg.Q1.validate();
+ pkg.Q_1.validate();
var hash = Crypto.SHA256(Crypto.SHA256(message, {asBytes: true}), {asBytes: true});
this.e = BigInteger.fromByteArrayUnsigned(hash).mod(n);
- if (!this.e.equals(pkg.e)) {
- throw new Error('We arrived at different values for e.');
- }
-
this.paillier = pkg.paillier;
this.alpha = pkg.alpha;
this.beta = pkg.beta;
+ var r_1 = pkg.Q_1.getX().toBigInteger().mod(n);
+ var testSig = Bitcoin.ECDSA.serializeSig(r_1, pkg.s_1.mod(n));
+ if (!Bitcoin.ECDSA.verify(hash, testSig, pkg.P_1.getEncoded())) {
+ throw new Error('Verification of s1 failed.');
+ }
+
+ // Verify that alpha and beta are valid by generating and verifying sigma_1n
+ var s_a_1 = this.paillier.multiply(this.alpha, this.e);
+ var s_b_1 = this.paillier.multiply(this.beta, r_1);
+ var sigma_1 = this.paillier.addCrypt(this.paillier.addCrypt(s_a_1, s_b_1), this.paillier.multiply(pkg.A, n));
+
+ var e = Crypto.SHA256(sigma_1.toByteArrayUnsigned(), {asBytes: true});
+ e = BigInteger.fromByteArrayUnsigned(e);
+ var sigma_1n = this.paillier.rerandomize(sigma_1, e);
+ ff('sigma_1n_b', sigma_1n);
+
+ var sigma_1_verify = this.paillier.encrypt(pkg.s_1, pkg.v_n);
+ if (!sigma_1n.equals(sigma_1_verify)) {
+ throw new Error('Sigma ciphertext did not match expected value.');
+ }
+
this.k2 = Bitcoin.ECDSA.getBigRandom(n);
ff('k2', this.k2);
this.z2 = this.k2.modInverse(n);
ff('z2', this.z2);
- var Q2 = G.multiply(this.k2);
- ff('q2', Q2);
+ var Q_2 = G.multiply(this.k2);
+ ff('q2', Q_2);
- var Q = pkg.Q1.multiply(this.k2);
+ var Q = pkg.Q_1.multiply(this.k2);
this.r = Q.getX().toBigInteger().mod(n);
ff('r', this.r);
@@ -143,26 +183,26 @@ self.onmessage = function (event) {
throw new Error('r must not be zero.');
}
- var c = Bitcoin.ECDSA.getBigRandom(this.paillier.n.divide(n));
- ff('c', c);
+ var B = Bitcoin.ECDSA.getBigRandom(this.paillier.n.divide(n));
+ ff('B', B);
var p = this.paillier;
var s_a = p.multiply(this.alpha, this.e.multiply(this.z2));
var s_b = p.multiply(this.beta, this.r.multiply(this.d2).multiply(this.z2));
- var sigma = p.add(p.addCrypt(s_a, s_b), c.multiply(n));
+ var sigma = p.add(p.addCrypt(s_a, s_b), B.multiply(n));
ff('sigma', sigma);
return {
- Q2: Q2,
+ Q_2: Q_2,
r: this.r,
sigma: sigma
};
};
Alice.prototype.step3 = function (pkg) {
- pkg.Q2.validate();
+ pkg.Q_2.validate();
- var Q = pkg.Q2.multiply(this.k1);
+ var Q = pkg.Q_2.multiply(this.k1);
this.r = Q.getX().toBigInteger().mod(n);
if (!this.r.equals(pkg.r)) {
@@ -219,11 +259,25 @@ self.onmessage = function (event) {
log("sig :", hex(sig));
log("sig/CHK:", hex(sigChk));
- log("ver :", Bitcoin.ECDSA.verify(hash, sig, pub));
+ var ver = Bitcoin.ECDSA.verify(hash, sig, pub);
+ log("ver :", ver);
log("ver/CHK:", Bitcoin.ECDSA.verify(hash, sigChk, pub));
log("ver/CTL:", Bitcoin.ECDSA.verify(hash, Bitcoin.ECDSA.sign(hash, dChk), pub));
+ ff("result", ver ? "SIGNATURE VALID" : "SIGNATURE INVALID");
var priv = Bitcoin.ECDSA.getBigRandom(n);
pub = G.multiply(priv).getEncoded();
log("ver/GEN:", Bitcoin.ECDSA.verify(hash, Bitcoin.ECDSA.sign(hash, priv), pub));
};
+
+self.onmessage = function (event) {
+ try {
+ start();
+ } catch(e) {
+ var stack = e.stack.replace(/^[^\(]+?[\n$]/gm, '')
+ .replace(/^\s+at\s+/gm, '')
+ .replace(/^Object.<anonymous>\s*\(/gm, '{anonymous}()@')
+ .split('\n');
+ log(e+'\n'+stack);
+ }
+};
diff --git a/src/paillier.js b/src/paillier.js
index 17fada6..32b1735 100644
--- a/src/paillier.js
+++ b/src/paillier.js
@@ -65,6 +65,14 @@ Bitcoin.Paillier = (function () {
return c.modPow(f, this.nSq);
};
+ Paillier.PublicKey.prototype.rerandomize = function (c, r) {
+ if (!r) {
+ var coprimeBitLength = this.n.bitLength() - Math.floor(Math.random()*10);
+ r = new BigInteger(coprimeBitLength, 1, rng);
+ }
+ return c.multiply(r.modPow(this.n, this.nSq)).mod(this.nSq);
+ };
+
Paillier.PrivateKey = function (n,g,l,m,nSq) {
this.l = l;
this.m = m;
@@ -73,12 +81,8 @@ Bitcoin.Paillier = (function () {
this.pub = new Paillier.PublicKey(n,g,this.nSq);
};
- Paillier.PrivateKey.prototype.encrypt = function (m) {
- return this.pub.encrypt(m);
- };
-
Paillier.PrivateKey.prototype.decrypt = function (c) {
- return c.modPow(this.l, this.nSq).mod(this.nSq).subtract(BigInteger.ONE)
+ return c.modPow(this.l, this.nSq).subtract(BigInteger.ONE)
.divide(this.n).multiply(this.m).mod(this.n);
};
@@ -93,5 +97,16 @@ Bitcoin.Paillier = (function () {
return rn.modPow(e, this.n);
};
+ function createProxyMethod(name) {
+ return function () {
+ return this.pub[name].apply(this.pub,
+ Array.prototype.slice.apply(arguments));
+ };
+ };
+ var a = ["add", "addCrypt", "multiply", "rerandomize", "encrypt"];
+ for (var i = 0, l = a.length; i < l; i++) {
+ Paillier.PrivateKey.prototype[a[i]] = createProxyMethod(a[i]);
+ }
+
return Paillier;
})();
diff --git a/test/test.js b/test/test.js
index 010dc5c..8d6e21b 100755
--- a/test/test.js
+++ b/test/test.js
@@ -98,3 +98,15 @@ test("Signing and Verifying", function () {
);
ok(Bitcoin.ECDSA.verify(hash2, sig_c, s2), "Verify constant signature");
});
+
+//
+// Testing Paillier
+// -----------------------------------------------------------------------------
+module("paillier");
+
+test("Classes", function () {
+ expect(3);
+ ok(Bitcoin.Paillier, "Bitcoin.Paillier");
+ ok(Bitcoin.Paillier.PublicKey, "Bitcoin.Paillier.PublicKey");
+ ok(Bitcoin.Paillier.PrivateKey, "Bitcoin.Paillier.PrivateKey");
+});