readme, benchmarks, fix verification

README.md

@@ -0,0 +1,31 @@
+# Riskless
+### Undergraduate dissertation
+
+**Paper: https://judesouthworth.xyz/Dissertation.pdf**
+
+## Running the demo
+
+1. `pip install -r requirements.txt`
+2. `python3 app.py`
+3. Navigate browser to `http://localhost:5000`
+
+### Running the benchmarks
+
+1. Press F12
+2. Type the name of the benchmark to run:
+   - `RSABench`
+   - `PaillierBench`
+   - `ZeroProofBench`
+   - `ZeroProofVerifierBench`
+   - `ZeroProofSizeBench`
+   - `Protocol4Bench`
+   - `Protocol4VerifierBench`
+   - `Protocol4SizeBench`
+3. Wait for results to output in milliseconds
+
+### Changing the key size
+
+The key size is determined by the variable `KEY_SIZE` in `random_primes.js`. This 
+variable dictates the size of _each_ prime used in the modulus, so the total key size
+is twice this value. Recommend selecting one of 512, 1024, or 2048. In fact, I recommend
+selecting 512 if you want the program to run in less than 3-5 business days.

static/js/lz-string.js

@@ -0,0 +1,309 @@
+var LZString = (function () {
+    var r = String.fromCharCode,
+        o = "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/=",
+        n = "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+-$",
+        e = {};
+    function t(r, o) {
+        if (!e[r]) {
+            e[r] = {};
+            for (var n = 0; n < r.length; n++) e[r][r.charAt(n)] = n;
+        }
+        return e[r][o];
+    }
+    var i = {
+        compressToBase64: function (r) {
+            if (null == r) return "";
+            var n = i._compress(r, 6, function (r) {
+                return o.charAt(r);
+            });
+            switch (n.length % 4) {
+                default:
+                case 0:
+                    return n;
+                case 1:
+                    return n + "===";
+                case 2:
+                    return n + "==";
+                case 3:
+                    return n + "=";
+            }
+        },
+        decompressFromBase64: function (r) {
+            return null == r
+                ? ""
+                : "" == r
+                ? null
+                : i._decompress(r.length, 32, function (n) {
+                      return t(o, r.charAt(n));
+                  });
+        },
+        compressToUTF16: function (o) {
+            return null == o
+                ? ""
+                : i._compress(o, 15, function (o) {
+                      return r(o + 32);
+                  }) + " ";
+        },
+        decompressFromUTF16: function (r) {
+            return null == r
+                ? ""
+                : "" == r
+                ? null
+                : i._decompress(r.length, 16384, function (o) {
+                      return r.charCodeAt(o) - 32;
+                  });
+        },
+        compressToUint8Array: function (r) {
+            for (
+                var o = i.compress(r),
+                    n = new Uint8Array(2 * o.length),
+                    e = 0,
+                    t = o.length;
+                e < t;
+                e++
+            ) {
+                var s = o.charCodeAt(e);
+                (n[2 * e] = s >>> 8), (n[2 * e + 1] = s % 256);
+            }
+            return n;
+        },
+        decompressFromUint8Array: function (o) {
+            if (null == o) return i.decompress(o);
+            for (var n = new Array(o.length / 2), e = 0, t = n.length; e < t; e++)
+                n[e] = 256 * o[2 * e] + o[2 * e + 1];
+            var s = [];
+            return (
+                n.forEach(function (o) {
+                    s.push(r(o));
+                }),
+                i.decompress(s.join(""))
+            );
+        },
+        compressToEncodedURIComponent: function (r) {
+            return null == r
+                ? ""
+                : i._compress(r, 6, function (r) {
+                      return n.charAt(r);
+                  });
+        },
+        decompressFromEncodedURIComponent: function (r) {
+            return null == r
+                ? ""
+                : "" == r
+                ? null
+                : ((r = r.replace(/ /g, "+")),
+                  i._decompress(r.length, 32, function (o) {
+                      return t(n, r.charAt(o));
+                  }));
+        },
+        compress: function (o) {
+            return i._compress(o, 16, function (o) {
+                return r(o);
+            });
+        },
+        _compress: function (r, o, n) {
+            if (null == r) return "";
+            var e,
+                t,
+                i,
+                s = {},
+                u = {},
+                a = "",
+                p = "",
+                c = "",
+                l = 2,
+                f = 3,
+                h = 2,
+                d = [],
+                m = 0,
+                v = 0;
+            for (i = 0; i < r.length; i += 1)
+                if (
+                    ((a = r.charAt(i)),
+                    Object.prototype.hasOwnProperty.call(s, a) ||
+                        ((s[a] = f++), (u[a] = !0)),
+                    (p = c + a),
+                    Object.prototype.hasOwnProperty.call(s, p))
+                )
+                    c = p;
+                else {
+                    if (Object.prototype.hasOwnProperty.call(u, c)) {
+                        if (c.charCodeAt(0) < 256) {
+                            for (e = 0; e < h; e++)
+                                (m <<= 1),
+                                    v == o - 1 ? ((v = 0), d.push(n(m)), (m = 0)) : v++;
+                            for (t = c.charCodeAt(0), e = 0; e < 8; e++)
+                                (m = (m << 1) | (1 & t)),
+                                    v == o - 1 ? ((v = 0), d.push(n(m)), (m = 0)) : v++,
+                                    (t >>= 1);
+                        } else {
+                            for (t = 1, e = 0; e < h; e++)
+                                (m = (m << 1) | t),
+                                    v == o - 1 ? ((v = 0), d.push(n(m)), (m = 0)) : v++,
+                                    (t = 0);
+                            for (t = c.charCodeAt(0), e = 0; e < 16; e++)
+                                (m = (m << 1) | (1 & t)),
+                                    v == o - 1 ? ((v = 0), d.push(n(m)), (m = 0)) : v++,
+                                    (t >>= 1);
+                        }
+                        0 == --l && ((l = Math.pow(2, h)), h++), delete u[c];
+                    } else
+                        for (t = s[c], e = 0; e < h; e++)
+                            (m = (m << 1) | (1 & t)),
+                                v == o - 1 ? ((v = 0), d.push(n(m)), (m = 0)) : v++,
+                                (t >>= 1);
+                    0 == --l && ((l = Math.pow(2, h)), h++),
+                        (s[p] = f++),
+                        (c = String(a));
+                }
+            if ("" !== c) {
+                if (Object.prototype.hasOwnProperty.call(u, c)) {
+                    if (c.charCodeAt(0) < 256) {
+                        for (e = 0; e < h; e++)
+                            (m <<= 1),
+                                v == o - 1 ? ((v = 0), d.push(n(m)), (m = 0)) : v++;
+                        for (t = c.charCodeAt(0), e = 0; e < 8; e++)
+                            (m = (m << 1) | (1 & t)),
+                                v == o - 1 ? ((v = 0), d.push(n(m)), (m = 0)) : v++,
+                                (t >>= 1);
+                    } else {
+                        for (t = 1, e = 0; e < h; e++)
+                            (m = (m << 1) | t),
+                                v == o - 1 ? ((v = 0), d.push(n(m)), (m = 0)) : v++,
+                                (t = 0);
+                        for (t = c.charCodeAt(0), e = 0; e < 16; e++)
+                            (m = (m << 1) | (1 & t)),
+                                v == o - 1 ? ((v = 0), d.push(n(m)), (m = 0)) : v++,
+                                (t >>= 1);
+                    }
+                    0 == --l && ((l = Math.pow(2, h)), h++), delete u[c];
+                } else
+                    for (t = s[c], e = 0; e < h; e++)
+                        (m = (m << 1) | (1 & t)),
+                            v == o - 1 ? ((v = 0), d.push(n(m)), (m = 0)) : v++,
+                            (t >>= 1);
+                0 == --l && ((l = Math.pow(2, h)), h++);
+            }
+            for (t = 2, e = 0; e < h; e++)
+                (m = (m << 1) | (1 & t)),
+                    v == o - 1 ? ((v = 0), d.push(n(m)), (m = 0)) : v++,
+                    (t >>= 1);
+            for (;;) {
+                if (((m <<= 1), v == o - 1)) {
+                    d.push(n(m));
+                    break;
+                }
+                v++;
+            }
+            return d.join("");
+        },
+        decompress: function (r) {
+            return null == r
+                ? ""
+                : "" == r
+                ? null
+                : i._decompress(r.length, 32768, function (o) {
+                      return r.charCodeAt(o);
+                  });
+        },
+        _decompress: function (o, n, e) {
+            var t,
+                i,
+                s,
+                u,
+                a,
+                p,
+                c,
+                l = [],
+                f = 4,
+                h = 4,
+                d = 3,
+                m = "",
+                v = [],
+                g = { val: e(0), position: n, index: 1 };
+            for (t = 0; t < 3; t += 1) l[t] = t;
+            for (s = 0, a = Math.pow(2, 2), p = 1; p != a; )
+                (u = g.val & g.position),
+                    (g.position >>= 1),
+                    0 == g.position && ((g.position = n), (g.val = e(g.index++))),
+                    (s |= (u > 0 ? 1 : 0) * p),
+                    (p <<= 1);
+            switch (s) {
+                case 0:
+                    for (s = 0, a = Math.pow(2, 8), p = 1; p != a; )
+                        (u = g.val & g.position),
+                            (g.position >>= 1),
+                            0 == g.position && ((g.position = n), (g.val = e(g.index++))),
+                            (s |= (u > 0 ? 1 : 0) * p),
+                            (p <<= 1);
+                    c = r(s);
+                    break;
+                case 1:
+                    for (s = 0, a = Math.pow(2, 16), p = 1; p != a; )
+                        (u = g.val & g.position),
+                            (g.position >>= 1),
+                            0 == g.position && ((g.position = n), (g.val = e(g.index++))),
+                            (s |= (u > 0 ? 1 : 0) * p),
+                            (p <<= 1);
+                    c = r(s);
+                    break;
+                case 2:
+                    return "";
+            }
+            for (l[3] = c, i = c, v.push(c); ; ) {
+                if (g.index > o) return "";
+                for (s = 0, a = Math.pow(2, d), p = 1; p != a; )
+                    (u = g.val & g.position),
+                        (g.position >>= 1),
+                        0 == g.position && ((g.position = n), (g.val = e(g.index++))),
+                        (s |= (u > 0 ? 1 : 0) * p),
+                        (p <<= 1);
+                switch ((c = s)) {
+                    case 0:
+                        for (s = 0, a = Math.pow(2, 8), p = 1; p != a; )
+                            (u = g.val & g.position),
+                                (g.position >>= 1),
+                                0 == g.position &&
+                                    ((g.position = n), (g.val = e(g.index++))),
+                                (s |= (u > 0 ? 1 : 0) * p),
+                                (p <<= 1);
+                        (l[h++] = r(s)), (c = h - 1), f--;
+                        break;
+                    case 1:
+                        for (s = 0, a = Math.pow(2, 16), p = 1; p != a; )
+                            (u = g.val & g.position),
+                                (g.position >>= 1),
+                                0 == g.position &&
+                                    ((g.position = n), (g.val = e(g.index++))),
+                                (s |= (u > 0 ? 1 : 0) * p),
+                                (p <<= 1);
+                        (l[h++] = r(s)), (c = h - 1), f--;
+                        break;
+                    case 2:
+                        return v.join("");
+                }
+                if ((0 == f && ((f = Math.pow(2, d)), d++), l[c])) m = l[c];
+                else {
+                    if (c !== h) return null;
+                    m = i + i.charAt(0);
+                }
+                v.push(m),
+                    (l[h++] = i + m.charAt(0)),
+                    (i = m),
+                    0 == --f && ((f = Math.pow(2, d)), d++);
+            }
+        },
+    };
+    return i;
+})();
+"function" == typeof define && define.amd
+    ? define(function () {
+          return LZString;
+      })
+    : "undefined" != typeof module && null != module
+    ? (module.exports = LZString)
+    : "undefined" != typeof angular &&
+      null != angular &&
+      angular.module("LZString", []).factory("LZString", function () {
+          return LZString;
+      });

static/js/modules/crypto/paillier.js

@@ -4,6 +4,25 @@ import { gcd, mod_exp } from "./math.js";
 const PAILLIER = 0;
 const JURIK = 1;
 
+function RSTransform(g, a, p) {
+    let plainText = p.toString(16);
+    if (plainText.length % 2 !== 0) {
+        plainText = "0" + plainText;
+    }
+
+    let aStr = a.toString(16);
+    if (aStr.length % 2 !== 0) {
+        aStr = "0" + aStr;
+    }
+
+    let hasher = new jsSHA("SHAKE256", "HEX");
+    hasher.update(g.toString(16));
+    hasher.update(plainText);
+    hasher.update(aStr);
+
+    return BigInt("0x" + hasher.getHash("HEX", { outputLen: 2048 }));
+}
+
 class Ciphertext {
     constructor(key, plainText, r, set) {
         if (set !== undefined) {
@@ -93,28 +112,12 @@ class Ciphertext {
         }
 
         let a = mod_exp(rp, this.pubKey.n, this.pubKey.n2);
-        let hasher = new jsSHA("SHAKE256", "HEX");
 
-        let plainText = this.plainText.toString(16);
+        let challenge = RSTransform(this.pubKey.g, a, this.plainText);
-        if (plainText.length % 2 !== 0) {
-            plainText = "0" + plainText;
-        }
-
-        let aStr = a.toString(16);
-        if (aStr.length % 2 !== 0) {
-            aStr = "0" + aStr;
-        }
-
-        hasher.update(this.pubKey.g.toString(16));
-        hasher.update(plainText);
-        hasher.update(aStr);
-
-        let challenge = BigInt("0x" + hasher.getHash("HEX", { outputLen: 2048 }));
 
         return {
             plainText: "0x" + this.plainText.toString(16),
             a: "0x" + a.toString(16),
-            challenge: "0x" + challenge.toString(16),
             proof:
                 "0x" +
                 (
@@ -195,11 +198,17 @@ export class ReadOnlyCiphertext {
     }
 
     verifyNI(statement) {
+        let challenge = RSTransform(
+            this.pubKey.g,
+            BigInt(statement.a),
+            BigInt(statement.plainText)
+        );
+
         let verifier = new ValueProofSessionVerifier(
             this,
             BigInt(statement.plainText),
             BigInt(statement.a),
-            BigInt(statement.challenge)
+            challenge
         );
 
         if (verifier.verify(BigInt(statement.proof))) {

static/js/modules/crypto/random_primes.js

@@ -1,6 +1,6 @@
 import { mod_exp } from "./math.js";
 
-export const KEY_SIZE = 512;
+export const KEY_SIZE = 2048;
 
 export function cryptoRandom(bits) {
     if (bits === undefined) {

static/js/modules/interface/proofs.js

@@ -135,7 +135,6 @@ export function verifyRegions(obj, key) {
                 let plaintext = c.verifyNI(verification.zeroProofs[regionName]);
 
                 if (plaintext !== 0n) {
-                    console.log(plaintext);
                     return false;
                 }
             }

templates/index.html

@@ -7,7 +7,8 @@
 
     <script src="https://cdn.socket.io/4.5.4/socket.io.min.js"></script>
     <script src="https://cdnjs.cloudflare.com/ajax/libs/crypto-js/4.1.1/crypto-js.min.js"></script>
-    <script src="{{ url_for('static', filename='js/sha3.js') }}" type="module"></script>
+    <script src="{{ url_for('static', filename='js/lz-string.js') }}"></script>
+    <script src="{{ url_for('static', filename='js/sha3.js') }}"></script>
     <script src="{{ url_for('static', filename='js/modules/interface/main.js') }}" type="module"></script>
     <script src="{{ url_for('static', filename='js/modules/crypto/main.js') }}" type="module"></script>
 </head>
@@ -188,6 +189,8 @@
         function RSABench() {
             console.log("Warming up")
 
+            const ROUNDS = 250;
+
             for (let i = 0n; i < 100n; i++) {
                 window.rsa.pubKey.encrypt(i);
             }
@@ -195,17 +198,19 @@
             console.log("Benching")
 
             performance.mark("rsa-start")
-            for (let i = 0n; i < 250n; i++) {
+            for (let i = 0n; i < BigInt(ROUNDS); i++) {
                 window.rsa.pubKey.encrypt(i);
             }
             performance.mark("rsa-end")
 
-            console.log(`Bench done. Duration: ${performance.measure("rsa-duration", "rsa-start", "rsa-end").duration}`)
+            console.log(`Bench done. Time per encrypt: ${performance.measure("rsa-duration", "rsa-start", "rsa-end").duration / ROUNDS}`)
         }
 
         function PaillierBench() {
             console.log("Warming up")
 
+            const ROUNDS = 250
+
             for (let i = 0n; i < 100n; i++) {
                 window.paillier.pubKey.encrypt(i);
             }
@@ -213,12 +218,162 @@
             console.log("Benching")
 
             performance.mark("paillier-start")
-            for (let i = 0n; i < 250n; i++) {
+            for (let i = 0n; i < BigInt(ROUNDS); i++) {
                 window.paillier.pubKey.encrypt(i);
             }
             performance.mark("paillier-end")
 
-            console.log(`Bench done. Duration: ${performance.measure("paillier-duration", "paillier-start", "paillier-end").duration}`)
+            console.log(`Bench done. Time per encrypt: ${performance.measure("paillier-duration", "paillier-start", "paillier-end").duration / ROUNDS}`)
+        }
+
+        function ZeroProofBench() {
+            console.log("Warming up")
+
+            const cipherText = paillier.pubKey.encrypt(0n)
+            const ROUNDS = 20;
+
+            for (let i = 0; i < 5; i++) {
+                cipherText.proveNI();
+            }
+
+            console.log("Benching")
+
+            performance.mark("paillier-start")
+            for (let i = 0; i < ROUNDS; i++) {
+                cipherText.proveNI()
+            }
+            performance.mark("paillier-end")
+
+            console.log(`Bench done. Time per proof: ${performance.measure("paillier-duration", "paillier-start", "paillier-end").duration / ROUNDS}`)
+        }
+
+        function ZeroProofVerifierBench() {
+            console.log("Warming up")
+            const ROUNDS = 20
+
+            const cipherText = paillier.pubKey.encrypt(1n)
+            const readOnly = cipherText.asReadOnlyCiphertext();
+            const proof = cipherText.proveNI()
+
+            for (let i = 0; i < 5; i++) {
+                readOnly.verifyNI(proof)
+            }
+
+            console.log("Benching")
+
+            performance.mark("paillier-start")
+            for (let i = 0; i < ROUNDS; i++) {
+                readOnly.verifyNI(proof)
+            }
+            performance.mark("paillier-end")
+
+            console.log(`Bench done. Time per verification: ${performance.measure("paillier-duration", "paillier-start", "paillier-end").duration / ROUNDS}`)
+        }
+
+        function Protocol4Bench() {
+            console.log("Warming up")
+
+            const regions = {
+                A: paillier.pubKey.encrypt(0n),
+                B: paillier.pubKey.encrypt(1n),
+                C: paillier.pubKey.encrypt(0n),
+                D: paillier.pubKey.encrypt(0n),
+                E: paillier.pubKey.encrypt(0n)
+            }
+            const ROUNDS = 20;
+
+            for (let i = 0; i < 5; i++) {
+                proveRegions(regions)
+            }
+
+            console.log("Benching")
+
+            performance.mark("paillier-start")
+            for (let i = 0; i < ROUNDS; i++) {
+                proveRegions(regions)
+            }
+            performance.mark("paillier-end")
+
+            console.log(`Bench done. Time per proof: ${performance.measure("paillier-duration", "paillier-start", "paillier-end").duration / ROUNDS}`)
+        }
+
+        function Protocol4VerifierBench() {
+            console.log("Warming up")
+
+            const ROUNDS = 20;
+            const regions = {
+                A: paillier.pubKey.encrypt(0n),
+                B: paillier.pubKey.encrypt(1n),
+                C: paillier.pubKey.encrypt(0n),
+                D: paillier.pubKey.encrypt(0n),
+                E: paillier.pubKey.encrypt(0n)
+            }
+            let proof = proveRegions(regions)
+
+            for (let i = 0; i < 5; i++) {
+                verifyRegions(proof, paillier.pubKey)
+            }
+
+            console.log("Benching")
+
+            performance.mark("paillier-start")
+            for (let i = 0; i < ROUNDS; i++) {
+                verifyRegions(proof, paillier.pubKey)
+            }
+            performance.mark("paillier-end")
+
+            console.log(`Bench done. Time per verification: ${performance.measure("paillier-duration", "paillier-start", "paillier-end").duration / ROUNDS}`)
+        }
+
+        // https://gist.github.com/kawanet/352a2ed1d1656816b2bc
+        function string_to_buffer(src) {
+          return (new Uint16Array([].map.call(src, function(c) {
+            return c.charCodeAt(0)
+          }))).buffer;
+        }
+
+        function Protocol4Size() {
+            const regions = {
+                A: paillier.pubKey.encrypt(0n),
+                B: paillier.pubKey.encrypt(1n),
+                C: paillier.pubKey.encrypt(0n),
+                D: paillier.pubKey.encrypt(0n),
+                E: paillier.pubKey.encrypt(0n)
+            }
+            let ROUNDS  = 10;
+
+            let size = 0;
+            let compressedSize = 0;
+
+            for (let x = 0; x < ROUNDS; x++) {
+                let s = JSON.stringify(proveRegions(regions));
+                size += string_to_buffer(s).byteLength;
+                compressedSize += LZString.compressToUint8Array(s).length;
+            }
+
+            return {
+                size: size / ROUNDS,
+                compressedSize: compressedSize / ROUNDS
+            };
+        }
+
+        function ZeroProofSize() {
+            const ROUNDS = 100;
+            const cipherText = paillier.pubKey.encrypt(0n)
+
+            let size = 0;
+            let compressedSize = 0;
+
+            for (let x = 0; x < ROUNDS; x++) {
+                let s = JSON.stringify(cipherText.proveNI());
+                size += string_to_buffer(s).byteLength;
+                compressedSize += LZString.compressToUint8Array(s).length;
+            }
+
+            return {
+                size: size / ROUNDS,
+                compressedSize: compressedSize / ROUNDS
+            };
         }
     </script>
 </body>

whitepaper/Dissertation.tex

@@ -8,6 +8,11 @@
 \usepackage{amsthm}
 \usepackage{tikz}
 \usepackage{minted}
+\usepackage{multirow}
+\usepackage{tabularx}
+\usepackage{booktabs}
+\usepackage{ragged2e}
+\usepackage[alph]{parnotes}
 
 \DeclareMathOperator{\lcm}{lcm}
 \DeclareMathOperator{\id}{id}
@@ -598,18 +603,52 @@ The other proofs do not translate so trivially to this structure however. In fac
 
 \subsection{Complexity results}
 
-All measurements taken on Brave 1.50.114 (Chromium 112.0.5615.49) 64-bit, using a Ryzen 5 3600 CPU.
+All measurements taken on Brave 1.50.114 (Chromium 112.0.5615.49) 64-bit, using a Ryzen 5 3600 CPU. Absolute timings are extremely dependent on the browser engine: for example Firefox 111.0.1 was typically 4 times slower than the results shown.
 
-\begin{center}
+\begin{table}[htp]
-	\begin{tabular}{|c|c|c|c|}
+	\fontsize{10pt}{10pt}\selectfont
-		\hline
+	\caption{Time to encrypt}
-		Modulus size & Na\"ive encrypt & Jacobi encrypt & RSA encrypt \\\hline
+	\begin{tabularx}{\textwidth}{c *3{>{\Centering}X}}
+		\toprule
+		Modulus size & Na\"ive encrypt & Jacobi encrypt & RSA encrypt \\
+		\midrule
 		$|n| = 1024$   & 6ms & 4ms & <1ms \\  
 		$|n| = 2048$   & 34ms & 22ms & <1ms \\
 		$|n| = 4096$   & 189ms & 128ms & <1ms \\
-		\hline
+		\bottomrule
-	\end{tabular}
+	\end{tabularx}
-\end{center}
+\end{table}
+
+\begin{table}[htp]
+	\fontsize{10pt}{10pt}\selectfont
+	\caption{Time to process proofs}
+	\begin{tabularx}{\textwidth}{c *4{>{\Centering}X}}
+		\toprule
+		\multirow{2}{*}{Modulus size} & \multicolumn{2}{c}{Proof-of-zero non-interactive} & \multicolumn{2}{c}{\hyperref[protocol1]{Protocol~\ref*{protocol1}} with $t = 24$} \tabularnewline \cmidrule(l){2-3}\cmidrule(l){4-5}
+		& Prover & Verifier & Prover & Verifier \\
+		\midrule
+		$|n| = 1024$ & 10ms & 18ms & 1,740ms & 2,190ms \\
+		$|n| = 2048$ & 44ms & 68ms & 8,170ms & 8,421ms \\
+		$|n| = 4096$ & 225ms & 292ms & 41,500ms & 34,405ms \\
+		\bottomrule
+	\end{tabularx}
+\end{table}
+
+\begin{table}[htp]
+	\fontsize{10pt}{10pt}\selectfont
+	\caption{Byte size\parnote{1 UTF-16 character, as used by ECMAScript, is 2 or more bytes} of encoded proofs}
+	\begin{tabularx}{\textwidth}{c *4{>{\Centering}X}}
+		\toprule
+		\multirow{2}{*}{Modulus size} & \multicolumn{2}{c}{Proof-of-zero non-interactive} & \multicolumn{2}{c}{\hyperref[protocol1]{Protocol~\ref*{protocol1}} with $t = 24$} \tabularnewline \cmidrule(l){2-3}\cmidrule(l){4-5}
+		& JSON & with LZ-String & JSON & with LZ-String \\
+		\midrule
+		$|n| = 1024$ & 1,617B & 576B (35.62\%) & 338,902B & 95,738B (28.25\%) \\
+		$|n| = 2048$ & 3,153B & 1,050B (33.30\%) & 662,233B & 187,333B (28.29\%) \\
+		$|n| = 4096$ & 6,226B & 1,999B (32.11\%) & 1,315,027B & 368,646B (28.03\%) \\
+		\bottomrule
+	\end{tabularx}
+	\parnotes
+\end{table}
 
 \subsection{Quantum resistance}