From bf96267f89d18f278e78cf02c97ab1e7513fb871 Mon Sep 17 00:00:00 2001 From: Ian Coleman Date: Thu, 1 Oct 2020 23:44:38 +0000 Subject: Remove bias from entropy in base 6 and base 10 --- src/index.html | 2 +- src/js/entropy.js | 345 +++++++++++++++++++++++++----------------------------- src/js/index.js | 8 +- 3 files changed, 165 insertions(+), 190 deletions(-) (limited to 'src') diff --git a/src/index.html b/src/index.html index 7eb123c..f66d4ed 100644 --- a/src/index.html +++ b/src/index.html @@ -86,7 +86,7 @@
- +
diff --git a/src/js/entropy.js b/src/js/entropy.js index 62b2711..3b62e10 100644 --- a/src/js/entropy.js +++ b/src/js/entropy.js @@ -16,7 +16,136 @@ window.Entropy = new (function() { - var TWO = new libs.BigInteger.BigInteger(2); + let eventBits = { + + "binary": { + "0": "0", + "1": "1", + }, + + // log2(6) = 2.58496 bits per roll, with bias + // 4 rolls give 2 bits each + // 2 rolls give 1 bit each + // Average (4*2 + 2*1) / 6 = 1.66 bits per roll without bias + "base 6": { + "0": "00", + "1": "01", + "2": "10", + "3": "11", + "4": "0", + "5": "1", + }, + + // log2(6) = 2.58496 bits per roll, with bias + // 4 rolls give 2 bits each + // 2 rolls give 1 bit each + // Average (4*2 + 2*1) / 6 = 1.66 bits per roll without bias + "base 6 (dice)": { + "0": "00", // equivalent to 0 in base 6 + "1": "01", + "2": "10", + "3": "11", + "4": "0", + "5": "1", + }, + + // log2(10) = 3.321928 bits per digit, with bias + // 8 digits give 3 bits each + // 2 digits give 1 bit each + // Average (8*3 + 2*1) / 10 = 2.6 bits per digit without bias + "base 10": { + "0": "000", + "1": "001", + "2": "010", + "3": "011", + "4": "100", + "5": "101", + "6": "110", + "7": "111", + "8": "0", + "9": "1", + }, + + "hexadecimal": { + "0": "0000", + "1": "0001", + "2": "0010", + "3": "0011", + "4": "0100", + "5": "0101", + "6": "0110", + "7": "0111", + "8": "1000", + "9": "1001", + "a": "1010", + "b": "1011", + "c": "1100", + "d": "1101", + "e": "1110", + "f": "1111", + }, + + // log2(52) = 5.7004 bits per card, with bias + // 32 cards give 5 bits each + // 16 cards give 4 bits each + // 4 cards give 2 bits each + // Average (32*5 + 16*4 + 4*2) / 52 = 4.46 bits per card without bias + "card": { + "ac": "00000", + "2c": "00001", + "3c": "00010", + "4c": "00011", + "5c": "00100", + "6c": "00101", + "7c": "00110", + "8c": "00111", + "9c": "01000", + "tc": "01001", + "jc": "01010", + "qc": "01011", + "kc": "01100", + "ad": "01101", + "2d": "01110", + "3d": "01111", + "4d": "10000", + "5d": "10001", + "6d": "10010", + "7d": "10011", + "8d": "10100", + "9d": "10101", + "td": "10110", + "jd": "10111", + "qd": "11000", + "kd": "11001", + "ah": "11010", + "2h": "11011", + "3h": "11100", + "4h": "11101", + "5h": "11110", + "6h": "11111", + "7h": "0000", + "8h": "0001", + "9h": "0010", + "th": "0011", + "jh": "0100", + "qh": "0101", + "kh": "0110", + "as": "0111", + "2s": "1000", + "3s": "1001", + "4s": "1010", + "5s": "1011", + "6s": "1100", + "7s": "1101", + "8s": "1110", + "9s": "1111", + "ts": "00", + "js": "01", + "qs": "10", + "ks": "11", + }, + + } // matchers returns an array of the matched events for each type of entropy. // eg @@ -51,48 +180,28 @@ window.Entropy = new (function() { } } - // Convert array of cards from ["ac", "4d", "ks"] - // to numbers between 0 and 51 [0, 16, 51] - function convertCardsToInts(cards) { - var ints = []; - var values = "a23456789tjqk"; - var suits = "cdhs"; - for (var i=0; i -1) { - newParts[i] = base.parts[i]; - newInts[i] = base.ints[i]; + newEvents[i] = base.events[i]; } else { - newParts[i] = "0"; - newInts[i] = 0; + newEvents[i] = "0"; } } base.str = "base 6 (dice)"; - base.ints = newInts; - base.parts = newParts; + base.events = newEvents; base.matcher = matchers.base6; } // Detect empty entropy - if (base.parts.length == 0) { + if (base.events.length == 0) { return { binaryStr: "", cleanStr: "", @@ -100,44 +209,23 @@ window.Entropy = new (function() { base: base, }; } - // Convert base.ints to BigInteger. - // Due to using unusual bases, eg cards of base52, this is not as simple as - // using BigInteger.parse() - var entropyInt = libs.BigInteger.BigInteger.ZERO; - for (var i=base.ints.length-1; i>=0; i--) { - var thisInt = libs.BigInteger.BigInteger.parse(base.ints[i]); - var power = (base.ints.length - 1) - i; - var additionalEntropy = libs.BigInteger.BigInteger.parse(base.asInt).pow(power).multiply(thisInt); - entropyInt = entropyInt.add(additionalEntropy); - } - // Convert entropy to binary - var entropyBin = entropyInt.toString(2); - // If the first integer is small, it must be padded with zeros. - // Otherwise the chance of the first bit being 1 is 100%, which is - // obviously incorrect. - // This is not perfect for non-2^n bases. - var expectedBits = Math.floor(base.parts.length * Math.log2(base.asInt)); - while (entropyBin.length < expectedBits) { - entropyBin = "0" + entropyBin; - } - // Calculate the number of bits per event - var bitsPerEvent = Math.log2(base.asInt); - // Cards binary must be handled differently, since they're not replaced - if (base.asInt == 52) { - var cardEntropy = processCardEntropy(base.parts); - entropyBin = cardEntropy.binaryStr; - bitsPerEvent = cardEntropy.bitsPerEvent; - } + // Convert entropy events to binary + var entropyBin = base.events.map(function(e) { + return eventBits[base.str][e.toLowerCase()]; + }).join(""); + // Get average bits per event + // which may be adjusted for bias if log2(base) is fractional + var bitsPerEvent = base.bitsPerEvent; // Supply a 'filtered' entropy string for display purposes - var entropyClean = base.parts.join(""); - var entropyHtml = base.parts.join(""); + var entropyClean = base.events.join(""); + var entropyHtml = base.events.join(""); if (base.asInt == 52) { - entropyClean = base.parts.join(" ").toUpperCase(); + entropyClean = base.events.join(" ").toUpperCase(); entropyClean = entropyClean.replace(/C/g, "\u2663"); entropyClean = entropyClean.replace(/D/g, "\u2666"); entropyClean = entropyClean.replace(/H/g, "\u2665"); entropyClean = entropyClean.replace(/S/g, "\u2660"); - entropyHtml = base.parts.join(" ").toUpperCase(); + entropyHtml = base.events.join(" ").toUpperCase(); entropyHtml = entropyHtml.replace(/C/g, "\u2663"); entropyHtml = entropyHtml.replace(/D/g, "\u2666"); entropyHtml = entropyHtml.replace(/H/g, "\u2665"); @@ -154,18 +242,6 @@ window.Entropy = new (function() { return e; } - function getSortedDeck() { - var s = []; - var suits = "CDHS"; - var values = "A23456789TJQK"; - for (var i=0; i= hexMatches.length / 2 && autodetect) || baseStr === "card") { - var ints = convertCardsToInts(cardMatches); return { ints: ints, - parts: cardMatches, + events: cardMatches, matcher: matchers.card, asInt: 52, + bitsPerEvent: (32*5 + 16*4 + 4*2) / 52, // see cardBits str: "card", } } @@ -199,9 +276,10 @@ window.Entropy = new (function() { var ints = diceMatches.map(function(i) { return parseInt(i) }); return { ints: ints, - parts: diceMatches, + events: diceMatches, matcher: matchers.dice, asInt: 6, + bitsPerEvent: (4*2 + 2*1) / 6, // see diceBits str: "dice", } } @@ -210,9 +288,10 @@ window.Entropy = new (function() { var ints = base6Matches.map(function(i) { return parseInt(i) }); return { ints: ints, - parts: base6Matches, + events: base6Matches, matcher: matchers.base6, asInt: 6, + bitsPerEvent: (4*2 + 2*1) / 6, // see diceBits str: "base 6", } } @@ -221,126 +300,22 @@ window.Entropy = new (function() { var ints = base10Matches.map(function(i) { return parseInt(i) }); return { ints: ints, - parts: base10Matches, + events: base10Matches, matcher: matchers.base10, asInt: 10, + bitsPerEvent: (8*3 + 2*1) / 10, // see b10Bits str: "base 10", } } var ints = hexMatches.map(function(i) { return parseInt(i, 16) }); return { ints: ints, - parts: hexMatches, + events: hexMatches, matcher: matchers.hex, asInt: 16, + bitsPerEvent: 4, str: "hexadecimal", } } - // Assume cards are NOT replaced. - // Additional entropy decreases as more cards are used. This means - // total possible entropy is measured using n!, not base^n. - // eg the second last card can be only one of two, not one of fifty two - // so the added entropy for that card is only one bit at most - function processCardEntropy(cards) { - // Track how many instances of each card have been used, and thus - // how many decks are in use. - var cardCounts = {}; - var numberOfDecks = 0; - // Work out number of decks by max(duplicates) - for (var i=0; i numberOfDecks) { - numberOfDecks = cardCounts[card]; - } - } - // Work out the total number of bits for this many decks - // See http://crypto.stackexchange.com/q/41886 - var gainedBits = 0; - // Equivalent of Math.log2(factorial(52*numberOfDecks)) - // which becomes infinity for numberOfDecks > 4 - for (var i=1; i<=52*numberOfDecks; i++) { - gainedBits = gainedBits + Math.log2(i); - } - var lostBits = 52 * Math.log2(factorial(numberOfDecks)); - var maxBits = gainedBits - lostBits; - // Convert the drawn cards to a binary representation. - // The exact technique for doing this is unclear. - // See - // http://crypto.stackexchange.com/a/41896 - // "I even doubt that this is well defined (only the average entropy - // is, I believe)." - // See - // https://github.com/iancoleman/bip39/issues/33#issuecomment-263021856 - // "The binary representation can be the first log(permutations,2) bits - // of the sha-2 hash of the normalized deck string." - // - // In this specific implementation, the first N bits of the hash of the - // normalized cards string is being used. Uppercase, no spaces; eg - // sha256("AH8DQSTC2H") - var totalCards = numberOfDecks * 52; - var percentUsed = cards.length / totalCards; - // Calculate the average number of bits of entropy for the number of - // cards drawn. - var numberOfBits = Math.floor(maxBits * percentUsed); - // Create a normalized string of the selected cards - var normalizedCards = cards.join("").toUpperCase(); - // Convert to binary using the SHA256 hash of the normalized cards. - // If the number of bits is more than 256, multiple hashes - // are used until the required number of bits is reached. - var entropyBin = ""; - var iterations = 0; - while (entropyBin.length < numberOfBits) { - var hashedCards = sjcl.hash.sha256.hash(normalizedCards + ":" + iterations); - var hashHex = sjcl.codec.hex.fromBits(hashedCards); - for (var i=0; i