tokens
sequence
label_ids
sequence
[ "The", "two", "unsupervised", "algorithms", "oSVM", "and", "SGD", "-", "oSVM", "are", "better", "generalised", "than", "Isolation", "Forest", ",", "even", "though", "their", "single", "domain", "performance", "is", "weaker", "." ]
[ 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ]
[ "Supervised", "vs", "Unsupervised", "Learning", "Models", ":", "%" ]
[ 0, 0, 0, 0, 1, 0, 0 ]
[ "It", "shows", "the", "average", "performance", "decay", "for", "each", "unsupervised", "learning", "model", "when", "evaluated", "on", "three", "datasets", "not", "used", "for", "the", "training", "compared", "to", "evaluation", "on", "the", "training", "dataset", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Explaining", "Generalisability" ]
[ 0, 1 ]
[ "The", "superiority", "of", "unsupervised", "learning", "models", "is", "consistent", "across", "all", "source", "domains", ",", "i.e.", ",", "the", "average", "per", "source", "domain", "decay", "(", "last", "column", "of", "Tables", "~", "and", "~", ")", "is", "significantly", "lower", "for", "unsupervised", "learning", "models", "for", "all", "the", "four", "datasets", "." ]
[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ]
[ "The", "unsupervised", "learning", "models", "generalise", "better", "with", "an", "overall", "average", "performance", "decay", "of", "28.33\\%", ",", "compared", "to", "the", "supervised", "learning", "models", "with", "an", "overall", "average", "performance", "decay", "of", "56.28\\%", "." ]
[ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "There", "is", "an", "average", "performance", "decay", "of", "28.33\\%", "when", "an", "unsupervised", "learning", "model", "is", "evaluated", "on", "a", "dataset", "other", "than", "its", "training", "dataset", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Using", "the", "chaining", "construction", ",", "one", "can", "extend", "the", "delays", "constructed", "via", "cubing", "." ]
[ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "Long", "delays", "achieved", "via", "long", "chains", "are", "still", "impractical", "since", "this", "chaining", "construction", "does", "not", "increase", "the", "ratio", "between", "encryption", "time", "and", "decryption", "time", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Among", "the", "few", "candidates", "we", "presented", ",", "we", "believe", "that", "using", "FPE", "methods", "with", "standard", "block", "ciphers", "is", "the", "most", "balanced", "option", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ]
[ "Card", "shuffling", "schemes", "should", "be", "used", "when", "the", "highest", "degree", "of", "security", "is", "needed", "." ]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Moreover", ",", "the", "bit", "sampling", "present", "in", "the", "shuffles", "should", "depend", "on", "the", "entire", "internal", "state", "of", "the", "algorithm", "." ]
[ 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "This", "is", "likely", "to", "result", "in", "the", "most", "unpredictable", "permutations", ",", "since", "neither", "the", "keys", "nor", "the", "coin", "flips", "can", "be", "computed", "before", "they", "are", "required", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Despite", "their", "frequent", "use", ",", "the", "algebraic", "methods", "represent", "the", "least", "secure", "option", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ]
[ "Since", "chaining", "can", "be", "done", "securely", ",", "in", "this", "section", "we", "analyse", "possible", "attacks", "to", "the", "cubing", "operation", "." ]
[ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "We", "focus", "on", "four", "different", "attack", "vectors", ":", "exponentiation", ",", "modular", "reduction", ",", "integer", "multiplication", "and", "hardware", "performance", "." ]
[ 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0 ]
[ "Chaining", "the", "delays", "%" ]
[ 1, 0, 0, 0 ]
[ "In", "the", "previous", "section", ",", "we", "showed", "how", "to", "construct", "a", "simple", "delay", "encryption", "based", "on", "cubing", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0 ]
[ "As", "we", "will", "discuss", "in", "Section", ",", "the", "larger", "the", "prime", "p", "is", "the", "easier", "it", "is", "to", "parallelise", "the", "underlying", "integer", "multiplication", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "The", "techniques", "described", "in", "this", "section", "will", "also", "apply", "to", "other", "time", "-", "lock", "puzzles", ",", "but", "we", "keep", "our", "focus", "on", "the", "cubing", "scheme", "from", "Section", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ]
[ "It", "follows", "that", "there", "is", "an", "upper", "bound", "on", "the", "delays", "one", "can", "achieve", "by", "cubing", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0 ]
[ "The", "purpose", "of", "this", "section", "is", "to", "mitigate", "this", "issue", "by", "reliably", "chaining", "together", "multiple", "delays", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "After", "generating", "a", "vector", "of", "parameters", "\\vec{k}", "=", "\\vector{k_1,", "\\dots,", "k_n}", ",", "we", "can", "construct", "the", "function", "h^{n}_p(x)", "=", "(f_p", "\\circ", "g_p(k_1)", "\\circ", "\\cdots", "\\circ", "f_p", "\\circ", "g_p(k_n))", "(x)", "as", "the", "new", "delay", "encryption", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "Although", "the", "ratio", "between", "encryption", "time", "and", "decryption", "time", "does", "not", "increase", ",", "using", "this", "construction", "one", "can", "achieve", "longer", "delays", "." ]
[ 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "Let", "p", "be", "a", "safe", "prime", "and", "define", "the", "function", "f_p", "\\colon", "\\mathbb", "Z_p", "\\to", "\\mathbb", "Z_p", "by", "f_p(x)", "=", "x^3", "\\mmod", "p", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "The", "main", "disadvantage", "of", "this", "approach", "is", "that", "the", "sender", "is", "required", "to", "compute", "extra", "safe", "primes", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "We", "have", "discovered", "only", "three", "different", "approaches", "to", "modular", "reduction", "in", "the", "literature", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ]
[ "The", "most", "common", "are", "Montgomery", "'s", "algorithm", "and", "Barret", "'s", "reduction", "." ]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0 ]
[ "The", "third", "approach", "is", "based", "on", "table", "lookup", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "Barret", "'s", "algorithm", "is", "based", "on", "the", "equality", "a", "\\mod", "b", "=", "a", "-", "\\floor{", "\\frac", "a", "b}", "b", "." ]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "The", "extra", "bit", "in", "q", "is", "unlikely", "to", "cause", "any", "performance", "hit", "." ]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "All", "of", "these", "involve", "m(+1)", "bit", "numbers", ",", "apart", "from", "the", "last", "subtraction", "." ]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ]
[ "Note", "that", "computing", "q", "may", "require", "a", "considerable", "amount", "of", "precomputation", ",", "but", "in", "our", "encryption", "scheme", "this", "needs", "to", "be", "done", "only", "once", "per", "``", "public", "key", "''", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ]
[ "Figure", "shows", "the", "pseudocode", "for", "Montgomery", "'s", "reduction", "." ]
[ 0, 0, 0, 1, 0, 0, 0, 1, 0 ]
[ "In", "order", "to", "use", "the", "algorithm", "of", "Figure", ",", "the", "modular", "exponentiation", "needs", "to", "be", "carried", "out", "in", "``", "Montgomery", "form", "''", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ]
[ "Therefore", "REDC", "(c_M^2)", "=", "c^2R", "\\mmod", "b", "." ]
[ 0, 1, 0, 0, 0, 0, 0, 0 ]
[ "The", "final", "result", "will", "also", "need", "to", "be", "converted", "back", "from", "the", "``", "Montgomery", "form", "''", ",", "which", "can", "be", "done", "by", "using", "{", "REDC", "}", "once", "more", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ]
[ "In", "order", "for", "{", "REDC", "}", "to", "work", "correctly", ",", "one", "needs", "a", "<", "Rb", "and", "(R,", "b)", "=", "1", "." ]
[ 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Comparing", "this", "with", "Barret", "'s", "reduction", ",", "we", "see", "that", "the", "overall", "complexity", "is", "very", "similar", "." ]
[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "As", "a", "result", ",", "we", "believe", "the", "post-processing", "part", "of", "Eker", "\\r{a}", "'s", "algorithm", "could", "be", "computed", "in", "a", "small", "amount", "of", "time", ",", "assuming", "that", "n", "is", "small", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Since", "the", "tests", "are", "independent", ",", "they", "can", "be", "run", "in", "parallel", "on", "different", "quantum", "computers", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "Another", "recent", "paper", "by", "Gidney", "and", "Eker", "\\r{a}", "{}", "analyses", "the", "real", "-", "world", "requirements", "for", "running", "quantum", "algorithms", "like", "the", "one", "described", "above", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ]
[ "In", "particular", ",", "they", "examined", "different", "quantum", "algorithms", ",", "among", "which", "Shor", "'s", "and", "Eker", "\\r{a}", "'s", ",", "in", "order", "to", "find", "the", "most", "efficient", "in", "terms", "of", "size", "and", "speed", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Since", "most", "of", "the", "quantum", "algorithms", "described", "in", "the", "literature", "are", "rather", "abstract", "like", "the", "one", "presented", "here", ",", "Gidney", "and", "Eker", "\\r{a}", "also", "develop", "their", "own", "algorithm", "by", "piecing", "together", "optimisations", "from", "many", "sources", "." ]
[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "The", "first", "interesting", "result", "is", "that", "``", "the", "cost", "of", "implementing", "the", "QFT", "[", "Quantum", "Fourier", "Transform", "]", "is", "negligible", "''", "{}", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0 ]
[ "Moreover", ",", "Gidney", "and", "Eker", "\\r{a}", "provided", "the", "Python", "script", "that", "was", "used", "to", "generate", "their", "estimate", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "The", "quantum", "experiment", "requires", "m", "+", "\\lfloor\\frac", "m", "s\\rfloor", "modular", "multiplications", "to", "compute", "g^a", "and", "\\lfloor", "\\frac", "m", "s", "\\rfloor", "to", "compute", "x^{-b}", "." ]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "In", "these", "calculations", "we", "minimise", "the", "size", "of", "the", "quantum", "computer", ",", "and", "we", "allow", "any", "percentage", "of", "failure", "as", "long", "as", "the", "expected", "runtime", "is", "less", "than", "a", "year", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "The", "failure", "percentage", "shown", "in", "the", "table", "is", "the", "probability", "of", "the", "quantum", "computer", "to", "fail", "and", "it", "is", "not", "correlated", "with", "the", "success", "probability", "of", "Eker", "\\r{a}", "'s", "algorithm", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "To", "represent", "the", "complexity", "of", "the", "quantum", "algorithm", ",", "both", "its", "size", "in", "mega", "qubits", "and", "its", "Tiffoli", "count", "are", "listed", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Moreover", ",", "using", "such", "large", "values", "for", "s", "is", "problematic", "as", "the", "probability", "of", "success", "of", "the", "post-processing", "is", "very", "low", "unless", "the", "quantum", "experiment", "is", "run", "multiple", "times", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ]
[ "Therefore", ",", "it", "is", "unlikely", "that", "in", "the", "foreseeable", "future", "quantum", "computers", "will", "be", "efficient", "enough", "to", "pose", "a", "threat", "to", "our", "delay", "encryption", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "To", "motivate", "the", "security", "of", "our", "scheme", ",", "we", "performed", "an", "extensive", "and", "practical", "analysis", "of", "the", "sequentiality", "of", "modular", "exponentiation", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "This", "part", "of", "our", "work", "is", "interesting", "in", "its", "own", "right", "since", "it", "naturally", "translates", "to", "the", "context", "of", "the", "RSA", "-", "based", "time", "-", "lock", "puzzle", "by", "Rivest", "\\etal", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ]
[ "Although", "this", "is", "a", "common", "technique", "in", "the", "construction", "of", "Verifiable", "Delay", "Functions", ",", "it", "has", "not", "been", "applied", "to", "the", "field", "of", "TRE", "and", "time", "-", "lock", "puzzles", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0 ]
[ "As", "a", "result", ",", "chaining", "is", "relatively", "inefficient", ",", "but", "it", "suits", "our", "use", "in", "cryptographic", "protocols", "such", "as", "those", "by", "Couteau", "\\etal", "or", "by", "Roscoe", "and", "Ryan", "." ]
[ 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Note", "that", "the", "use", "of", "chaining", "does", "not", "increase", "the", "ratio", "between", "decryption", "time", "and", "encryption", "time", "." ]
[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0 ]
[ "In", "Section", ",", "we", "outline", "the", "encryption", "scheme", "proposed", "and", "analyse", "what", "parameters", "should", "be", "used", "to", "make", "it", "secure", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "In", "Section", ",", "we", "look", "at", "the", "additional", "attacks", "that", "a", "quantum", "computer", "could", "carry", "against", "our", "scheme", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ]
[ "We", "then", "proceed", "to", "analyse", "ways", "in", "which", "our", "scheme", "can", "be", "extended", "to", "longer", "delays", "via", "chaining", "in", "Section", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ]
[ "In", "this", "study", ",", "we", "point", "out", "that", "a", "resourceful", "malicious", "entity", "can", "obtain", "a", "considerable", "speedup", "in", "computing", "the", "exponentiation", "as", "the", "modulus", "grows", "larger", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ]
[ "Another", "source", "of", "memory", "consumption", "of", "the", "algorithms", "from", "Figures", "and", "is", "that", "the", "reduction", "modulo", "a", "prime", "p", "is", "done", "using", "lookup", "tables", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Moreover", ",", "using", "such", "\\alpha", "the", "circuit", "will", "have", "depth", "O(n", "\\log", "n)", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ]
[ "However", ",", "large", "enough", "primes", "should", "be", "used", "to", "discourage", "the", "construction", "of", "ad", "hoc", "boolean", "circuits", "and", "other", "uses", "of", "lookup", "tables", "which", ",", "in", "theory", ",", "could", "pose", "a", "serious", "threat", "to", "our", "scheme", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Therefore", ",", "it", "is", "essential", "to", "understand", "whether", "an", "efficient", "logarithm", "oracle", "could", "be", "built", "in", "the", "real", "world", "as", "we", "understand", "it", "today", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Computing", "the", "discrete", "logarithm" ]
[ 0, 0, 0, 1 ]
[ "Classical", "algorithms", "for", "the", "discrete", "logarithm", "%" ]
[ 0, 0, 0, 0, 0, 1, 0 ]
[ "The", "fastest", "classical", "algorithm", "to", "compute", "the", "discrete", "logarithm", "is", "the", ",", "so", "-", "called", ",", "number", "field", "sieve", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Despite", "being", "an", "exponential", "algorithm", ",", "Adrian", "\\", "etal", "show", "how", "enough", "precomputation", "can", "be", "used", "to", "compute", "the", "discrete", "logarithm", "of", "a", "DHKE", "group", "in", "roughly", "one", "minute", "." ]
[ 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0 ]
[ "Let", "the", "DHKE", "protocol", "have", "parameters", "p,", "g", "where", "p", "is", "the", "prime", "modulo", "and", "g", "the", "generator", "." ]
[ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ]
[ "In", "the", "same", "paper", ",", "the", "authors", "estimate", "that", "to", "be", "able", "to", "compute", "the", "discrete", "logarithm", "in", "a", "1024", "-", "bits", "DHKE", "group", "in", "almost", "``", "real", "time", "''", ",", "the", "malicious", "entity", "would", "need", "to", "invest", "a", "dozen", "billion", "dollars", "and", "a", "couple", "of", "years", "of", "precomputation", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "On", "the", "theoretical", "side", ",", "h^n_p", "is", "a", "permutation", "\\mathbb", "Z_p", "\\to", "\\mathbb", "Z_p", "and", "it", "can", "be", "represented", "using", "a", "Lagrange", "polynomial", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "As", "noted", "in", ",", "computing", "a", "polynomial", "gcd", "is", "sufficient", "to", "solve", "this", "problem", "." ]
[ 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ]
[ "In", "detail", ",", "to", "invert", "an", "injective", "polynomial", "f(x)", "at", "a", "point", "c", "is", "enough", "to", "compute", "\\gcd\\left(f(x)-c,", "x^p", "-x\\right)", "since", "\\inv", "f(c)", "is", "the", "only", "common", "root", "to", "both", "polynomials", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "Luckily", ",", "determining", "the", "Lagrange", "polynomial", "for", "h^n_p", "can", "be", "problematic", "given", "the", "size", "of", "p", "if", "h^n_p", "is", "unpredictable", "enough", "." ]
[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Therefore", ",", "we", "aim", "to", "use", "appropriate", "pseudo-random", "permutations", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "Given", "the", "size", "of", "p", ",", "constructing", "a", "truly", "random", "permutation", "becomes", "prohibitive", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ]
[ "As", "a", "result", ",", "we", "believe", "that", "the", "use", "of", "the", "``", "swapping", "neighbors", "\"", "function", "should", "be", "avoided", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ]
[ "Algebraic", "methods" ]
[ 0, 1 ]
[ "It", "follows", "that", "g_p", "must", "be", "an", "invertible", "function", "which", "cannot", "be", "represented", "with", "a", "simple", "closed", "form", "in", "\\mathbb", "Z_p", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "For", "the", "sake", "of", "completeness", ",", "we", "present", "two", "algebraic", "methods", "used", "in", "the", "context", "of", "VDFs", ",", "although", "they", "lack", "any", "security", "guarantee", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "In", "particular", ",", "Lenstra", "and", "Wesolowski", "prove", "that", "h^n_p", "is", "secure", "when", "g_p", "is", "a", "permutation", "picked", "at", "random", "and", "the", "keys", "are", "all", "equal", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "In", "the", "setting", "of", "verifiable", "delay", "functions", ",", "a", "similar", "problem", "was", "studied", "by", "Lenstra", "and", "Wesolowski", "and", "by", "Boneh", "\\", "etal", "%" ]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Chaining", "candidates" ]
[ 1, 0 ]
[ "Therefore", ",", "it", "is", "possible", "to", "compute", "\\inv", "{(h_p^n)}", "by", "finding", "the", "root", "of", "a", "polynomial", "in", "\\mathbb", "Z_p", "." ]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Another", "important", "family", "of", "multiplication", "algorithms", "is", "based", "on", "Fourier", "Transforms", "." ]
[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0 ]
[ "The", "idea", "behind", "these", "algorithms", "is", "similar", "to", "the", "principle", "behind", "the", "Toom", "-", "Cook", "algorithms", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "The", "Fast", "Fourier", "Transform", "(", "FFT", ")", "is", "used", "to", "evaluate", "these", "polynomials", "at", "the", "root", "of", "unity", "of", "some", "field", "." ]
[ 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "One", "of", "the", "most", "famous", "FFT", "algorithms", "is", "the", "Sch\\\"", "onhage", "-", "Strassen", "algorithm", "which", "achieves", "a", "time", "complexity", "of", "O(n", "\\log", "n", "\\log", "\\log", "n)", "." ]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "This", "algorithm", "is", "very", "appealing", "as", "it", "works", "in", "the", "polynomial", "ring", "(\\mathbb", "Z/\\langle2^k", "+1\\rangle)[x]", "/", "(x^T", "+1)", "where", "both", "k", "and", "T", "are", "powers", "of", "2", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "This", "allows", "the", "algorithm", "to", "avoid", "floating", "-", "point", "calculations", "and", "their", "related", "precision", "issues", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ]
[ "This", "is", "likely", "the", "most", "popular", "FFT", "-", "based", "integer", "multiplication", "algorithm", ",", "and", "it", "is", "the", "one", "implemented", "within", "the", "GMP", "library", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "Another", "FFT", "algorithm", "worth", "mentioning", "is", "the", "one", "proposed", "by", "Harvey", "and", "van", "der", "Hoeven", "since", "its", "time", "complexity", "is", "O(n", "\\log", "n)", "." ]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ]
[ "This", "algorithm", "works", "in", "a", "multidimensional", "polynomial", "ring", "over", "the", "complex", "numbers", ",", "so", "any", "implementation", "must", "be", "careful", "to", "handle", "floating", "-", "point", "numbers", "with", "enough", "precision", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Since", "the", "FFT", "algorithms", "follow", "the", "same", "structure", "as", "the", "Toom", "-", "Cook", "family", ",", "they", "could", "also", "be", "parallelised", "relatively", "easily", "." ]
[ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "As", "we", "will", "see", "in", "the", "next", "section", ",", "these", "algorithms", "are", "the", "most", "common", "option", "when", "trying", "to", "parallelise", "integer", "multiplication", "on", "a", "GPU", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "Essentially", ",", "the", "above", "approach", "reduces", "the", "exponentiation", "process", "to", "computing", "only", "the", "repeated", "squaring", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0 ]
[ "This", "shows", "that", "it", "is", "unreasonable", "to", "assume", "that", "the", "exponentiation", "a^b", "requires", "more", "than", "\\lfloor", "\\log", "b", "\\rfloor", "sequential", "multiplications", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "On", "the", "other", "hand", ",", "the", "lack", "of", "algorithms", "that", "can", "outperform", "the", "process", "of", "repeated", "squaring", "is", "a", "favourable", "sign", "for", "our", "Conjecture", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Under", "the", "assumption", "that", "repeated", "squaring", "cannot", "be", "parallelised", ",", "the", "only", "attack", "vector", "left", "is", "improving", "each", "singular", "modular", "multiplication", "." ]
[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0 ]