tokens
sequence
label_ids
sequence
[ "There", "are", "mainly", "two", "approaches", "to", "compute", "a", "modular", "multiplication", "ab", "\\mmod", "c", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ]
[ "This", "latter", "approach", "can", "be", "much", "more", "memory", "efficient", "than", "the", "former", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ]
[ "On", "the", "other", "hand", ",", "the", "naive", "approach", "seems", "to", "be", "the", "most", "widely", "used", "as", "one", "can", "take", "advantage", "of", "the", "extensive", "research", "on", "both", "integer", "multiplication", "and", "modular", "reduction", "." ]
[ 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, 1, 0 ]
[ "Since", "memory", "restrictions", "are", "not", "a", "concern", "in", "our", "setting", ",", "we", "will", "focus", "on", "the", "fastest", "methods", "for", "squaring", "and", "modular", "reduction", "." ]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0 ]
[ "Modular", "reduction" ]
[ 0, 1 ]
[ "In", "this", "section", "we", "look", "at", "the", "algorithms", "for", "computing", "modular", "reduction", "and", "how", "to", "use", "those", "during", "modular", "exponentiation", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "Our", "aim", "is", "to", "find", "the", "best", "approach", "within", "the", "setting", "of", "repeated", "squaring", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "Let", "b", "be", "an", "integer", "of", "n", "bits", "and", "a", "an", "integer", "of", "m", "bits", "with", "m", "\\geq", "n", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ]
[ "The", "suggestion", "of", "Field", "Programmable", "Gate", "Arrays", "(", "FPGA", ")", "often", "appears", "in", "the", "literature", "." ]
[ 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0 ]
[ "The", "use", "of", "this", "technology", "seems", "to", "allow", "for", "theoretical", "improvements", ",", "but", "we", "do", "not", "believe", "that", "commercially", "available", "FPGAs", "could", "have", "such", "impact", "." ]
[ 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 ]
[ "As", "an", "example", ",", "Woo", "\\etal", "implements", "three", "different", "multiplication", "algorithms", "on", "FPGAs", "and", "these", "algorithms", "can", "multiply", "two", "8", "-", "bit", "numbers", "with", "a", "latency", "of", "5", "cycles", "." ]
[ 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, 1, 0, 0, 0, 0 ]
[ "A", "recent", "architecture", "like", "AMD", "Zen", "2", "using", "GMP", "requires", "around", "1.75", "clock", "cycles", "per", "64", "-", "bits", "when", "multiplying", "numbers", "." ]
[ 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Therefore", ",", "using", "a", "processor", "like", "the", "AMD", "Ryzen", "7", "3800X", ",", "which", "has", "a", "base", "clock", "speed", "of", "3.9", "GHz", ",", "one", "can", "multiply", "two", "64", "-", "bits", "numbers", "in", "roughly", "0.4", "ns", "." ]
[ 0, 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 ]
[ "In", "addition", "to", "the", "performance", "gap", ",", "the", "AMD", "processor", "mentioned", "above", "costs", "around", "Â", "£", "300", ",", "which", "is", "likely", "to", "be", "much", "less", "than", "the", "cost", "of", "a", "custom", "FPGA", "." ]
[ 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 ]
[ "Therefore", ",", "we", "do", "not", "believe", "that", ",", "at", "the", "time", "of", "writing", ",", "FPGAs", "are", "a", "viable", "way", "to", "achieve", "any", "performance", "boost", "." ]
[ 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 ]
[ "Another", "common", "type", "of", "hardware", "used", "in", "the", "literature", "are", "GPUs", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "The", "literature", "reveals", "many", "attempts", "to", "parallelise", "multiplication", "algorithms", "on", "many", "-", "core", "GPUs", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0 ]
[ "However", ",", "we", "could", "not", "find", "an", "exhaustive", "comparison", "of", "different", "GPU", "and", "CPU", "architectures", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0 ]
[ "Using", "G", "MP", "and", "an", "Intel", "i7", "-", "6920", "HQ", "we", "perform", "the", "same", "multiplication", "in", "roughly", "5", "ms", "." ]
[ 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "On", "the", "other", "hand", ",", "Ochoa", "-", "Jiménez", "\\etal", "implement", "the", "RSA", "scheme", "on", "both", "CPU", "and", "GPU", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0 ]
[ "After", "testing", "their", "implementations", "using", "key", "sizes", "up", "to", "3072", "bits", ",", "they", "conclude", "that", "``", "in", "spite", "of", "its", "massive", "parallelism", ",", "we", "observe", "that", "GPU", "implementations", "of", "RSA", "are", "considerably", "slower", "than", "their", "CPU", "counterparts", "'", "'", "." ]
[ 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, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ]
[ "When", "considering", "the", "lookup", "table", "approaches", ",", "the", "most", "recent", "results", "we", "could", "find", "are", "by", "Cao", "\\", "etal", "and", "by", "Will", "and", "Ko", "." ]
[ 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 ]
[ "This", "leads", "to", "an", "addition", "-", "based", "algorithm", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "As", "a", "result", ",", "throughout", "this", "paper", "we", "will", "assume", "that", "the", "complexity", "of", "the", "modular", "reduction", "consists", "of", "two", "multiplications", "of", "\\lfloor", "\\log", "a\\rfloor", "bits", "and", "two", "additions", "of", "\\lfloor", "\\log", "a", "\\rfloor", "bits", "." ]
[ 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "Finally", ",", "we", "need", "to", "analyse", "how", "to", "best", "use", "modular", "reduction", "within", "the", "exponentiation", "process", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0 ]
[ "Let", "p", "be", "an", "n", "-", "bit", "prime", "and", "let", "x,", "y", "\\leq", "p", "be", "integers", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "If", "the", "computation", "is", "done", "in", "Montgomery", "form", ",", "then", "the", "reduction", "should", "be", "interleaved", "with", "the", "multiplication", "to", "prevent", "the", "factor", "R", "from", "growing", "." ]
[ 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 ]
[ "Otherwise", ",", "one", "may", "choose", "to", "take", "the", "reduction", "modulo", "p", "every", "m", "steps", "for", "any", "m\\geq", "1", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Let", "M(n)", "be", "the", "time", "required", "to", "multiply", "two", "n", "-", "bit", "numbers", "and", "let", "R(n)", "be", "the", "time", "required", "to", "reduce", "an", "n", "-", "bit", "number", "modulo", "p", "." ]
[ 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, 1, 0, 0 ]
[ "Interleaving", "multiplication", "and", "reduction", "leads", "to", "a", "time", "complexity", "of", "k(M(n)", "+", "R(2n))", "for", "some", "k", "which", "represents", "the", "number", "of", "modular", "multiplications", "needed", "." ]
[ 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ]
[ "Card", "Shuffling" ]
[ 0, 1 ]
[ "Card", "shuffling", "schemes", "are", "essentially", "families", "of", "algorithms", "to", "construct", "ciphers", "where", "the", "domain", "can", "be", "chosen", "at", "will", "." ]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "An", "oblivious", "shuffle", "is", "a", "shuffle", "where", "the", "trajectory", "of", "a", "card", "can", "be", "computed", "without", "calculating", "the", "paths", "of", "the", "other", "cards", "." ]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "In", "the", "literature", ",", "we", "could", "find", "only", "a", "few", "FPE", "schemes", "based", "on", "oblivious", "card", "shuffling", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0 ]
[ "The", "earliest", "card", "shuffle", "present", "in", "the", "literature", "is", "the", "Thorp", "shuffle", "described", "in", "Figure", "." ]
[ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ]
[ "As", "with", "the", "other", "card", "shuffles", ",", "in", "order", "to", "guarantee", "high", "security", ",", "this", "cipher", "will", "need", "R", "\\in", "O(\\log", "p)", "." ]
[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "The", "swap", "-", "or", "-", "not", "shuffle", "by", "Hoang", "\\etal", "follows", "a", "similar", "structure", ":", "pair", "cards", "x", "and", "K-x\\mmod", "N", "and", "swap", "the", "cards", "according", "to", "a", "coin", "flip", "." ]
[ 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 ]
[ "Hoang", "\\", "etal", "\\", "suggest", "the", "use", "of", "round", "functions", "F_i", "that", "should", "be", "applied", "to", "\\max(x,", "y)", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Eker", "\\r{a}", "algorithm" ]
[ 0, 0, 1 ]
[ "Eker", "\\r{a}", "showed", "how", "one", "can", "improve", "Shor", "'s", "algorithm", "by", "reducing", "the", "number", "of", "group", "operations", "needed", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "The", "algorithm", "can", "be", "split", "in", "two", "parts", ":", "a", "quantum", "experiment", "that", "produces", "a", "set", "of", "pairs", "and", "a", "classical", "post-processing", "algorithm", "that", "yields", "the", "result", "." ]
[ 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, 0 ]
[ "The", "quantum", "experiment", "can", "be", "seen", "in", "Figure", "." ]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0 ]
[ "The", "crucial", "detail", "that", "allows", "Eker", "{\\aa", "}'s", "algorithm", "to", "be", "useful", "in", "our", "setting", "is", "that", "the", "size", "of", "the", "quantum", "register", "b", "can", "be", "bounded", "using", "the", "parameter", "s", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Therefore", ",", "only", "the", "computation", "on", "x", "and", "the", "Fourier", "transform", "must", "be", "performed", "during", "the", "``", "delay", "period", "''", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Computing", "u", "from", "v", "is", "an", "instance", "of", "the", "closest", "vector", "problem", "(", "CVP", ")", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0 ]
[ "The", "CVP", "is", "considered", "infeasible", "in", "high", "dimensions", ",", "therefore", "the", "parameter", "n", "should", "be", "chosen", "carefully", "." ]
[ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "The", "simplest", "algorithm", "to", "solve", "CVP", "is", "Babai", "'s", "nearest", "plane", "algorithm", "{", "}", "(", "see", "Figure", ")", "." ]
[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ]
[ "Once", "the", "reduced", "basis", "and", "the", "Gram", "-", "Schmidt", "orthogonalisation", "are", "computed", ",", "the", "algorithm", "is", "relatively", "fast", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "The", "step", "that", "dominates", "the", "complexity", "of", "the", "algorithm", "is", "the", "LLL", "reduction", "of", "the", "basis", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ]
[ "Nguyen", "and", "Sthelé", "propose", "a", "complex", "method", "to", "LLL", "reduce", "a", "lattice", "basis", "in", "time", "O(d^2n(d+\\log", "B)M(d)", "\\log", "B", ")", "where", "d", "is", "the", "dimension", "of", "the", "lattice", ",", "n", "the", "dimension", "of", "the", "underlying", "vector", "space", ",", "B", "the", "largest", "norm", "of", "a", "vector", "in", "the", "basis", "and", "M(d)", "indicates", "the", "time", "required", "to", "multiply", "two", "d", "-", "bits", "numbers", "." ]
[ 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, 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, 0 ]
[ "In", "addition", ",", "this", "algorithm", "computes", "the", "values", "\\mu_{i,j}", "=", "\\frac{", "b_i", "\\cdot", "b_j^*}", "{||b_j^*||^2}", "and", "r_{i,i}", "=", "||b_i^*||^2", "which", "can", "be", "used", "to", "speed", "up", "the", "computation", "of", "the", "Gram", "-", "Schmidt", "orthogonal", "basis", "." ]
[ 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, 0, 1, 0 ]
[ "Integer", "multiplication" ]
[ 0, 1 ]
[ "Although", "repeated", "squaring", "is", "a", "sequential", "process", ",", "integer", "multiplication", "is", "definitely", "not", "." ]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ]
[ "This", "makes", "the", "study", "of", "integer", "multiplication", "crucial", "for", "understanding", "where", "a", "malicious", "entity", "could", "gain", "an", "unfair", "advantage", "." ]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "In", "this", "section", ",", "we", "analyse", "the", "main", "known", "integer", "multiplication", "algorithms", "in", "order", "to", "find", "the", "one", "most", "suited", "to", "our", "application", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Let", "a,", "b", "be", "two", "n", "-", "bit", "numbers", ",", "we", "aim", "to", "compute", "ab", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ]
[ "The", "first", "family", "of", "multiplication", "algorithms", "that", "we", "must", "mention", "are", "the", "so", "-", "called", "Toom", "-", "Cook", "algorithms", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "In", "that", "paper", ",", "the", "famous", "Karatsuba", "-", "Ofman", "multiplication", "is", "nothing", "but", "Toom", "-", "2", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0 ]
[ "The", "asymptotic", "complexity", "of", "the", "Toom", "-", "k", "algorithm", "is", "O\\left(n^{\\frac", "{\\log(2k-1)}{\\log", "k}}\\right)", "." ]
[ 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ]
[ "This", "algorithm", "can", "be", "parallelised", "by", "computing", "each", "v_i", "on", "concurrent", "threads", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "We", "attempted", "modifying", "GMP", "'s", "implementation", "of", "this", "algorithm", "by", "parallelising", "it", "with", "low", "-", "level", "software", ",", "but", "this", "was", "not", "sufficient", "to", "achieve", "any", "performance", "improvements", "when", "multiplying", "numbers", "of", "hundreds", "of", "thousands", "of", "bits", "or", "smaller", "." ]
[ 0, 0, 0, 1, 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, 1, 0, 0, 0 ]
[ "As", "an", "example", ",", "the", "GMP", "library", "lets", "the", "user", "optimise", "the", "thresholds", "for", "picking", "k", "." ]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "During", "our", "testing", "on", "numbers", "with", "70034", "bits", ",", "we", "found", "that", "the", "algorithm", "with", "k=8", "gave", "the", "best", "performance", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "In", "Section", "we", "will", "discuss", "in", "detail", "the", "purpose", "of", "s", ",", "in", "short", "\\lambda", "can", "be", "regarded", "as", "the", "key", "length", "for", "the", "encryption", "scheme", "." ]
[ 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 ]
[ "Removal", "of", "the", "padding", "requires", "no", "knowledge", "of", "the", "random", "seed", "used", "." ]
[ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Figure", "illustrates", "the", "decryption", "algorithm", "." ]
[ 0, 0, 0, 0, 1, 0 ]
[ "The", "proposed", "scheme", "works", "as", "a", "delay", "encryption", "because", "the", "decryption", "process", "has", "time", "complexity", "O((\\log", "p)^2", "\\log", "\\log", "p)", "while", "the", "encryption", "process", "only", "O(\\log", "p", "\\log", "\\log", "p)", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "However", ",", "this", "holds", "true", "only", "assuming", "that", "repeated", "squaring", "is", "an", "inherently", "sequential", "process", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "This", "assumption", "has", "been", "widely", "studied", "since", "the", "first", "proposal", "of", "a", "time", "-", "lock", "puzzle", "by", "Rivest", "\\etal", "and", "today", "it", "is", "still", "widely", "used", "." ]
[ 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 ]
[ "On", "the", "practical", "side", ",", "we", "tested", "some", "timings", "on", "an", "Intel", "i7", "-", "6920", "HQ", "quad", "-", "core", "with", "Hyper-", "Threading", "enabled", "." ]
[ 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", "GMP", "library", "focuses", "on", "high", "performance", "thanks", "to", "its", "low", "-", "level", "assembly", "implementations", "of", "many", "primitives", "." ]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ]
[ "Moreover", ",", "the", "GMP", "community", "invests", "a", "significant", "amount", "of", "effort", "in", "producing", "code", "that", "is", "tailored", "for", "many", "processor", "pipelines", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "The", "C", "code", "was", "compiled", "with", "Apple", "clang", "version", "12.0.0", "and", "optimisation", "flag", "-", "Ofast", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0 ]
[ "Using", "the", "70034", "-", "bit", "prime", "p", "=", "2566851867", "\\times", "2^{70002}", "-1", "and", "picking", "random", "messages", ",", "we", "measured", "that", ",", "discounting", "padding", "costs", ",", "decryption", "takes", "roughly", "19", "s", "while", "encryption", "requires", "only", "0.8", "ms", "." ]
[ 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, 0, 1, 0, 0, 0, 0, 0 ]
[ "When", "making", "concrete", "computations", ",", "we", "will", "use", "the", "above", "prime", "." ]
[ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "Our", "concentration", "on", "this", "single", "prime", "is", "somewhat", "excused", "by", "Section", ",", "where", "we", "will", "show", "how", "the", "delay", "created", "by", "a", "single", "cubing", "puzzle", "can", "be", "multiplied", "by", "a", "small", "integer", "factor", "without", "changing", "the", "prime", "." ]
[ 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, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "This", "shows", "that", "it", "is", "possible", "to", "utilise", "our", "encryption", "scheme", "to", "delay", "a", "message", "efficiently", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ]
[ "Delay", "by", "cubing" ]
[ 0, 0, 1 ]
[ "\\begin{conjecture}", "[", "Sequentiality", "]", "%", "Given", "x,", "y,", "p", "with", "x,", "y", "\\in", "\\mathbb", "Z_p", "and", "p", "prime", ",", "computing", "x^y", "\\mod", "p", "requires", "computing", "at", "least", "\\lfloor", "\\log", "y", "\\rfloor", "modular", "multiplications", "sequentially", "." ]
[ 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 ]
[ "In", "this", "section", ",", "s", "we", "propose", "a", "time", "-", "lock", "puzzle", "based", "on", "the", "empirical", "hypothesis", "that", "repeated", "squaring", "cannot", "be", "parallelised", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0 ]
[ "The", "preparation", "phase", "is", "equivalent", "to", "the", "key", "-", "generation", "step", "in", "a", "traditional", "scheme", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ]
[ "The", "result", "of", "this", "process", "is", "a", "tuple", "of", "parameters", "(p,", "b,", "T)", "which", "should", "be", "treated", "as", "a", "public", "key", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "Figure", "shows", "the", "pseudocode", "for", "the", "setup", "phase", "." ]
[ 0, 0, 0, 1, 0, 0, 0, 0, 0 ]
[ "Note", "also", "that", "if", "p", "is", "a", "safe", "prime", ",", "then", "the", "cubing", "operation", "is", "invertible", "in", "\\mathbb", "Z_p", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ]
[ "The", "prime", "p", "is", "picked", "to", "be", "a", "``", "safe", "prime", "'", "'", "(", "i.e.", "\\@", "\\frac", "{p-1}", "2", "is", "also", "prime", ",", "often", "called", "a", "Sophie", "Germain", "prime", ")", "in", "order", "to", "reduce", "possible", "attacks", "on", "the", "system", "." ]
[ 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, 0, 0, 0, 0, 0, 1, 0 ]
[ "To", "decrypt", "a", "message", "you", "need", "at", "least", "n-1", "sequential", "modular", "multiplications", "." ]
[ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "To", "encrypt", "a", "message", "you", "need", "2", "sequential", "modular", "multiplications", ",", "where", "we", "neglect", "the", "cost", "of", "the", "padding", "scheme", "used", "." ]
[ 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ]
[ "It", "follows", "that", ",", "in", "the", "delay", "period", "(", "i.e.", "\\@", "n-1", "modular", "multiplications", ")", ",", "Bob", "can", "perform", "C", "\\frac{n-1}", "2", "encryptions", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "Assuming", "that", "the", "seed", "is", "drawn", "at", "random", "from", "the", "set", "of", "all", "the", "l", "-", "bit", "strings", ",", "then", "Bob", "has", "probability", "C", "\\frac", "{n-1}", "{2^{l+1}}", "of", "guessing", "the", "correct", "seed", "." ]
[ 0, 0, 0, 1, 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, 1, 0 ]
[ "This", "scenario", "is", "somewhat", "equivalent", "to", "Bob", "trying", "to", "guess", "an", "l", "-", "bit", "key", "for", "a", "symmetric", "cipher", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0 ]
[ "It", "is", "important", "to", "point", "out", "that", ",", "in", "a", "post-quantum", "scenario", ",", "Grover", "'s", "algorithm", "essentially", "halves", "the", "bit", "size", "of", "search", "space", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "However", ",", "Bob", "could", "perform", "an", "extensive", "precomputation", "before", "receiving", "Alice", "'s", "message", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ]
[ "Assume", "that", "Bob", "has", "access", "to", "C", "concurrent", "computers", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "Since", "our", "scheme", "uses", "primes", "with", "thousands", "of", "bits", ",", "there", "is", "practically", "no", "upper", "bound", "on", "the", "number", "l", "that", "Alice", "can", "pick", "." ]
[ 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 ]
[ "The", "above", "theorem", "proves", "that", ",", "using", "safe", "primes", ",", "we", "minimise", "the", "number", "of", "elements", "whose", "cube", "root", "is", "easy", "to", "compute", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
[ "In", "particular", ",", "it", "shows", "that", "computing", "the", "cube", "root", "requires", "an", "exponentiation", "where", "the", "exponent", "has", "a", "very", "similar", "bitsize", "to", "the", "prime", "modulus", "." ]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0 ]
[ "When", "using", "a", "deterministic", "padding", "scheme", ",", "the", "scheme", "proposed", "is", "not", "safe", "as", "any", "attacker", "can", "``", "guess", "''", "the", "plaintext", "if", "the", "message", "space", "is", "limited", "." ]
[ 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 ]
[ "Alice", "and", "Bob", "are", "using", "delay", "encryption", "to", "ensure", "neither", "of", "them", "can", "decide", "their", "move", "after", "seeing", "their", "opponent", "'s", "choice", "." ]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
[ "It", "follows", "that", "Alice", "must", "use", "some", "random", "seed", "in", "her", "encryption", "to", "stop", "Bob", "from", "guessing", "what", "she", "encrypted", "and", "encrypt", "the", "result", "to", "confirm", "it", "." ]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0 ]
[ "Note", "that", "Bob", "can", "perform", "an", "expensive", "precomputation", "step", "before", "receiving", "the", "message", "from", "Alice", "and", "that", "checking", "multiple", "guesses", "can", "be", "performed", "in", "parallel", "." ]
[ 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 ]