@Article{         74bc889d-a174-3206-83b7-759756dec6b5,
  Author        = "Bernstein, Daniel J. and Sorenson, Jonathan P.",
  Abstract      = "Fix pairwise coprime positive integers $p\_1 ,p\_2 ,....,p\_s $ . We propose representing integers u modulo m, where m is any positive integer up to roughly $\sqrt {p\_1 ,p\_2 ,....,p\_s } $ , as vectors $(u\bmod p\_1 ,u\bmod p\_2 ,....,u\bmod p\_s )$ . We use this representation to obtain a new result on the parallel complexity of modular exponentiation: there is an algorithm for the Common CRCW PRAM that, given positive integers x, e, and m in binary, of total bit length n, computes $x^e $ mod m in time $O(n/{\l}g {\l}g n)$ using $n^{O(1)} $ processors. For comparison, a parallelization of the standard binary algorithm takes superlinear time; Adleman and Kompella gave an $O(({\l}g n)^3 )$ expected time algorithm using $O(({\l}g n)^3 )$ processors; von zur Gathen gave an NC algorithm for the highly special case that m is polynomially smooth.",
  date-added    = "2023-10-06 15:17:55 +0200",
  date-modified = "2023-10-06 15:17:55 +0200",
  ISSN          = "00255718, 10886842",
  Journal       = "Mathematics of Computation",
  Number        = "257",
  Pages         = "443--454",
  Publisher     = "American Mathematical Society",
  Title         = "Modular Exponentiation via the Explicit Chinese Remainder Theorem",
  URL           = "http://www.jstor.org/stable/40234386",
  URLDate       = "2023-10-06",
  Volume        = "76",
  Year          = "2007",
  bdsk-url-1    = "http://www.jstor.org/stable/40234386",
  File          = "Modular Exponentiation via the Explicit Chinese Remainder Theorem - Bernstein-ModularExponentiationvia-2007.pdf"
}

@Article{ 74bc889d-a174-3206-83b7-759756dec6b5, Author = "Bernstein, Daniel J. and Sorenson, Jonathan P.", Abstract = "Fix pairwise coprime positive integers $p_1 ,p_2 ,....,p_s $ . We propose representing integers u modulo m, where m is any positive integer up to roughly $\sqrt {p_1 ,p_2 ,....,p_s } $ , as vectors $(u\bmod p_1 ,u\bmod p_2 ,....,u\bmod p_s )$ . We use this representation to obtain a new result on the parallel complexity of modular exponentiation: there is an algorithm for the Common CRCW PRAM that, given positive integers x, e, and m in binary, of total bit length n, computes $x^e $ mod m in time $O(n/{\l}g {\l}g n)$ using $n^{O(1)} $ processors. For comparison, a parallelization of the standard binary algorithm takes superlinear time; Adleman and Kompella gave an $O(({\l}g n)^3 )$ expected time algorithm using $O(({\l}g n)^3 )$ processors; von zur Gathen gave an NC algorithm for the highly special case that m is polynomially smooth.", date-added = "2023-10-06 15:17:55 +0200", date-modified = "2023-10-06 15:17:55 +0200", ISSN = "00255718, 10886842", Journal = "Mathematics of Computation", Number = "257", Pages = "443--454", Publisher = "American Mathematical Society", Title = "Modular Exponentiation via the Explicit Chinese Remainder Theorem", URL = "http://www.jstor.org/stable/40234386", URLDate = "2023-10-06", Volume = "76", Year = "2007", bdsk-url-1 = "http://www.jstor.org/stable/40234386", File = "Modular Exponentiation via the Explicit Chinese Remainder Theorem - Bernstein-ModularExponentiationvia-2007.pdf" }