Algorithms for quantum computation: discrete logarithms and factoring

1994
11 pages
  • Published: 1994

Citations per year

20062011201620212025050100150
Abstract: (IEEE)
A computer is generally considered to be a universal computational device; i.e., it is believed able to simulate any physical computational device with a cost in computation time of at most a polynomial factor: It is not clear whether this is still true when quantum mechanics is taken into consideration. Several researchers, starting with David Deutsch, have developed models for quantum mechanical computers and have investigated their computational properties. This paper gives Las Vegas algorithms for finding discrete logarithms and factoring integers on a quantum computer that take a number of steps which is polynomial in the input size, e.g., the number of digits of the integer to be factored. These two problems are generally considered hard on a classical computer and have been used as the basis of several proposed cryptosystems. We thus give the first examples of quantum cryptanalysis.<>
  • Quantum computing
  • Quantum mechanics
  • Polynomials
  • Computational modeling
  • Physics computing
  • Computer simulation
  • Costs
  • Mechanical factors
  • Cryptography
  • Circuit simulation
  • [1]

    Quantum mechanical Hamiltonian models of Turing machines

    • P. Benioff
      • J.Statist.Phys. 29 (1982) 515-546
  • [2]

    Quantum mechanical Hamiltonian models of Turing machines that dissipate no energy

    • P. Benioff
      • Phys.Rev.Lett. 48 (1982) 1581-1585
  • [3]

    Logical reversibility of computation

    • C.H. Bennett
      • IBM J.Res.Dev. 17 (1973) 525-532
  • [4]

    What is feasible on a quantum computer

    • C.H. Bennett
      ,
    • E. Bernstein
      ,
    • G. Brassard
      ,
    • U. Vazirani
  • [5]

    Quantum complexity theory

    • E. Bernstein
      ,
    • U. Vazirani
  • [6]

    The quantum challenge to structural complexity theory

    • A. Berthiaume
      ,
    • G. Brassard
  • [7]

    Oracle quantum computing

    • A. Berthiaume
      ,
    • G. Brassard
  • [8]
    An approximate Fourier transform useful in quantum factoring,. Order fulltext via DESY Google Scholar
    • D. Coppersmith
  • [9]

    Quantum theory the Church-Turing principle and the universal quantum computer

      • Proc.Roy.Soc.Lond.A A 400 (1985) 96-117
  • [10]

    Quantum computational networks

      • Proc.Roy.Soc.Lond.A A 425 (1989) 73-90
  • [11]

    Rapid solution of problems by quantum computation

      • Proc.Roy.Soc.Lond.A A 439 (1992) 553-558
  • [12]

    Two-bit gates are universal for quantum computation

    • D.P. DiVincenzo
  • [13]

    Simulating physics with computers

    • R. Feynman
  • [14]
  • [15]

    Are there interactive protocols for co-NP languages?

    • L. Fortnow
      ,
    • M. Sipser
      • Inform.Proc.Lett. 28 (1988) 249-251
  • [16]

    Discrete logarithms in GF(p) using the number field sieve

    • D.M. Gordon
      • SIAM J.Discrete Math. 6 (1993) 124-139
  • [17]
    An Introduction to the Theory of Numbers, New York:Oxford University Press,. Order fulltext via DESY Google Scholar
    • G.H. Hardy
      ,
    • E.M. Wright
  • [18]

    Is quantum mechanics useful?

    • R. Landauer
  • [19]
    The Development of the Number Field Sieve
  • [20]

    A rigorous time bound for factoring integers

    • H.W. Lenstra
      ,
    • C. Pomerance
      • J.Am.Math.Soc. 5 (1992) 483-516
  • [22]

    Envisioning a quantum supercomputer

    • S. Lloyd
      • Science 263 (1994) 695
  • [23]

    Riemann's hypothesis and tests for primality

    • G.L. Miller
      • J.Comput.Syst.Sci. 13 (1976) 300-317
  • [24]

    An improved algorithm for computing discrete logarithms over GF(p) and its cryptographic significance

    • S. Pohlig
      ,
    • M. Hellman
  • [25]

    Fast rigorous factorization and discrete logarithm algorithms

    • C. Pomerance