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Open problems in number theoretic complexity, II

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Book cover Algorithmic Number Theory (ANTS 1994)

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References

  1. Leonard M. Adleman and Jonathan DeMarrais. A subexponential algorithm for discrete logarithms over all finite fields. In Douglas R. Stinson, editor, Advances in Cryptology: Crypto '93, volume 773 of Lecture Notes in Computer Science, pages 147–158, New York, 1993. Springer-Verlag.

    Google Scholar 

  2. Leonard M. Adleman and Jonathan DeMarrais. A subexponential algorithm for discrete logarithms over all finite fields. Mathematics of Computation, 61:1–15, 1993. Extended abstract in [AD93a].

    Google Scholar 

  3. Leonard M. Adleman. Factoring numbers using singular integers. In Proceedings of the 23th Annual Symposium on Theory of Computing, pages 64–71, 1991.

    Google Scholar 

  4. Leonard M. Adleman. The function field sieve. In Proceedings of the 1994 Algorithmic Number Theory Symposium, Lecture Notes in Computer Science. Springer-Verlag, to appear.

    Google Scholar 

  5. Leonard M. Adleman, Dennis Estes, and Kevin S. McCurley. Solving bivariate quadratic congruences in random polynomial time. Mathematics of Computation, 48:17–28, 1987.

    Google Scholar 

  6. Leonard M. Adleman and Ming-Deh Huang. Primality testing and two dimensional Abelian varieties over finite fields, volume 1512 of Lecture Notes in Mathematics. Springer-Verlag, 1992.

    Google Scholar 

  7. Leonard M. Adleman, Ming-Deh A. Huang, and Jonathan DeMarrais. A subexponential algorithm for discrete logarithms in the rational subgroup of the Jacobian of a hyperelliptic curve over a finite field. In Proceedings of the 1994 Algorithmic Number Theory Symposium, Lecture Notes in Computer Science. Springer-Verlag, to appear.

    Google Scholar 

  8. Alan Aho, John Hopcroft, and Jeffrey Ullman. The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading, MA, 1974.

    Google Scholar 

  9. Leonard M. Adleman and Kireeti Kompella. Using smoothness to achieve parallelism. In Proceedings of the 20th ACM Symposium on Theory of Computing, pages 528–538, 1988.

    Google Scholar 

  10. Leonard M. Adleman and H. W. Lenstra, Jr. Finding irreducible polynomials over finite fields. In Proceedings of the 18th Annual Symposium on Theory of Computing, pages 350–355, New York, 1986. Association for Computing Machinery.

    Google Scholar 

  11. Leonard M. Adleman and K. Manders. Reducibility, randomness, and intractibility. In Proc. 9th Annual ACM Symposium On Theory Of Computing, pages 151–163, New York, 1977. Association for Computing Machinery.

    Google Scholar 

  12. Leonard M. Adleman and R. McDonnell. An application of higher reciprocity to computational number theory. In Proceedings of the 22nd Annual Symposium on Foundations of Computer Science, pages 100–106. IEEE Computer Society, 1982.

    Google Scholar 

  13. Leonard M. Adleman and Kevin S. McCurley. Open problems in numbertheoretic complexity. In Discrete Algorithms and Complexity (Proceedings of the Japan-US Joint Seminar on Discrete Algorithms and Complexity Theory), pages 237–262. Academic Press, 1986.

    Google Scholar 

  14. Leonard M. Adleman, K. Manders, and Gary L. Miller. On taking roots in finite fields. In Proceedings of the 18th Annual Symposium on Foundations of Computer Science, pages 175–178, Rhode Island, 1977. IEEE Computer Society.

    Google Scholar 

  15. N. Ankeny. The least quadratic nonresidue. Annals of Mathematics, 55:65–72, 1952.

    Google Scholar 

  16. Leonard M. Adleman, Carl Pomerance, and Robert Rumely. On distinguishing prime numbers from composite numbers. Annals of Mathematics, 117:173–206, 1983.

    Google Scholar 

  17. Eric Bach. Discrete logarithms and factoring. Technical Report UCB/CSD 84/186, University of California, Computer Science Division (EECS). University of California, Berkely, California, June 1984.

    Google Scholar 

  18. Eric Bach. Analytic Methods in the Analysis and Design of Number Theoretic Algorithms. MIT Press, Cambridge, 1985.

    Google Scholar 

  19. Eric Bach. How to generate factored random numbers. SIAM Journal of Computing, 17:179–193, 1988.

    Google Scholar 

  20. Eric Bach. Explicit bounds for primality testing and related problems. Mathematics of Computation, 55:355–380, 1990.

    Google Scholar 

  21. Lenore Blum, Manuel Blum, and Michael Shub. A simple unpredictable pseudo-random number generator. SIAM Journal of Computing, 15:364–383, 1986.

    Google Scholar 

  22. Elwyn Berlekamp. Factoring polynomials over finite fields. Bell System Technical Journal, 46:1853–1859, 1967.

    Google Scholar 

  23. Elwyn Berlekamp. Algebraic Coding Theory. McGraw-Hill, New York, 1968.

    Google Scholar 

  24. Elwyn Berlekamp. Factoring polynomials over large finite fields. Mathematics of Computation, 24:713–735, 1970.

    Google Scholar 

  25. Richard Brent and H. Kung. Systolic VLSI arrays for linear time gcd computation. In F. Anceau and E. Aas, editors, VLSI 83, pages 145–154. IFIP, Elsevevier, 1983.

    Google Scholar 

  26. Manuel Blum and Silvio Micali. How to generate cryptographically strong sequences of pseudorandom bits. SIAM Journal of Computing, 13:850–864, 1984.

    Google Scholar 

  27. Ernest F. Brickell and Kevin S. McCurley. An interactive identification scheme based on discrete logarithms and factoring. Journal of Cryptology, 5:29–40, 1992.

    Google Scholar 

  28. Eric Bach, Gary L. Miller, and Jeffrey O. Shallit. Sums of divisors, perfect numbers, and factoring. In Proceedings of the 16th Annual Symposium on Theory of Computing, New York, 1984. Association for Computing Machinery.

    Google Scholar 

  29. Enrico Bombieri. Le grand crible dans la théorie analytique des nombres. Avec une sommaire en anglais. Astérisque, 18, 1974.

    Google Scholar 

  30. Johannes Buchmann and Victor Shoup. Constructing non-residues in finite fields and the extended Riemann hypothesis. In Proceedings of the 23th Annual Symposium on Theory of Computing, pages 72–79, 1991.

    Google Scholar 

  31. Johannes Buchmann. Complexity of algorithms in algebraic number theory. In R.A. Mollin, editor, Proceedings of the First Conference of the Canadian Number Theory Association, pages 37–53, Berlin, 1990. De Gruyter.

    Google Scholar 

  32. Johannes Buchmann and Hugh C. Williams. On the existence of a short proof for the value of the class number and regulator of a real quadratic field. In Richard A. Mollin, editor, Proceedings of the NATO Advanced Study Institute on Number Theory and Applications, pages 327–345, The Netherlands, 1989. Kluwer.

    Google Scholar 

  33. B. Chor and O. Goldreich. An improved parallel algorithm for integer GCD. Algorithmica. To Appear.

    Google Scholar 

  34. A. L. Chistov. The complexity of constructing the ring of integers of a global field. Dokl Akad. Nauk. SSSR, 306:1063–1067, 1989. English translation: Soviet Math. Dokl. 39 (1989), 597–600.

    Google Scholar 

  35. H. Cohen and H. W. Lenstra, Jr. Primality testing and Jacobi sums. Mathematics of Computation, 42:297–330, 1984.

    Google Scholar 

  36. Stephen Cook. Towards a complexity theory of synchronous parallel computation. Enseignment Math., 27:99–124, 1981.

    Google Scholar 

  37. Stephen Cook. A taxonomy of problems with fast parallel algorithms. Information and Control, 64:2–22, 1985.

    Google Scholar 

  38. Don Coppersmith. Fast evaluation of discrete logarithms in fields of characteristic two. IEEE Transactions on Information Theory, 30:587–594, 1984.

    Google Scholar 

  39. Don Coppersmith. Modifications to the number field sieve. Technical Report RC16264, IBM TJ Watson Research Center, Yorktown Heights, New York, 1990.

    Google Scholar 

  40. Don Coppersmith, Andrew Odlyzko, and Richard Schroeppel. Discrete logarithms in GF(p). Algorithmica, 1:1–15, 1986.

    Google Scholar 

  41. Jean-Marc Couveignes. Computing a square root for the number field sieve. In A. K. Lenstra and H. W. Lenstra, Jr., editors, The development of the number field sieve, number 1554 in Lecture Notes in Mathematics, pages 95–102. Springer-Verlag, 1993.

    Google Scholar 

  42. H. Cramér. On the order of magnitude of the difference between consecutive prime numbers. Acta Arithmetica, 2:23–46, 1936.

    Google Scholar 

  43. David Cantor and Hans Zassenhaus. A new algorithm for factoring polynomials over finite fields. Mathematics of Computation, 36:587–592, 1981.

    Google Scholar 

  44. Bert den Boer. Diffie-Hellman is as strong as discrete log for certain primes. In Advances in Cryptology: Proceedings of Crypto '88, volume 403 of Lecture Notes in Computer Science, pages 530–539, New York, 1990. Springer-Verlag.

    Google Scholar 

  45. W. Diffie and M. E. Hellman. New directions in cryptography. IEEE Transactions on Information Theory, 22:644–654, 1976.

    Google Scholar 

  46. John D. Dixon. Asymptotically fast factorization of integers. Mathematics of Computation, 36:255–260, 1981.

    Google Scholar 

  47. P. D. T. A. Elliot and H. Halberstam. The least prime in an arithmetic progression. In Studies in Pure Mathematics, pages 69–61. Academic Press, London, 1971.

    Google Scholar 

  48. Taher ElGamal. A public key cryptosystem and a signature scheme based on discrete logarithms. IEEE Transactions on Information Theory, 31:469–472, 1985.

    Google Scholar 

  49. P. Erdös. Remarks on number theory, I. Mat. Lapok, 12:10–17, 1961.

    Google Scholar 

  50. S. A. Evdokimov. Factoring a solvable polynomial over a finite field and generalized Riemann hypothesis. Zapiski Nauchn. Semin. Leningr. Otdel. Matem. Inst. Acad. Sci. USSR, 176:104–117, 1989. In Russian.

    Google Scholar 

  51. S. A. Evdokimov. Factorization of polynomials over finite fields. In Proceedings of the 1994 Algorithmic Number Theory Symposium, Lecture Notes in Computer Science, Berlin, to appear. Springer-Verlag.

    Google Scholar 

  52. M. A. Frumkin. Complexity questions in number theory. J. Soviet Math., 29:1502–1517, 1985. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, vol. 118 (1982), 188–210.

    Google Scholar 

  53. Martin Fürer. Deterministic and Las Vegas primality testing algorithms. In Proceedings of ICALP 1985, 1985.

    Google Scholar 

  54. Karl Friedrich Gauss. Disquisitiones Arithmeticæ. Springer-Verlag, New York, 1986. Reprint of the 1966 English translation by Arthur A. Clarke, S.J., Yale University Press, revised by William C. Waterhouse. Original 1801 edition published by Fleischer, Leipzig.

    Google Scholar 

  55. John Gill. Computational complexity of probabilistic Turing machines. SIAM Journal of Computing, 4:675–695, 1977.

    Google Scholar 

  56. Michael Garey and David Johnson. Computers and Intractibility: A Guide to the Theory of NP-Completeness. W. H. Freeman, San Francisco, 1979.

    Google Scholar 

  57. Shafi Goldwasser and Joe Kilian. Almost all primes can be quickly certified. In Proceedings of the 18th Annual Symposium on Theory of Computing, pages 316–329, New York, 1986. Association for Computing Machinery.

    Google Scholar 

  58. Martin Grötschel, László Lovász, and Alexander Schrijver. Geometrie Algorithms and Combinatorial Optimization Springer-Verlag, Berlin, 1988.

    Google Scholar 

  59. Shafi Goldwasser and Silvio Micali. Probabilistic encryption & how to play mental poker keeping secret all partial information. In Proceedings of the 14th Annual Symposium on Theory of Computing, pages 365–377, New York, 1982. Association for Computing Machinery.

    Google Scholar 

  60. Shafi Goldwasser and Silvio Micali. Probabilistic encryption. Journal of Computer and System Science, 28:270–299, 1984.

    Google Scholar 

  61. Dorian Goldfeld. Gauss' class number problem for imaginary quadratic fields. Bulletin of the American Mathematical Society, 13:23–38, 1985.

    Google Scholar 

  62. Daniel M. Gordon. Discrete logarithms in GF(p) using the number field sieve. SIAM Journal of Discrete Mathematics, 6:124–138, 1993.

    Google Scholar 

  63. Richard Guy. How to factor a number. Congressus Numeratium, XXVII:49–89, 1977. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics, University of Manitoba.

    Google Scholar 

  64. D. R. Heath-Brown. Almost-primes in arithmetic progressions and short intervals. Mathematical Proceedings of the Cambridge Philosophical Society, 83:357–375, 1978.

    Google Scholar 

  65. James Lee Hafner and Kevin S. McCurley. On the distribution of running times of certain integer factoring algorithms. Journal of Algorithms, 10:531–556, 1989.

    Google Scholar 

  66. James Lee Hafner and Kevin S. McCurley. A rigorous subexponential algorithm for computation of class groups. Journal of the American Mathematical Society, 2:837–850, 1989.

    Google Scholar 

  67. Ming-Deh A. Huang. Riemann hypothesis and finding roots over finite fields. In Proceedings of the 17th Annual Symposium on Theory of Computing, pages 121–130, New York, 1985. Association for Computing Machinery.

    Google Scholar 

  68. Ming-Deh A. Huang. Generalized riemann hypothesis and factoring polynomials over finite fields. Journal of Algorithms, 12:464–481, 1991.

    Google Scholar 

  69. William Kantor. Polynomial-time algorithms for finding elements of prime order and Sylow subgroups. Journal of Algorithms, 4:478–514, 1985.

    Google Scholar 

  70. Ravi Kannan, Gary L. Miller, and L. Rudolph. Sublinear parallel algorithm for computing the greatest common divisor of two integers. In Proceedings of the 25th Annual Symposium on Foundations of Computer Science, pages 7–11. IEEE Computer Society, 1984.

    Google Scholar 

  71. Ravi Kannan, Gary L. Miller, and L. Rudolph. Sublinear parallel algorithm for computing the greatest common divisor of two integers. SIAM Journal of Computing, 16:7–16, 1987. Extended abstract in [KMR84].

    Google Scholar 

  72. Donald E. Knuth. Seminumerical Algorithms, volume 2 of The Art of Computer Programming. Addison-Wesley, Reading, Massachusetts, second edition, January 1981.

    Google Scholar 

  73. Neal Koblitz. A Course in Number Theory and Cryptography. Number 114 in Graduate Texts in Mathematics. Springer-Verlag, New York, 1987.

    Google Scholar 

  74. Neal Koblitz. Elliptic curve cryptosystems. Mathematics of Computation, 48:203–209, 1987.

    Google Scholar 

  75. Neal Koblitz. Primality of the number of points on an elliptic curve over a finite field. Pacific Journal of Mathematics, 131:157–165, 1988.

    Google Scholar 

  76. Neal Koblitz. A family of Jacobians suitable for discrete log cryptosystems. In S. Goldwasser, editor, Advances in Cryptology: Proceedings of Crypto '88, volume 403 of Lecture Notes in Computer Science, pages 94–99, Berlin, 1990. Springer-Verlag.

    Google Scholar 

  77. Neal Koblitz. Constructing elliptic curve cryptosystems in characteristic 2. In A. J. Menezes and S. A. Vanstone, editors, Advances in Cryptology: Proceedings of Crypto '90, volume 537 of Lecture Notes in Computer Science, pages 156–167, Berlin, 1991. Springer-Verlag.

    Google Scholar 

  78. Sergei Konyagin and Carl Pomerance. On primes recognizable in deterministic polynomial time. preprint, May 1994.

    Google Scholar 

  79. Jeffrey C. Lagarias. Succinct certificates for the solvability of binary quadratic diophantine equations. In Proceedings of the 20th Annual Symposium on Foundations of Computer Science, pages 47–54. IEEE Computer Society, 1979.

    Google Scholar 

  80. Jeffrey C. Lagarias. On the computational complexity of determining the solvability or unsolvability of the equation x 2 − dy 2 = −1. Transactions of the American Mathematical Society, 260:485–508, 1980.

    Google Scholar 

  81. Jeffrey C. Lagarias. Worst-case complexity bounds for algorithms in the theory of integral quadratic forms. Journal of Algorithms, 1:142–186, 1980.

    Google Scholar 

  82. Jeffrey C. Lagarias. The computational complexity of simultaneous diophantine approximation problems. SIAM Journal of Computing, 14:196–209, 1985.

    Google Scholar 

  83. Susan Landau. Polynomial time algorithms for Galois groups. In J. Fitch, editor, Proceedings of EUROSAM '84, volume 174 of Lecture Notes in Computer Science, pages 225–236, New York, 1985. Springer-Verlag.

    Google Scholar 

  84. Susan Landau. Some remarks on computing the square parts of integers. Information and Computation, 78:246–253, 1988.

    Google Scholar 

  85. D. H. Lehmer. Computer technology applied to the theory of numbers. In Studies in Number Theory, pages 117–151. Mathematical Association of America, 1969. Distributed by Prentice Hall, Englewood Cliffs, NJ.

    Google Scholar 

  86. H. W. Lenstra, Jr. Primality testing algorithms (after Adleman, Rumely, and Williams). In Séminaire Bourbaki 1980/81, Exposé 576, number 901 in Lecture Notes in Mathematics, pages 243–257. Springer-Verlag, Berlin, 1981.

    Google Scholar 

  87. H. W. Lenstra, Jr. Integer programming with a fixed number of variables. Mathematics of Operations Research, 8:538–548, 1983.

    Google Scholar 

  88. H. W. Lenstra, Jr. Factoring integers with elliptic curves. Annals of Mathematics, 126:649–673, 1987.

    Google Scholar 

  89. H. W. Lenstra, Jr. Algorithms for finite fields. In Number Theory and Cryptography, volume 154 of London Mathematical Society Lecture Note Series, pages 76–85. Cambridge University Press, Cambridge, 1990.

    Google Scholar 

  90. H. W. Lenstra, Jr. Algorithms in algebraic number theory. Bulletin of the American Mathematical Society, 26:211–244, 1992.

    Google Scholar 

  91. Arjen K. Lenstra and H. W. Lenstra, Jr., editors. The development of the number field sieve. Number 1554 in Lecture Notes in Mathematics. Springer-Verlag, 1993.

    Google Scholar 

  92. Arjen K. Lenstra, H. W. Lenstra, Jr., and László Lovász. Factoring polynomials with rational coefficients. Mathematische Annalen, 261:515–534, 1982.

    Google Scholar 

  93. Jeffrey C. Lagarias, H. W. Lenstra, Jr., and Claus-Peter Schaorr. Korkine-Zolotarev bases and successive minima of a lattice and its reciprocal lattice. Combinatorica, 10:333–348, 1990.

    Google Scholar 

  94. Susan Landau and Gary L. Miller. Solvability by radicals is in polynomial time. Journal of Computer and System Science, 30:179–208, 1985.

    Google Scholar 

  95. Rudolf Lidl and Harald Niederreiter. Finite Fields. Addison-Wesley, Reading, MA, 1983.

    Google Scholar 

  96. László Lovász. An Algorithmic Theory of Numbers, Graphs, and Convexity. Number 50 in CBMS-NSF Regional Conference Series in Applied Mathematics. Society of Industrial and Applied Mathematicians, Philadelphia, PA, 1986.

    Google Scholar 

  97. Renet Lovorn. Rigorous Subexponential Algorithms for Discrete Logarithms over Finite Fields. PhD thesis, University of Georgia, May 1992.

    Google Scholar 

  98. H. W. Lenstra, Jr. and Carl Pomerance. A rigorous time bound for factoring integers. Journal of the American Mathematical Society, 5:483–516, 1992.

    Google Scholar 

  99. K. Manders and Leonard M. Adleman. NP-complete decision problems for binary quadratics. Journal of Computer and System Science, 16:168–184, 1978.

    Google Scholar 

  100. Michael Morrison and John Brillhart. A method of factoring and the factorization of F 7. Mathematics of Computation, 29:183–205, 1975.

    Google Scholar 

  101. Kevin S. McCurley. A key distribution system equivalent to factoring. Journal of Cryptology, 1:95–105, 1988.

    Google Scholar 

  102. Kevin S. McCurley. Cryptographic key distribution and computation in class groups. In Richard A. Mollin, editor, Proceedings of the NATO Advanced Study Institute on Number Theory and Applications, pages 459–479, The Netherlands, 1989. Kluwer.

    Google Scholar 

  103. Kevin S. McCurley. The discrete logarithm problem. In Pomerance [Pom90], pages 49–74.

    Google Scholar 

  104. Kevin S. McCurley. Odds and ends from cryptology and computational number theory. In Pomerance [Pom90], pages 145–166.

    Google Scholar 

  105. Gary L. Miller. Riemann's hypothesis and tests for primality. Journal of Computer and System Science, 13:300–317, 1976.

    Google Scholar 

  106. Victor Miller. Use of elliptic curves in cryptography. In Advances in Cryptology: Proceedings of Crypto '85, volume 218 of Lecture Notes in Computer Science, pages 417–426, Berlin, 1986. Springer-Verlag.

    Google Scholar 

  107. Alfred J. Menezes, Tatsuaki Okamoto, and Scott A. Vanstone. Reducing elliptic curve logarithms to logarithms in a finite field. IEEE Transactions in Information Theory, ???, 1994. Extended abstract in Proceedings of the 23rd ACM Symposium on Theory of Computing, 1991, ACM, pp. 80–89.

    Google Scholar 

  108. U.S. Department of Commerce. A proposed federal information processing standard for digital signature standard. In Federal Register, volume 56, no. 169, pages 42980–42982. U.S. GPO, August 1991.

    Google Scholar 

  109. Andrew Odlyzko. The discrete logarithm problem and its cryptographic significance. In Advances in Cryptology: Proceedings of Eurocrypt '84, volume 209 of Lecture Notes in Computer Science, pages 224–314, Berlin, 1985. Springer-Verlag.

    Google Scholar 

  110. Andrew Odlyzko. Discrete logarithms and smooth polynomials. In Gary L. Mullen and Peter Shiue, editors, Finite Fields: Theory, Applications, and Algorithms, Contemporary Mathematics Series, Providence, RI, 1994. American Mathematical Society.

    Google Scholar 

  111. H. Ong, Claus-Peter Schnorr, and Adi Shamir. An efficient signature scheme based on quadratic equations. In Proceedings of the 16th Annual Symposium on Theory of Computing, pages 208–216, New York, 1984. Association for Computing Machinery.

    Google Scholar 

  112. Stephen Pohlig and Martin Hellman. An improved algorithm for computing discrete logarithms over GF(p) and its cryptographic significance. IEEE Transactions on Information Theory, 24:106–110, 1978.

    Google Scholar 

  113. Jonathan Pila. Frobenius maps of abelian varieties and finding roots of unity in finite fields. Mathematics of Computation, 55:745–763, 1990.

    Google Scholar 

  114. D. A. Plaisted. Fast verification, testing, and generation of large primes. Theoretical Computer Science, 9:1–16, 1979.

    Google Scholar 

  115. Carl Pomerance. Analysis and comparison of some integer factoring methods. In H. W. Lenstra, Jr. and R. Tijdeman, editors, Computational Methods in Number Theory, Part I, number 154 in Math. Centre Tract, pages 89–139. Math. Centre, Amsterdam, 1982.

    Google Scholar 

  116. Carl Pomerance. Fast, rigorous factorization and discrete logarithm algorithms. In Discrete Algorithms and Complexity (Proceedings of the Japan-US Joint Seminar on Discrete Algorithms and Complexity Theory), pages 119–143. Academic Press, 1986.

    Google Scholar 

  117. Carl Pomerance, editor. Cryptography and Computational Number Theory, volume 42 of Proceedings of Symposia in Applied Mathematics. American Mathematical Society, Providence, 1990.

    Google Scholar 

  118. Carl Pomerance. Recent developments in primality testing. Mathematical Intelligencer, 3:97–105, 1980/81.

    Google Scholar 

  119. Vaughn Pratt. Every prime has a succinct certificate. SIAM Journal of Computing, 4:214–220, 1975.

    Google Scholar 

  120. John Pollard and Claus-Peter Schnorr. Solution of x 2 + ky 2 ≡ m (mod n), with application to digital signatures. IEEE Transactions on Information Theory, 22:702–709, 1987.

    Google Scholar 

  121. Janos Pintz, William L. Steiger, and Endre Szemerédi. Two infinite sets of primes with fast primality tests. In Proceedings of the 20th Annual Symposium on Theory of Computing, pages 504–509. Association for Computing Machinery, 1988. Journal version in [PSS89].

    Google Scholar 

  122. Janos Pintz, William L. Steiger, and Endre Szemerédi. Infinite sets of primes with fast primality tests and quick generation of large primes. Mathematics of Computation, 53:399–406, 1989. Extended abstract in [PSS88].

    Google Scholar 

  123. Michael O. Rabin. Digitalized signatures and public-key functions as intractible as factorization. Technical Report TR-212, Massachussetts Institute of Technology, Laboratory for Computer Science, 1979.

    Google Scholar 

  124. Michael O. Rabin. Probabilistic algorithm for testing primality. Journal of Number Theory, 12:128–138, 1980.

    Google Scholar 

  125. Michael O. Rabin. Probabilistic algorithms in finite fields, SIAM Journal of Computing, 9:273–280, 1980.

    Google Scholar 

  126. Hans Riesel. Modern factorization methods. BIT, 25:205–222, 1985.

    Google Scholar 

  127. Hans Riesel. Prime Numbers and Computer Methods for Factorization. Birkhäuser, Boston, 1985.

    Google Scholar 

  128. L. Rónyai. Factoring polynomials over finite fields. Journal of Algorithms, 9:391–400, 1988.

    Google Scholar 

  129. L. Rónyai. Galois groups and factoring polynomials over finite fields. In Proceedings of the 30th Annual Symposium on Foundations of Computer Science, pages 99–104. IEEE Computer Society, 1989.

    Google Scholar 

  130. Ronald Rivest, Adi Shamir, and Leonard M. Adleman. A method for obtaining digital signatures and public key cryptosystems. Communications of the ACM, 21:120–126, 1978.

    Google Scholar 

  131. Rene School. Quadratic fields and factorisation. In H. W. Lenstra, Jr. and R. Tijdeman, editors, Computational Methods in Number Theory, Part I, number 154 in Math. Centre Tract. Math. Centre, Amsterdam, 1982.

    Google Scholar 

  132. R. Schoof. Elliptic curves over finite fields and the computation of square roots modulo p. Mathematics of Computation, 44:483–494, 1985.

    Google Scholar 

  133. Claus-Peter Schnorr. Efficient signature generation by smart cards. Journal of Cryptology, 4:161–174, 1991.

    Google Scholar 

  134. Oliver Schirokauer. Discrete logarithms and local units. Philisophical Transactions of the Royal Society of London (A), 345:409–423, 1993.

    Google Scholar 

  135. Martin Seysen. A probabilistic factorization algorithm with quadratic forms of negative discriminant. Mathematics of Computation, 48:757–780, 1987.

    Google Scholar 

  136. Daniel Shanks. Class number, a theory of factorization, and genera. In Analytic Number Theory, volume 20 of Proceedings of Symposia in Pure Mathematics, pages 415–440. American Mathematical Society, 1971.

    Google Scholar 

  137. Daniel Shanks. Five number-theoretic algorithms. In Proceedings of the Second Manitoba Conference on Numerical Mathematics, pages 51–70, 1972.

    Google Scholar 

  138. Jeffrey O. Shallit. An exposition of Pollard's algorithm for quadratic congruences. Technical Report 84-006, University of Chicago, Department of Computer Science, December 1984.

    Google Scholar 

  139. Victor Shoup. New algorithms for finding irreducible polynomials over finite fields. Mathematics of Computation, 54:435–447, 1990. Extended abstract in Proceedings of the 29th Annual IEEE Symposium on Foundations of Computer Science (1988), pp. 283–290.

    Google Scholar 

  140. Victor Shoup. On the deterministic complexity of factoring polynomials over finite fields. Information Processing Letters, 33:261–267, 1990.

    Google Scholar 

  141. Victor Shoup. Searching for primitive roots in finite fields. In Proceedings of the 22th Annual Symposium on Theory of Computing, pages 546–554, 1990.

    Google Scholar 

  142. Peter W. Shor. Polynomial time algorithms for discrete logarithms and factoring on a quantum computer. In Proceedings of the 1994 Algorithmic Number Theory Symposium. Springer-Verlag, to appear.

    Google Scholar 

  143. Joseph H. Silverman. The Arithmetic of Elliptic Curves, volume 106 of Graduate Texts in Mathematics. Springer-Verlag, 1986.

    Google Scholar 

  144. Claus-Peter Schnorr and H. W. Lenstra, Jr. A Monte Carlo factoring algorithm with linear storage. Mathematics of Computation, 43:289–311, 1984.

    Google Scholar 

  145. Jonathan Sorenson. Counting the integers factorable via cyclotomic methods. Technical Report 919, University of Wisconsin, Deparment of Computer Sciences, 1990.

    Google Scholar 

  146. Alfred Schönhage and Volker Strassen. Schnelle Multiplikation grosser Zahlen. Computing, 7:281–292, 1971.

    Google Scholar 

  147. R. Solovay and Volker Strassen. A fast Monte-Carlo test for primality. SIAM Journal of Computing, 6:84–85, 1977.

    Google Scholar 

  148. Jeffrey O. Shallit and Adi Shamir. Number-theoretic functions which are equivalent to number of divisors. Information Processing Letters, 20:151–153, 1985.

    Google Scholar 

  149. C. Thiel. Verifying the class number belongs to NP∩coNP. In Proceedings of the 1994 Algorithmic Number Theory Symposium. Springer-Verlag, to appear.

    Google Scholar 

  150. A. Tonelli. Bemerkung ber die aufl sung quadratischer Congruenzen. Göttinger Nachrichten, pages 314–346, 1891.

    Google Scholar 

  151. P. van Emde Boas. Another NP-complete partition problem and the complexity of computing short vectors in lattices. Technical Report 81-04, Mathematics Department, University of Amsterdam, 1981.

    Google Scholar 

  152. Paul van Oorschot. A comparison of practical public key cryptosystems based on integer factorization and discrete logarithms. In Gustavus J. Simmons, editor, Contemporary Cryptology: The Science of Information Integrity, IEEE Proceedings, pages 280–322. IEEE, 1991.

    Google Scholar 

  153. Joachim von zur Gathen, Marek Karpinski, and Igor Shparlinski. Counting curves and their projections. preprint, March 1993.

    Google Scholar 

  154. Y. Wang. On the least primitive root of a prime. Scientia Sinica, 10:1–14, 1961.

    Google Scholar 

  155. Hugh C. Williams. Primality testing on a computer. Ars Combinatorica, 5:127–185, 1978.

    Google Scholar 

  156. Hugh C. Williams. Factoring on a computer. Mathematical Intelligencer, 6:29–36, 1984.

    Google Scholar 

  157. Andrew C. Yao. Theory and applications of trapdoor functions. In Proceedings of the 23rd Annual Symposium on Foundations of Computer Science, pages 80–91. IEEE Computer Society, 1982.

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, University of Southern California, 90089-0782, Los Angeles, CA, USA

    Leonard M. Adleman

  2. Organization 1423, MS 1110, Sandia National Laboratories, 87185, Albuquerque, NM, USA

    Kevin S. McCurley

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  1. Leonard M. Adleman
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  2. Kevin S. McCurley
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Leonard M. Adleman Ming-Deh Huang

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© 1994 Springer-Verlag Berlin Heidelberg

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Adleman, L.M., McCurley, K.S. (1994). Open problems in number theoretic complexity, II. In: Adleman, L.M., Huang, MD. (eds) Algorithmic Number Theory. ANTS 1994. Lecture Notes in Computer Science, vol 877. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58691-1_70

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  • DOI: https://doi.org/10.1007/3-540-58691-1_70

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  • Online ISBN: 978-3-540-49044-9

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