Monte Carlo methods for index computation (𝑚𝑜𝑑𝑝)
Open accessAbstract
We describe some novel methods to compute the index of any integer relative to a given primitive root of a prime p. Our first method avoids the use of stored tables and apparently requires O ( p 1 / 2 ) O(p^{1/2}) operations. Our second algorithm, which may be regarded as a method of catching kangaroos, is applicable when the index is known to lie in a certain interval; it requires O ( w 1 / 2 ) O(w^{1/2}) operations for an interval of width w , but does not have complete certainty of success. It has several possible areas of application, including the factorization of integers.
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The asymptotically best currently known classical algorithms for computing discrete logarithms in subgroups of this form are generic cycle-finding algorithms, such as Pollard's ρ- and λ-algorithms, that run in time O(√r) and O(√d), respectively, and the general number field sieve (GNFS), that runs in subexponential time in the bit length n of N.
“The asymptotically best currently known classical algorithms for computing discrete logarithms in subgroups of this form are generic cycle-finding algorithms, such as Pollard's ρ - and λ -algorithms [69], that run in time O ( √ r ) and O ( √ d ) , respectively, and the general number field sieve (GNFS), that runs in subexponential time in the bit length n of N .”
From How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits (2021) · cites this paper as [69]
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Research works printed but not linked: 14
- D. E. KNUTH, Seminumerical Algorithms, The Art of Computer Programming, Vol. 2, Addison-Wesley, Reading, Mass., 1969. MR 44 #3531.
- A. RAPOPORT, "Cycle distributions in random nets," BulL Math. Biophys., v. 10, 1948, pp. 145-157.
- B. HARRIS, "Probability distributions related to random mappings," Ann. Math. Statist., v. 31, 1960, pp. 1045-1062.
- J. M. POLLARD, "A Monte Carlo method for factorization," BIT, v. 15, 1975, No. 3, pp. 331-335. MR 52 #13611.
- R. K. GUY, "How to factor a number," Proc. Fifth Manitoba Conf. on Numerical Math. (Winnipeg, Man., 1975), Univ. of Calgary, Dept. of Math., Statist, and Comp. Sei., Research paper no. 301, Dec. 197 5.
- J. S. DEVITT, Aliquot Sequences, M. Sc. Thesis, Dept. of Math, and Stat., Univ. of Calgary, May 1976.
- D. E. KNUTH, Sorting and Searching, The Art of Computer Programming, Vol. 3, Addi- son-Wesley, Reading, Mass., 1973.
- D. SHANKS, "Class number, a theory of factorization, and genera," Proc. Sympos. Pure Math., vol. 20, Amer. Math. Soc, Providence, R. I., 1970, pp. 415-440. MR 47 #4932.
- J. M. POLLARD, "Theorems on factorization and primality testing," Proc Cambridge Philos. Soc, V. 76, 1974, pp. 521-528. MR 50 #6992.
- W. DIFFIE & M. E. HELLMAN, "New directions in cryptography," IEEE Trans. Infor- mation Theory, vol. IT-22, No. 6, Nov. 1976, pp. 644-654.
- J. C. P. MILLER, "On factorization, with a suggested new approach," Math. Comp., v. 29, 1975, pp. 155-172.
- MICHAEL A. MORRISON & JOHN BRILLHART, "A method of factoring and the factorization of F7," Math. Comp., v. 29, 1975, pp. 183-205.
- D. H. LEHMER, "An announcement concerning the Delay Line Sieve DLS 127," Math. Comp., v. 20, 1966, pp. 645-646.
- D. H. LEHMER & EMMA LEHMER, "A new factorization technique using quadratic forms," Math. Comp., v. 28, 1974, pp. 625-635.
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- Authors
- Pollard, J. M.
- Corresponding author
- Pollard, J. M.
- Venue
- Mathematics of Computation
- Publisher
- American Mathematical Society (AMS)
- Published
- 1978
- Volume
- 32
- Issue
- 143
- Pages
- 918-924
- ISSN
- 0025-5718 (ISSN-L)
- 1088-6842
- OpenAlex
- W2019816098
- Cited by
- 825
- References
- 0
- Citations by year
- 2026: 1 · 2025: 7 · 2024: 11 · 2023: 4 · 2022: 11 · 2021: 31 · 2020: 38 · 2019: 24 · 2018: 39 · 2017: 27 · 2016: 34 · 2015: 28 · 2014: 45 · 2013: 32 · 2012: 48
- FWCI
- 7.27
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Mathematics of Computation (publishedVersion)
- Topics
- Physical Sciences › Computer Science › Information Systems › Cryptography and Residue Arithmetic
- Physical Sciences › Mathematics › Algebra and Number Theory › Analytic Number Theory Research
- Physical Sciences › Mathematics › Modeling and Simulation › Mathematical and Computational Methods
- Keywords
- discrete logarithm problem, index computation, primitive roots, prime modulus, randomized algorithms, integer factorization, modular arithmetic
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