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# Integer Factoring Using Small Algebraic Dependencies

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Manindra Agrawal, Nitin Saxena, and Shubham Sahai Srivastava. Integer Factoring Using Small Algebraic Dependencies. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.MFCS.2016.6

## Abstract

Integer factoring is a curious number theory problem with wide applications in complexity and cryptography. The best known algorithm to factor a number n takes time, roughly, exp(2*log^{1/3}(n)*log^{2/3}(log(n))) (number field sieve, 1989). One basic idea used is to find two squares, possibly in a number field, that are congruent modulo n. Several variants of this idea have been utilized to get other factoring algorithms in the last century. In this work we intend to explore new ideas towards integer factoring. In particular, we adapt the AKS primality test (2004) ideas for integer factoring. In the motivating case of semiprimes n=pq, i.e. p<q are primes, we exploit the difference in the two Frobenius morphisms (one over F_p and the other over F_q) to factor n in special cases. Specifically, our algorithm is polynomial time (on number theoretic conjectures) if we know a small algebraic dependence between p,q. We discuss families of n where our algorithm is significantly faster than the algorithms based on known techniques.
##### Keywords
• integer
• factorization
• factoring integers
• algebraic dependence
• dependencies

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## References

1. Manindra Agrawal, Neeraj Kayal, and Nitin Saxena. PRIMES is in P. Annals of Math, 160(2):781-793, 2004.
2. Shi Bai, Pierrick Gaudry, Alexander Kruppa, Emmanuel Thome, and Paul Zimmermann. Factorization of RSA-220 with CADO-NFS. 2016.
3. Daniel Julius Bernstein. Introduction to post-quantum cryptography. In Post-quantum cryptography, pages 1-14. Springer, 2009.
4. Dan Boneh et al. Twenty years of attacks on the RSA cryptosystem. Notices of the AMS, 46(2):203-213, 1999.
5. Joe Peter Buhler, Hendrik Willem Lenstra Jr, and Carl Pomerance. Factoring integers with the number field sieve. In The development of the number field sieve, pages 50-94. Springer, 1993.
6. Yingpu Deng and Yanbin Pan. An algorithm for factoring integers. Cryptology ePrint Archive, Report 2012/097, 2012.
7. John D Dixon. Asymptotically fast factorization of integers. Mathematics of computation, 36(153):255-260, 1981.
8. Carl Friedrich Gauss. Disquisitiones Arithmeticae. 1801. Article 329.
9. Joseph Gerver. Factoring large numbers with a quadratic sieve. Mathematics of Computation, 41(163):287-294, 1983.
10. Rajiv Gupta and Maruti Ram Murty. A remark on artin’s conjecture. Inventiones mathematicae, 78(1):127-130, 1984.
11. F.R.S. Horsley, Rev. Samuel. The sieve of eratosthenes. being an account of his method of finding all the prime numbers. Philosophical Transactions (1683-1775), 62:327-347, 1772.
12. Ravi Kannan. Algorithmic geometry of numbers. Annual review of computer science, 2(1):231-267, 1987.
13. Thorsten Kleinjung, Kazumaro Aoki, Jens Franke, Arjen Klaas Lenstra, Emmanuel Thomé, Joppe W Bos, Pierrick Gaudry, Alexander Kruppa, Peter Lawrence Montgomery, Dag Arne Osvik, et al. Factorization of a 768-bit RSA modulus. In Advances in Cryptology-CRYPTO'10, pages 333-350. 2010.
14. R Sherman Lehman. Factoring large integers. Mathematics of Computation, 28(126):637-646, 1974.
15. Arjen Klaas Lenstra, Hendrik Willem Lenstra Jr., Mark Steven Manasse, and John M. Pollard. The number field sieve. In Proceedings of the Twenty-second Annual ACM Symposium on Theory of Computing, pages 564-572, 1990.
16. Arjen Klaas Lenstra, Hendrik Willem Lenstra, and Lászlo Lovász. Factoring polynomials with rational coefficients. Math. Ann., 261:515-534, 1982.
17. Hendrik Willem Lenstra Jr. Factoring integers with elliptic curves. Annals of mathematics, pages 649-673, 1987.
18. James McKee. Turning euler’s factoring method into a factoring algorithm. Bulletin of the London Mathematical Society, 28(133):351-355, 1996.
19. Pieter Moree. Artin’s primitive root conjecture—a survey. INTEGERS, 10(6):1305-1416, 2012.
20. Oystein Ore. Number theory and its history. Courier Corporation, 2012.
21. John M Pollard. Theorems on factorization and primality testing. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 76 of Cambridge Univ Press, pages 521-528, 1974.
22. John M Pollard. A monte carlo method for factorization. BIT Numerical Mathematics, 15(3):331-334, 1975.
23. Carl Pomerance. The quadratic sieve factoring algorithm. In Advances in cryptology, pages 169-182, 1985.
24. Carl Pomerance. A tale of two sieves. Biscuits of Number Theory, 85, 2008.
25. Arnold Schönhage. Factorization of univariate integer polynomials by diophantine approximation and improved basis reduction algorithm. ICALP, 172:436-447, 1984.
26. Peter Williston Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput., 26(5):1484-1509, 1997.
27. Lawrence Clinton Washington. Introduction to cyclotomic fields, volume 83. Springer, 2012.
28. Hugh Cowie Williams. A p+1 method of factoring. Mathematics of Computation, 39(159):225-234, 1982.
29. Hugh Cowie Williams and Jeffrey Outlaw Shallit. Factoring integers before computers. Mathematics of computation, 48:481-531, 1994. (1943-1993, Fifty Years of Computational Mathematics (W. Gautschi, ed.), Proc. Sympos. Appl. Math.).