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My professor has given me an RSA factoring problem as an assignment. The given modulus is 30 decimal digits long. I have been searching a lot about factoring algorithms. But it has been quite a headache to choose one for my given requirements. Which algorithms give the best performance for 30 decimal digit numbers?

Note: So far I have read about Brute force approach and Quadratic Sieve. The latter is complex and the former time consuming.

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  • $\begingroup$ Factoring a 30 digits number is an easy task. As an example, you can type Factorization(10263280077814176196883978050069); at magma.maths.usyd.edu.au/calc and see the result. You can change my number and see the decumentation for details of factorization methods. $\endgroup$ – Meysam Ghahramani Apr 7 '20 at 7:13
  • $\begingroup$ @MeysamGhahramani Thank you for answering it. However in a night's search I have found pollard-rho to be the one that suits my requirement. nevertheless, once again thank you. $\endgroup$ – rajiv Apr 7 '20 at 8:28
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    $\begingroup$ Use Linux factor command? Also Wolfram alpha can do that for you. $\endgroup$ – kelalaka Apr 7 '20 at 9:55
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Which algorithms give the best performance for 30 decimal digit numbers?

30 decimal digits is 100 binary digits.

  • Trial division will require around $2^{50}$ divisions.
  • Pollard-Rho-Factoring will require around $2^{25}$ checks - this may be a good pick for you if you want to implement it yourself quickly and don't mind having to wait a bit on an optimized execution result.
  • Most decent quadratic sieve implementations will do this number within seconds.
  • Most ECM implementations will do that too. If you need something more optimized than Pollard-Rho, using ECM is probably a good pick. Refer to this database for optimal curve representation choices and the Handbook of Applied Cryptography (Ch14, PDF) for fast scalar multiplication strategies.
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