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Public-key Cryptography: Theory and Practice by C. E. Veni Madhavan, Abhijit Das

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4.3. The Integer Factorization Problem

The integer factorization problem (IFP) (Problems 4.1, 4.2 and 4.3) is one of the most easily stated and yet hopelessly difficult computational problem that has attracted researchers’ attention for ages and most notably in the age of electronic computers. A huge number of algorithms varying widely in the basic strategy, mathematical sophistication and implementation intricacy have been suggested, and, in spite of these, factoring a general integer having only 1000 bits seems to be an impossible task today even using the fastest computers on earth.

It is important to note here that even proving rigorous bounds on the running times of the integer-factoring algorithms is quite often a very difficult task. ...

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