Cryptography

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2 GNFS examples: RSA ´ 130 p130 digitsq “ 18070820886874048059516561644059055662781025167694013491 70127021450056662540244048387341127590812303371781887966 563182013214880557 “ 396859994595974542901611261628837 86067576449112810064832555157243 ˆ 455344986467359721884036868972744 08864356301263205069600999044599. RSA ´ 140 p140 digitsq “ 2129024631825875754749788201627151749780670396327721627 8233383215381949984056495911366573853021918316783107387 995317230889569230873441936471 “ 33987174230284385545301236276138758 35633986495969597423490929302771479 ˆ 62642001874012850961516549482644422 19302037178623509019111660653946049.

R. Simon, On the power of quantum computation. SIAM J. Comput. 26(5), 1474–1483 (1997) [41] M. Sipser, Introduction to the Theory of Computation, 2nd edn. (Thomson, Boston, 2006) [42] W. Trappe, L. Washington, Introduction to Cryptography with Coding Theory, 2nd edn. (Prentice-Hall, Englewood Cliffs, 2006) [43] A. Turing, On computable numbers, with an application to the Entscheidungsproblem. Proc. Lond. Math. Soc. Ser. 2 42, 230–260 (1937); 43, 544–546 (1937) 30 1. P. H. Clearwater, in Explorations in Quantum Computation.

Let α P C be an algebraic number that is the root of f pxq, and denote the set of polynomials in α with integer coefficients as Zrαs. 3. Define the mapping (ring homomorphism): Φ : Zrαs ÞÑ Zn via Φpαq “ m which ensures that for any f pxq P Zrxs, we have Φpf pαqq ” f pmq p mod nq. 4. Find a finite set U of coprime integers pa, bq such that ź ź pa ´ bαq “ β 2 , pa ´ bmq “ y 2 pa,bqPU pa,bqPU for β P Zrαs and y P Z. Let x “ Φpβq. Then x2 ” ΦpβqΦpβq ” Φpβ 2 q ¨ ˛ ź ” Φ˝ pa ´ bαq‚ pa,bqPU ” ź Φpa ´ bαq pa,bqPU ” ź pa ´ bmq pa,bqPU ” y2 pmod nq which is of the required form of the factoring congruence, and hopefully, a factor of n can be found by calculating gcdpx ˘ y, nq.

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