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Shor Algorithm (continued)

Shor Algorithm (continued). Use of number theory and reductions. Anuj Dawar. Reductions. Solve RSA. Factor big integers. Find period. Estimate Phase. Fourier Transform. RCA ENCRYPTION. Easy to multiply but difficult to factor big integers. Review of Number Theory.

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Shor Algorithm (continued)

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  1. Shor Algorithm (continued) Use of number theory and reductions Anuj Dawar

  2. Reductions Solve RSA Factor big integers Find period Estimate Phase Fourier Transform

  3. RCA ENCRYPTION

  4. Easy to multiply but difficult to factor big integers.

  5. Review of Number Theory

  6. Shor knows number theory and uses it!!! • In many cases, we can use the knowledge from other areas of research in a new and creative way. • You do not have to invent everything from scratch. You just reuse something that was invented by other people. • If the two areas are not obviously linked, your invention can be very important. • This is exactly what was done by Shor. We introduced modular arithmetic in last lecture as a general tool for algorithms and hardware Now we will show how creatively Shor used it in his algorithm.

  7. Assume: We want to find the smallest r such that the above is true

  8. We want to find the smallest r such that the above is true Now we substitute m1 * m2 for N

  9. Greatest common denominator More interesting case

  10. We want to find the smallest r such that the above is true Finding the smallest period r

  11. But we had some additional assumptions on last slide, what if not satisfied? Do not worry now, we are not mathematicians

  12. So now we are quite optimistic! So now what remains is to be able to find period, but this is something well done with spectral transforms.

  13. Reductions Solve RSA Factor big integers We are here Find period Estimate Phase This was done earlier Fourier Transform

  14. Going Back to Phase Estimation We will use phase estimation to find period

  15. Choosing the operator U It requires modulo multiplication in modular arithmetic Not trivial Potential research how to do this efficiently

  16. Choosing the initial state for operator U In general not easy But hopefully we find a special case Potential research how to do this efficiently for arbitrary cases

  17. Phase is 1/r

  18. Now the problem is reduced to creation of certain quantum state. We published papers – see David Rosenbaum

  19. Final circuit for period finding We find this Easy initialization Number to be factorized a r = 1 mod N U |x |a*x mod N 

  20. Now we use a classical computer. Therefore, using the QPE algorithm, we can efficiently calculate k -- r where k and r are unknown 2. If k and r are co-prime, then canceling to an irreducible fraction will yield r. 3. If k and r are not co-prime, we try again.

  21. Summary of Shor Algorithm 1. We want to find m1 * m2 = N where N is the number to factorize 2. We prove that this problem is equivalent to solving a r = 1 mod N 3. We use the QPE circuit initialized to |0 |1 4. We calculate each of the circuits U, U2, … U 2^2n 5. We apply the Quantum Phase Estimation Algorithm. 6. We use standard computer for verification and we repeat QPE if required.

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