## How Shor's Algorithm Factors 314191

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This video explains how Shor’s Algorithm factors the pseudoprime number 314191 into its prime factors using a quantum computer. The quantum computation relies on the number-theoretic analysis of the factoring problem via modular arithmetic mod N (where N is the number to be factored), and finding the order or period of a random coprime number mod N. The exponential speedup comes in part from the use of the quantum fast fourier transform which achieves interference among frequencies that are not related to the period (period-finding is the goal of the QFT FFT).

REFERENCES

RSA Numbers (sample large numbers to try factoring)
https://en.wikipedia.org/wiki/RSA_numbers

IBM on RSA https://www.ibm.com/support/knowledgecenter/en/SSB23S1.1.0.13/gtps7/s7pkey.html

Modulo Multiplication Group Tables http://mathworld.wolfram.com/ModuloMultiplicationGroup.html

Difference of squares factorization https://en.wikipedia.org/wiki/Difference_of_two_squares

Euclid’s Algorithm https://en.wikipedia.org/wiki/Euclideanalgorithm

Rational sieve for factoring https://en.wikipedia.org/wiki/Rational_sieve

General Number field Sieve https://en.wikipedia.org/wiki/Generalnumberfieldsieve

Scott Aaronson blog post about Shor’s Algorithm https://www.scottaaronson.com/blog/?p=208

Experimental implementation of Shor’s Algorithm (factoring 15, 21, and 35) https://arxiv.org/pdf/1903.00768.pdf

Adiabatic Quantum Computation factoring the number 291311 https://arxiv.org/pdf/1706.08061.pdf

Scott Aaronson course notes https://www.scottaaronson.com/qclec/ https://www.scottaaronson.com/qclec/combined.pdf

Shor’s Algorithm on Quantiki https://www.quantiki.org/wiki/shors-factoring-algorithm

TLS And SSL use RSA encryption https://en.wikipedia.org/wiki/TransportLayerSecurity