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We can be clever and only look at SQRT(N_bits).-Classical factoring algorithm is of order O(e^log(N_bits)^3) and called General Number Field Sieve.-Number of particles in universe: 10^80 ~ 2^256
-Coprime meaning no common factor other than 1. GCD(x,N)=1.-Exponential number of a’s to check since we don’t know it. a is from 0-q-1-r has to be even or above won’t work.-GCD of last factors and N will yield non-trivial factor of N.
-take O(M^3) steps, where M is the number of bits N is.
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A Complexity Analysis of Shor’s Quantum Factoring Algorithm J. Caleb Wherry Austin Peay State University Department of Computer Science
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Introduction 3 Relativity Computer QuantumComputing Closed Timelike Curve Computation DNA Computing S. Aaronson and J. Watrous. Closed Timelike Curves Make Quantum and Classical Computing Equivalent, Proceedings of the Royal Society A 465:631-647, 2009. arXiv:0808.2669. 3
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Classical Factoring 4 Composite Number For RSA, p & q are prime. Why is Factoring Hard? Example 1: N being 16 bits long Example 2: N being 32 bits long 216 = 65,536 possible values 232 = 4,294,967,296 possible values RSA Normal RSA public key (above N) is around 1024-2048 bits.
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Quantum Computing 5 Orthonormal Basis Set Superposition of 0 & 1 |0 + |1 |0 |1 |0 | |1 E.g. = Qubits: Photons, Electrons, Ions, etc. *Spin of above particles. | Bloch Sphere
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Shor’s Quantum Factoring Algorithm 6 We have an integer N that we want to factor. To factor, we have to find the period of this function: Where x < N and coprime to N. Using rules of modular arithmetic to yield: Set up quantum side: Quantum Register:
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Shor’s Quantum Factoring Algorithm 7 Quantum Register: State After Transformation: Measure Register 1: New State: Quantum Fourier Transform What does this do? “Peaks” values in Register 1 around multiples of: Measure this register to get one of these values, then compute classically r with continued fractions.
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Shor’s Quantum Factoring Algorithm Once we have r, computing above will yield a non-trivial factor of N. QFT takes no more than O(M3), where M is the number of bits N is. *Caveat: Shor’s Algorithm resides in BQP, so the answer could be wrong. Re-run to get another answer or to verify first answer. Note though, running polynomial algorithm multiple times is much better than running 1 exponential algorithm.
References 11 Arora, Sanjeev;Barak, Boaz. “Computational Complexity: A Modern Approach.” New York: Cambridge University Press. 2009. Print. Hayward, Matthew. “Quantum Computing and Shor’s Algorithm.” Sydney: Macquarie University Mathematics Department. 2008. Print. Nielsen, Michael A.; Chuang, Isaac L. “Quantum Computation and Quantum Information.” New York: Cambridge University Press. 2000. Print. Shor, Peter W. “Algorithms for Quantum Computation: Discrete Logarithms and Factoring.” Proc. 35th Annual Symposium on Foundations of Computer Science. Ed. ShafiGoldwasser. IEEE Computer Society Press, 1994. 124-136. Print.
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