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Using the sequences v 2m v − a0 0 = 2m 1 an = a0 = n−1 n = 1 n−1 − an p0 = a0 p1 pn q0 q1 qn = = = = = a1 a0 + 1 an pn−1 + pn−2 1 a1 an qn−1 + qn−2 , compute the first fraction qpnn such that qn < M ≤ qn+1 . See any standard number theory text, such as Hardy and Wright [1979], for why this procedure works. In the high probability case when 2vm is within M1 2 of a multiple rj of 1r , the fraction obtained from the above procedure is ACM Computing Surveys, Vol. 32, No. 3, September 2000. 333 j r , because it has denominator less than M .

1992. Quantum cryptography using any two nonorthogonal states. Physical Review Letters 68, 3121–3124. BENNETT, C. , AND VAZIRANI, U. V. 1997. Strengths and weaknesses of quantum computing. Society for Industrial and Applied Mathematics Journal on Computing 26, 5, 1510–1523. gov/abs/quant-ph/9701001. BENNETT, C. H. AND BRASSARD, G. 1987. Quantum public key distribution reinvented. SIGACT News (ACM Special Interest Group on Automata and Computability Theory) 18, 51–53. BENNETT, C. , AND EKERT, A.

In the high-probability case that v is M2 m within 12 of j 2r , this fraction will be rj . The unique fraction with denominator less than M that is within M1 2 of 2vm can be obtained efficiently from the continued fraction expansion of 2vm as follows. Using the sequences v 2m v − a0 0 = 2m 1 an = a0 = n−1 n = 1 n−1 − an p0 = a0 p1 pn q0 q1 qn = = = = = a1 a0 + 1 an pn−1 + pn−2 1 a1 an qn−1 + qn−2 , compute the first fraction qpnn such that qn < M ≤ qn+1 . See any standard number theory text, such as Hardy and Wright [1979], for why this procedure works.

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