Introduction to Cryptography
Introduction to Cryptography CIS 628
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This 1 page Class Notes was uploaded by Sam Rau on Wednesday October 21, 2015. The Class Notes belongs to CIS 628 at Syracuse University taught by Staff in Fall. Since its upload, it has received 52 views. For similar materials see /class/225603/cis-628-syracuse-university in Computer & Information Science at Syracuse University.
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Date Created: 10/21/15
The question was raised in class ifgcda b 61 then for any positive integer k does gcdkakb kd Recall from the fundamental theorem of arithmetic that any positive integer can be factored into primes with a unique representation that is for any positive integer x 7 m1 m2 m3 quot391 xipl 39P2 P3 Pj This can equivalently be written quot391 x2m2 3m3 5m5 mpj if we allow some of the m to be 0 In particular all of a b and d can be represented thus a 7 2m2 3m sms mp1 7 pj b72n23quot35quot5 quotpf 7 pl 7 1 1 1 F 51722 33 55 p In this representation each It minmn That is the exponent for each prime in the factorization of d is the minimum of the two exponents in the factorizations of a and b d 2miquotm2 7712 3miquotm3 7713 5miquotm57quot5 Williquotmp1 npf 1 Now k can also be factored the same way ie 7 02 03 05 0p kiZ 3 5 pj zka 2m202 3m303 5m505 pZ39FJrop kb 2n202 3n303 5quot505 prlpJroj and gcdkakb z 39lt 7n202 j 39 lt 5 mm 3 39lt 1 was 2 quot quot quotF FW F F 202 303 505 pjp Zminm2n2 3minm3n3 5minm5n5 mprrtinmpjynpj 202303505 0 d b p gc a kgcdab D
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