Problem Set 5.pdf - Problem Set#5 Brandon DeAlmeida 1 For...

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Problem Set #5 Brandon DeAlmeida 1) For the past two weeks we ve been continuing learning about the RSA algorithm. While covering more about the cryptosystem we ve learned new techniques for factorization that are relevant to RSA. These, such as Sieving, have allowed us to learn about other topics such as B-smooth. 2) We wish to factor N = 493 by sieving prime powers up to B = 11 on the values of F(23) to F(38) where F(t) = t 2 − N . Starting with 23 we get: 23 2 N 36 ≡ N 2 2 ⋅ 3 2 24 2 N 83 25 2 N 132 ≡ N 2 2 ⋅ 3 ⋅ 11 26 2 N 183 ≡ N 3 ⋅ 61 27 2 N 236 ≡ N 2 2 ⋅ 59 28 2 N 291 ≡ N 3 ⋅ 97 29 2 N 348 ≡ N 2 2 ⋅ 3 ⋅ 29 30 2 N 407 ≡ N 11 ⋅ 37 31 2 N 468 ≡ N 2 2 ⋅ 3 2 ⋅ 13 32 2 N 38 ≡ N 2 ⋅ 19 33 2 N 103 34 2 N 170 ≡ N 2 ⋅ 5 ⋅ 17 35 2 N 239 36 2 N 310 ≡ N 2 2 ⋅ 5 ⋅ 31 37 2 N 383 38 2 N 458 ≡ N 2 ⋅ 229 39 2 N 42 ≡ N 2 ⋅ 3 ⋅ 7 40 2 N 121 ≡ N 11 2 Using the fact that 40 2 N 11 2 we can write 40 2 − 11 2 N (40 + 11)(40 − 11) ≡ N 0 . Finally, the GCD of 40 ± 11 and N will give us a factor of N: 493 − 9 ⋅ 51 = 34 51 − 34 = 17 34 − 2 ⋅ 17 = 0 ∴ 17 divides 493
3) We wish to compute the following values of Ψ(X, B) which is the number of B-smooth numbers from 2 through X: 3.1) Ψ(25,3) 2 = 2 3 = 3 4 = 2 2 5 = 5 6 = 2 ⋅ 3 7 = 7 8 = 2 3 9 = 3 2 10 = 2 ⋅ 5 11 = 11 12 = 2 2 ⋅ 3 13 = 13 14 = 2 ⋅ 7 15 = 3 ⋅ 5 16 = 2 4 17 = 17 18 = 2 ⋅ 3 2 19 = 19 20 = 2 2 ⋅ 5 21 = 3 ⋅ 7 22 = 2 ⋅ 11 23 = 23 24 = 2 3 ⋅ 3 25 = 5 2 ∴ Ψ(25,3) = 10 3.2) Ψ(35,5)

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