1. Math
  2. Advanced Math
  3. cryptography...

Question: cryptography...

Question details
Cryptography
(1) (2 pts each) Use the Euclidean Algorithm to find in each of the cases below the greatest common divisor ged(a, b). You may use a calculator to perform the division with remaider, but show your work by writing down all divisions with remainder explicitly. Furthermore, count the number of divisions you need to perform to find the greatest common divisor, and compare this number with the estimate 12 log2(b) + 2] that was given in class. (i) a = 72, b = 52 ii) a 720, b 522 (iii a 54321, b 12345. Remarks. 1. When you compute with a calculatororr.., where ro E Nand for i 0, then q-o and rb.0.rr23... E [0,..,9) 2. Please enumerate the equations as in the following example with a useful when you do problem 2 below 34 and b 22, This will be (1) 34 2212 (2) 22 12 10 (3) 12 = 10+2 (4) 10-5-2 +0 → gcd(34,22)=2 In(b (2) (2 pts each) For each pair (a, b) in problem 1 find integers s,t such that ged(a,b)sa +tb. Do so by running the Euclidean Algorithm backwards, i.e., from bottom to top. When you replace a remainder by the linear combination using equation () (1,2,3,.. from problem1, indicate this by writing (a) over the corresponding equality sign. (i) ged(72,52)s 72t 52. (ii) ged (720,522) s 720 +t 522 (iii) ged (54321, 12345) 54321 t 12345 (3) (2 pts each) In which of the following cases are a and m coprime? If so, find b e Z such that ab 1 (m) (i) m = 15, a = 7 (iii) m 151, a-52 (iv) m = 187, a = 143.
Solution by an expert tutor
Blurred Solution
This question has been solved
Subscribe to see this solution