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68 CHAPTER 4. THE EUCLIDEAN ALGORITHM
9. How do we use the Euclidean Algorithm to find a multiplicative inverse
10. How many Linear Ciphers are there modulo 26
11. What information is needed to decipher a Decimation Cipher A Linear
Cipher
12. Are Decimation Ciphers secure or not Why
13. Are Linear Ciphers secure or not Why
14. Can a frequency count help break a Decimation or Linear Cipher Explain.
4.8 Exercises
1. Use the Euclidean Algorithm to find the following gcd’s.
(a) gcd(32, 54).
(b) gcd(45, 39).
(c) gcd(78, 53).
(d) gcd(102, 78).
2. Use the Euclidean Algorithm to find the following gcd’s.
(a) gcd(61, 56).
(b) gcd(121, 77).
(c) gcd(735, 140).
(d) gcd(297, 201).
3. Use the Euclidean Algorithm to find the following gcd’s, and then find a
solution to the related modular equation.
(a) Find gcd(25, 85). Find a solution to 25x ≡ gcd (mod 85).
(b) Find gcd(27, 17). Find a solution to 17x ≡ gcd (mod 27).
(c) Find gcd(24, 102). Find a solution to 24x ≡ gcd (mod 102).
(d) Find gcd(149, 78). Find a solution to 78x ≡ gcd (mod 149).
4. Use the Euclidean Algorithm to find the following gcd’s, and then find a
solution to the related modular equation.
(a) Find gcd(16, 110). Find a solution to 16x ≡ gcd (mod 110).
(b) Find gcd(31, 79). Find a solution to 31x ≡ gcd (mod 79).
(c) Find gcd(298, 79). Find a solution to 79x ≡ gcd (mod 298).
(d) Find gcd(306, 192). Find a solution to 192x ≡ gcd (mod 306).