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- a. Excluding the identity cipher, how many different translation ciphers are there using an alphabet of n "letters"? b. Excluding the identity cipher, how many different affine ciphers are there using an alphabet of n "letters," where n is a prime?z is a modulo under mult, so do i just list elements as multiplication like {a^n|n Z}Let's say that Alice and Bob are exchanging keys using Diffie-Hellman key exchange using multiplicative group Z*499 and generator g=7. Let's say that Alice chooses secret exponent x=23 and receives the number 234 from Bob. What number should Alice send to Bob? _____ What number do Alice and Bob compute as their shared secret? ______
- Suppose that (n, e) is an RSA encryption key, with n = pq, where p and q are large primes and gcd(e, (p − 1)(q − 1)) = 1. Furthermore, suppose that d is an inverse of e modulo (p − 1)(q − 1). Suppose that C ≡ Me (mod pq). In the text we showed that RSA decryption, that is, the congruence Cd ≡ M (mod pq) holds when gcd(M, pq) = 1. Show that this decryption congruence also holds when gcd(M, pq) > 1.What is the order of the element $(\overline{2}, \overline{9})$ in $Z_{4} \times U_{10}$ is ( $Z_{4}$ is the additive group modulo 4 and $U_{10}$ is Euler group)If a (mod 0) = a, How would you interpret the congruence a ≡ b (mod 0)?
- If gcd(m,n) = 1 is a given condition, how can you prove that the congruences x ≡ a (mod m) and x ≡ b (mod n) have a solution no matter what. Is there any example that shows that gcd(m,n) = 1 is a necessary condition?The affine cipher E is given by E(x) ≡ 15x − 7 (mod 26). The conversion table for letters and numbers is attached. Show that 7 is a multiplicative inverse of 15 modulo 26.Suppose that the most common letter and the second most common letter in a long ciphertext produced by encrypting a plaintext using an affine cipher f (p) = (ap + b) mod 26 are Z and J, respectively. What are the most likely values of a and b?
- 1): For all integers a, b, and c, if a | b or a | c, then a | bc A): True b): False 2): Compute the value of the following expressions without a calculator using the Congruence Modulo Laws 4630 mod 9 A): 1 B): 0 C): 67390 D): too large to compute 3): Group the following numbers according to congruence mod 13. That is, put two numbers in the same group if they are equivalent mod 13. {−63, -54, -41, 11, 13, 76, 80, 130, 132, 137} A): Congruent to 0 mod 13: {13, 130} Congruent to 2 mod 13: {-63, 80, 132 }Congruent to 7 mod 13: {137 }Congruent to 11 mod 13: {-54, -41, 11, 76} b): Congruent to 0 mod 13: {13, 130} Congruent to 2 mod 13: {80, 132 }Congruent to 7 mod 13: {137 }Congruent to 11 mod 13: { 11, 76} Congruent to -7 mod 13 {-63} Congruent to 3 mod 13 {-41} C): Congruent to 0 mod 13: {13, 130} Congruent to 2 mod 13: {-63, 80, 132 }Congruent to 7 mod 13: {137 } Congruent to -2 Mod 13 {-54,=41} Congruent to 11 mod 13: { 11, 76} D): Congruent to 0 mod…If a ≡ b (mod n1) and a ≡ b (mod n2), must it be true that a ≡ b (mod n1n2)? What if gcd(n1, n2) = 1? In each case either give a proof or counterexample.Show that the set {5, 15, 25, 35} is a group under multiplication modulo 40. What is the identity element of this group? Can you see any relationship between this group and U(8)?