Suppose a community of RSA users share the same encryption exponent e but each user i has their own encryption modulus nį. Furthermore, suppose that distinct nį are pairwise relatively prime (if gcd(nį, nj) > 1, then i’s and j's systems are broken!). Now, let us assume that Alice sends the same message x (0 < x < nį for all i) to k different entities in the community, i.e., Alice sends y₁ = x² (mod n;) (0 < Yi < nį) to user i for 1 ≤ i ≤ k. Show that if k > e, then a cryptanalyst can recover the plaintext x without factoring any of the nį.

Suppose the alphabet consists of a through z, in natural order, followed by a blank and then the digits 0 through 9, in natural order. Associate these "letters" with the numbers 0,1,2,...,36, respectively, thus forming a 37-letter alphabet, D. Use the affine cipher to decipher the message X01916R916546M9CN1L6B1LL6X0RZ6UII if you know that the plaintext message begins with "t" followed by "h". Write out the affine mapping f and its inverse.

Suppose that in a long ciphertext message the letter occurred most frequently, followed in frequency by. Using the fact that in the -letter alphabet, described in Example, "blank" occurs most frequently, followed in frequency by, read the portion of the message enciphered using an affine mapping on. Write out the affine mapping and its inverse. Example 2 Translation Cipher Associate the letters of the "alphabet" with the integers. Let and define the mapping by where is the key, the number of positions from the plaintext to the ciphertext. If our alphabet consists of through, in natural order, followed by a blank, then we have "letters" that we associate with the integers as follows:

Use the alphabet C from the preceding problem and the affine cipher with key a=11andb=7 to decipher the message RRROAWFPHPWSUHIFOAQXZC:Q.ZIFLW/O:NXM and state the inverse mapping that deciphers this ciphertext. Exercise 7: Suppose the alphabet consists of a through z, in natural order, followed by a colon, a period, and then a forward slash. Associate these "letters" with the numbers 0,1,2,...,28, respectively, thus forming a 29-letter alphabet, C. Use the affine cipher with key a=3andb=22 to decipher the message OVVJNTTBBBQ/FDLWLFQ/GATYST and state the inverse mapping that deciphers this ciphertext.

Find all pairs of integers keys (a, b) for affine ciphers for which the encryption function c = (ap + b) mod 26 is the same as the corresponding decryption function.

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?

What is the decryption function for an affine cipher if the encryption function is c = (15p + 13) mod 26?

Recall that an affine cipher, fa,b(n) = an + b( mod 26), where a, b ∈ {0, 1, 2, . . . , 25}, is said to be valid if gcd(a, 26) = 1. (a)Use the key (a, b) = (3, 0) to come up with the cipher text corresponding to the following plaintext DO WELL.  (b) Now use the key (a, b) = (5, 1) to come up with the cipher text for DO WELL.  (c) The key (a, b) = (5, 1) was used to generate the following cipher text. Decrypt the message. gbidxv vbfev

Recall that an affine cipher, fa,b(n) = an + b( mod 26), where a, b ∈ {0, 1, 2, . . . , 25}, is said to be valid if gcd(a, 26) = 1. (a)Use the key (a, b) = (3, 0) to come up with the cipher text corresponding to the following plaintext DO WELL. (b)Now use the key (a, b) = (6, 3) to generate multiple (at least two) cipher texts corresponding to the same message DO WELL.

Encrypt the word "WATER" by using the cipher function f(p) = (3p+7)mod 26.