(d) Suppose that Bob has used two distinct primes p and q to create the publicmodulus n = pq and public encryption key e for use with RSA. Suppose also thatd is an integer such thated = 1 (mod X(n)).Show that xed = x (mod n) for any non-negative integer x.

RSA algorithm: 1. Choose two distinct prime numbers p and q. Take n = pq. 2. Compute λ(n), where λ…

Answered: Suppose that Bob has used two distinct…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.8: Introduction To Cryptography (optional)

Problem 20E: Suppose that in an RSA Public Key Cryptosystem. Encrypt the message "pascal" using the -letter...

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Suppose that in an RSA Public Key Cryptosystem, the public key is e=13,m=77. Encrypt the message "go for it" using two-digit blocks and the 27-letter alphabet A from Example 2. What is the secret key d? Example 2 Translation Cipher Associate the n letters of the "alphabet" with the integers 0,1,2,3.....n1. Let A={ 0,1,2,3.....n-1 } and define the mapping f:AA by f(x)=x+kmodn where k is the key, the number of positions from the plaintext to the ciphertext. If our alphabet consists of a through z, in natural order, followed by a blank, then we have 27 "letters" that we associate with the integers 0,1,2,...,26 as follows: Alphabet:abcdef...vwxyzblankA:012345212223242526
Suppose that in an RSA Public Key Cryptosystem, the public key is. Encrypt the message "pay me later” using two-digit blocks and the -letter alphabet from Example 2. What is the secret key? 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:
30. Prove that any positive integer is congruent to its units digit modulo .
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:
Suppose that the check digit is computed as described in Example . Prove that transposition errors of adjacent digits will not be detected unless one of the digits is the check digit. Example Using Check Digits Many companies use check digits for security purposes or for error detection. For example, an the digit may be appended to a -bit identification number to obtain the -digit invoice number of the form where the th bit, , is the check digit, computed as . If congruence modulo is used, then the check digit for an identification number . Thus the complete correct invoice number would appear as . If the invoice number were used instead and checked, an error would be detected, since .
Show that the check digit in bank identification numbers satisfies the congruence equation .
Encrypt the word "WATER" by using the cipher function f(p) = (3p+7)mod 26.
Find all integer values of x for which 3x = 4 (mod 8).
b) Suppose that the encryption function f(x) = (x-a) mod 26,0 ≤ x ≤ 25 and 0 ≤ a ≤ 25, encrypt the letter "F" by "A". Use the function f to decrypt the message "ZSVXO".
Solve both the questions
Two cryptographers encrypt a secret password consisting only of nonnegative integers less than min(11, 17) using an affine cipher. They agree to use the same positive integer a in the following way: the affine key of the first cryptographer is x→ax (mod 11) and the affine key of the second is x→ax (mod 17) (where their result is the least residue modulo 11, 17, respectively). For example, if their affine keys are → 3x (mod 5) and x→ 3x (mod 7), and the secret password is "1,2,3", then they would produce the ciphertext respectively. Assume that we know that the first digit of the secret password is 1, and suppose that the encrypted message of the first cryptographer is of the form and the encrypted message of the second cryptographer is of the form 3, 1, 1 and 3, 6, 2, (where the asterisks are numbers that we could not intercept). Enter the least positive integer a that could be used in their keys. Type your answer... 5 ***...*** 7***...***
Show that if a ≡ b (mod m) and c ≡ d (mod m), then a + c ≡ b + d (mod m).

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Suppose that Bob has used two distinct primes p and q to create the public modulus n = pq and public encryption key e for use with RSA. Suppose also that d is an integer such that ed = 1 (mod X(n)). Show that xed = x (mod n) for any non-negative integer x.