Discrete structure ASAP No. 1 Congruence is popular in cryptography applications

Question

Discrete structure ASAP No. 1 Congruence is popular in cryptography applications. The oldest example is Caesar Cipher used by Julius Caesar to communicate with his generals. In general, for Caesar Cipher, let p = plain text, c = cipher text, k = encryption key, the encryption algorithm is c = p + k mod 26, and decryption algorithm is p = c - k mod 26. Both parties must share a common secret key k. For example, LOVE rightarrow ORYH (circular shift by 3 places, k = 3) Suppose k is the sum of the eight digits of your EMPLID, what will be the cipher text for the plain text "CS"? A slightly more sophisticated version of the cipher is that the encryption algorithm is c = a* p + k mod 26, and decryption is to solve for p in the linear congruence equation (a* p + k) mod 26 = c, or a * p c - k (mod 26). Suppose k is the sum of the eight digits of your EMPLID, a is 3, and the cipher text is "X", decrypt the cipher text to recover the plain text letter. The letters A, B, C, ..., Y, Z, could be treated as numbers 0, 1, 2, ..., 23, 24, 25. Design a divide-and-conquer algorithm to find the minimum of a sequence of integers. Write pseudocode to describe this algorithm. Set up a recurrence relation for the previous algorithm, and construct a big-O estimate for the number of comparison operations needed. Let f be an increasing function that satisfies the recurrence relation f(n) = af(n/b) + c whenever n is divisible by b, where a greaterthanorequalto 1, b is an integer greater than 1, and c is a positive real number. Then f(n) is{O (n^log_b^a) if a > 1, O (log (n) if a = 1.

Explanation / Answer

k is sum of 8 digits number of your EMPLID :

so I take random 8 digit such as( 1+2+3+4+5+6+7+8)=36

so value of k = 36 then,

apply algorithm to is for making plain text into cipher is

for encrypation c=P+k mod 26

C=P + 10 means letter +10;

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z( this will become )

K L M N O P Q R S T U V W X Y Z A B C D E F G H I J( change value)

so we have to change CS => MC

for decryption c=P-k mod 26

C=P - 10 means letter -10;

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z( this will become )

Q R S T U V W X Y Z A B C D E F G H I J K L M N O P( change value)

so we have to change MC => CS

sorry first time i did counting mistake, now this answer is correct

I hope it will help you.

Thanks

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