Explanation Given information: The given cipher text is “08 05 15”. Formula used: Division algorithm: Let a be an integer and d be a positive integer. Then, there are unique integers q and r with 0 ≤ r < d such that a = d q + r , here, q is the quotient and r is the remainder. So, q = a div d and r = a mod d . Know that the technique is x m ( mod y ) = [ x ( mod y ) ] m ( mod y ) . RSA Cipher Work: The formula of the RSA cryptography is M = C d mod p q . To decrypt the numbers, use the formula M = C d mod p q . …… (1) Where, d = 27 and p q = 55 . To decrypt 08 , use the formula (1). M = 08 27 mod 55 Its known that, 27 as a sum of powers of 2. 27 = 2 4 + 2 3 + 2 1 + 2 0 = 16 + 8 + 2 + 1 Rewrite the statement as: M = 08 ( 16 + 8 + 2 + 1 ) mod 55 = ( 08 16 ⋅ 08 8 ⋅ 08 2 ⋅ 08 1 ) mod 55 = ( 08 16 mod 55 ⋅ 08 8 mod 55 ⋅ 08 2 mod 55 ⋅ 08 1 mod 55 ) mod 55 = ( 08 16 mod 55 ⋅ ( 08 4 ) 2 mod 55 ⋅ 64 mod 55 ⋅ 8 ) mod 55 = ( 08 16 mod 55 ⋅ ( 08 4 mod 55 ) 2 mod 55 ⋅ 9 ⋅ 8 ) mod 55 Simplify the above equation. ( 08 16 mod 55 ⋅ ( 08 4 mod 55 ) 2 mod 55 ⋅ 9 ⋅ 8 ) mod 55 = ( 08 16 mod 55 ⋅ ( ( 08 2 ) 2 mod 55 ) 2 mod 55 ⋅ 9 ⋅ 8 ) mod 55 = ( 08 16 mod 55 ⋅ ( ( 08 2 mod 55 ) 2 mod 55 ) 2 mod 55 ⋅ 9 ⋅ 8 ) mod 55 = ( 08 16 mod 55 ⋅ ( 9 2 mod 55 ) 2 mod 55 ⋅ 9 ⋅ 8 ) mod 55 = ( ( 08 8 ) 2 mod 55 ⋅ 26 2 mod 55 ⋅ 9 ⋅ 8 ) mod 55 Again, simplify the above equation. ( ( 08 8 ) 2 mod 55 ⋅ 26 2 mod 55 ⋅ 9 ⋅ 8 ) mod 55 = ( 16 2 mod 55 ⋅ 16 ⋅ 9 ⋅ 8 ) mod 55 = ( 256 mod 55 ⋅ 16 ⋅ 9 ⋅ 8 ) mod 55 = ( 36 ⋅ 16 ⋅ 9 ⋅ 8 ) mod 55 = 41472 mod 55 = 2 Thus, 08 27 mod 55 = 2 . To decrypt 05 , use the formula (1). M = 05 27 mod 55 Its known that, 27 as a sum of powers of 2. 27 = 2 4 + 2 3 + 2 1 + 2 0 = 16 + 8 + 2 + 1 Rewrite the statement as: M = 05 ( 16 + 8 + 2 + 1 ) mod 55 = ( 05 16 ⋅ 05 8 ⋅ 05 2 ⋅ 05 1 ) mod 55 = ( 05 16 mod 55 ⋅ 05 8 mod 55 ⋅ 05 2 mod 55 ⋅ 05 1 mod 55 ) mod 55 = ( 05 16 mod 55 ⋅ ( 05 4 ) 2 mod 55 ⋅ 25 mod 55 ⋅ 5 ) mod 55 = ( 05 16 mod 55 ⋅ ( 05 4 mod 55 ) 2 mod 55 ⋅ 25 ⋅ 5 ) mod 55 Simplify the above equation

5th Edition

ISBN: 9781337694193

Author: EPP, Susanna S.

Publisher: Cengage Learning,

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Chapter 8.4, Problem 23ES

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Decrypt the cipher text and write the numeric into letters.

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Discrete Mathematics With Applications

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