Question

Alice and Bob create a symmetric cipher as follows. Their private key k is a large integer and their messages (plaintexts) are d-digit integers M = {m ∈ Z : 0 ≤ m < 10^d}. To encrypt a message, Alice computes √k to d decimal places, throws away the part to the left of the decimal point, and keeps the remaining d digits. Let α be this d-digit number. (For example, if k = 87 and d = 6, then √87 = 9.32737905 ... and α = 327379.) Alice encrypts a message m as c ≡ m + α (mod 10^d). Since Bob knows k, he can also find α, and then he decrypts c by computing m ≡ c − α (mod 10^d). (a) Alice and Bob choose the secret key k = 11 and use it to encrypt 6-digit integers (i.e., d = 6). Bob wants to send Alice the message m = 328973. What is the ciphertext that he sends? (b) Alice and Bob use the secret key k = 23 and use it to encrypt 8-digit integers. Alice receives the ciphertext c = 78183903. What is the plaintext m?

          Alice and Bob create a symmetric cipher as follows. Their private key k is a large integer and their messages (plaintexts) are d-digit integers M = {m ∈ Z : 0 ≤ m < 10^d}. To encrypt a message, Alice computes √k to d decimal places, throws away the part to the left of the decimal point, and keeps the remaining d digits. Let α be this d-digit number. (For example, if k = 87 and d = 6, then √87 = 9.32737905 ... and α = 327379.) Alice encrypts a message m as c ≡ m + α (mod 10^d). Since Bob knows k, he can also find α, and then he decrypts c by computing m ≡ c − α (mod 10^d). (a) Alice and Bob choose the secret key k = 11 and use it to encrypt 6-digit integers (i.e., d = 6). Bob wants to send Alice the message m = 328973. What is the ciphertext that he sends? (b) Alice and Bob use the secret key k = 23 and use it to encrypt 8-digit integers. Alice receives the ciphertext c = 78183903. What is the plaintext m?

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