这个作业是回答密码学相关的问题

Math 435

Introduction to Cryptography

Homework 1

Due Sunday, June 21, 2020

1. The ciphertext SEOYKJOEJ has been generated with a shift cipher. Determine the key

and the plaintext.

2. Show that the encryption key of a cryptosystem is always injective. In other words, if

ek(x) = ek(y), then x = y. [Hint: try decrypting.]

3. Use the affine cipher ek(x) = 3x + 1 to encipher the plaintext BADGERS. What is the

decrypting function dk(x)?

4. Find the affine cipher (if it exists) that encrypts the plaintext BC into the ciphertext

AD. If no such affine cipher exists, show why it does not exist.

5. Suppose you encrypt using an affine cipher, then encrypt the encryption using another

affine cipher. Is there any advantage to doing this, rather than using a single affine

cipher? Why or why not? [Hint: for example, if f(x) = 3x + 1 and g(x) = 5x + 2,

what does f(g(x)) look like?]

6. Use the Euclidean algorithm to compute 7−1

in Z60 or explain why no such value exists.

7. The ciphertext CRWWZ was encrypted with an affine cipher. We know that the plantext

starts with HA. Decrypt the message.

8. Using MAMA as the key for a Vigen´ere cipher, encrypt BE COOL. What is the minimum

block-length of this cipher?

9. The ciphertext YIFZMA was encrypted by a Hill cipher with matrix

9 2

13 3

. Find

the plaintext.

10. How many keys are there for each of the following ciphers?

(a) Shift ciphers

(b) Affine ciphers

(c) General substitution ciphers

(d) Vigen´ere ciphers where the keyword has length 3

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