Problem 1
Name the four main cryptographic applications. For each of them write a brief description (one sentence will suffice).

Problem 2
(1) Define the greatest common divisor of the integers a, b.
(2) Compute the gcd(-18,21).
(3) Recall that every integer is a product of primes. Assuming a=p² qr³, b=pr² s where p,q,r,s are primes, write gcd(a,b).
(4) Use the previous result to compute gcd(1500,350).

Problem 3
We know that if d=gcd(a,b) then we can find x0,y0 such that d=ax0+by0.
(1) Assume d>1. Prove that for any integer k, numbers (bk)/d and (ak)/d are integers.
(2) Assume d>1. Prove that any pair of numbers xk=x0 +(bk)/d, yk=y0 –(ak)/d is also an integer solution of the equation (1).

Problem 4
The following ciphertext CRWWZ was encrypted using an affine cipher mod 26. The plaintext starts ha.
(1) Name the method of attack you will use.
(2) Using the method from item (1), compute the decryption function.
(3) Decrypt the message.

Problem 5
Assume we are using a four letter alphabet a,b,c,d. A message has been encrypted using substitution cipher and the encryption key is bcad. Compute the decryption key.

Problem 6
Name two main weaknesses of the Playfair cipher.

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These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

After reading the presentation from Trappe’s book (page 9), the main four cryptographic applications can be synthetized as below:
- Confidentiality – ensures that only the intended destination party can access the original message safely;
- Integrity – ensures that sent message was not altered or modified (either due to unauthorized parties or media errors);
Authentication – ensures the proof-of-origin or proof-of-identity (who) for the transmitted messages...

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