MATH 470 Exam 2 All Answers Correct
The 5 cryptographic algorithms - RSA public key cryptosystem - ElGamal public key cryptosystem - RSA digital signatures - ElGamal digital signatures - Diffine-Hellman key exchange
The 3 primality testing algorithms - Fermat primality test - Soloway-Strassen p...
The 5 cryptographic algorithms ✅- RSA public key cryptosystem
- ElGamal public key cryptosystem
- RSA digital signatures
- ElGamal digital signatures
- Diffine-Hellman key exchange
The 3 primality testing algorithms ✅- Fermat primality test
- Soloway-Strassen primality test
- Miller-Rabin primality test
The 2 factorization algorithms ✅- Fermat factorization method
- Quadratic sieve
The 4 discrete logarithm algorithms ✅- Definitions
- Pohlig-Hellman algorithm
- Baby step-giant step algorithm
- Index calculus
RSA public key cryptosystem ✅1. Bob chooses secret primes p and q to compute n = pq
2. Bob chooses e with gcd(e, (p-1)(q-1)) = 1
3. Bob computes d with de≡1 (mod (p-1)(q-1))
4. Bob makes n and e public, keeping p, q, and d a secret
5. Alice can encrypt message m with c≡m^e (mod n)
6. Bob can decrypt message m with m≡c^d (mod n)
, ElGamal public key cryptosystem ✅1. Bob chooses a large prime p and a primative root α
2. Bob chooses a secret integer 'a' and computes β≡α^a (mod p)
3. Bob makes (p, α, β) public
4. Alice picks a secret random integer k and computes r≡α^k (mod p)
5. Alice encrypts using t≡(β^k)(m) (mod p)
6. Alice sends the pair (r, t) to Bob
7. Bob decrypts using (t)(r^-a)≡m (mod p)
RSA digital signature ✅Alice creates her signature by...
1. Taking two large primes p and q and creating n=pq
2. Choosing Ea such that 1 < Ea < θ(n) with gcd(Ea, θ(n)) = 1
3. Calculating Da such that EaDa≡1 (mod θ(n))
4. Alice publishes (Ea, n) and keeps Da, p, and q secret
4. Signature is y≡m^Da (mod n)
5. (m, y) is made public
Bob can verify Alice's signature by...
1. Downloading (Ea, n)
2. Calculating z≡y^Ea (mod n). If z=m, the signature is valid
ElGamal digital signature ✅For Alice to sign a message m, she must...
1. Select a secret random k so that gcd(k, p-1)=1
2. Compute r≡α^k (mod p)
3. Compute s≡k^-1(m-ar) (mod (p-1))
4. Signed message is the triple (m, r, s)
Bob can verify the signature by...
1. Downloading Alice's public key (p, α, β)
2. Computing v₁≡(β^r)(r^s) (mod p)
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