1. Public Key Cryptosystem
Introduced in 1976 by Diffie and Hellman [2]
In PKC different keys are used for encryption and decryption
1978: First Two Implementations
RSA:
Rivest-Shamir-Adleman
ACM 2002 Turing Award
Based on integer
factorization
Merkle-Hellman
Knapsack Cryptosystem [1]
Based on the
subset-sum problem,
variant of knapsack problem
Additive
Knapsack
Cryptosystem
Multiplicative
Knapsack
Cryptosystem
Multiply-Iterated
Knapsack
Cryptosystem

2. Public Key Cryptosystems
Elliptic curve cryptography (ECC) is an approach to PKC based on the algebraic structure of elliptic curves over finite fields. The use of elliptic curves in cryptography was suggested independently by Neal Koblitz and Victor S. Miller in 1985.
The NTRUEncrypt public key cryptosystem, also known as the NTRU encryption algorithm, is a lattice-based alternative to RSA and ECC and is based on the shortest vector problem in a lattice.

3. Public Key Cryptosystems
ElGamal encryption system is an asymmetric key encryption algorithm for PKC which is based on the Diffie–Hellman key exchange. It was described by Taher Elgamal in 1985.
The Digital Signature Algorithm (DSA) is a United States Federal Government standard or FIPS for digital signatures.

4. RSA Alice: 1. Chooses secret (private) key
2. Create and publishes public key
3. Receives ciphertext from Bob
4. Decrypts ciphertext using secret key to recover
the plaintext – original message
Bob
1. Uses Alice Public Key to encrypt the message
2. Sends ciphertext – encrypted message to Alice

5. ALICE Preparation Step – Creates Private and Public Key
Selects two primes p and q
Calculates
m = pq
n = ( p – 1 ) ( q – 1 )
Selects e (1 < e < n), that relatively prime to n, gcd (e, n) = 1
Finds the multiplicative inverse of e mod n. Denotes the multiplicative inverse by d:
ed  1 (mod n)
Publishes e and m – this is the public key

6. BOB - Encryption Plaintext is number x
0 < x  m -1
gcd (x, m) =1
Ciphertext y is calculated as follows:
y = xe mod m
y is sent to Alice

7. Alice - Decryption Alice receives the ciphertext y
Calculates yd mod m and this is a plaintext x, so the final formula is: x  yd ( mod m )