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. Links http://www.nytimes.com/keyword/adi-shamir http://simson.net/ref/1988/Landau_Zero_Knowledge.pdf http://en.wikipedia.org/wiki/Public-key_cryptography http://en.wikipedia.org/wiki/Zero-knowledge_proof

5. 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

6. ALICE Preparation Step – Creates Private and Public Key Selects two primes p and q Calculates m = pq n = ( p – 1 ) ( q – 1 ) Selects e 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

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

8. Alice - Decryption Alice receives the ciphertext y Calculates y d mod m and this is a plaintext x, so the final formula is: x  y d ( mod m ) Why is it working? e -1  d ( mod n ) means: ed  1 (mod n) means: ed = 1 + kn = 1 + k ( p -1 )( q -1 ) (for some integer k ) y d  (x e mod m ) d  x ed  x 1 + k(p-1)(q-1)  x (mod m) (Theorem 4.3.2 from your textbook, x has to be relatively prime to m = pq)