1. T TT The Cryptography Istituto Tecnico Industriale “E.Divini” San Severino Marche

2. The cryptography is an old science, it ciphers information to make messages clear between sender and recipient, but incomprehensible by other people The message is called Plaintex (clear text) Ciphertext (coded text) The Cryptanalysis is the science of the "breaking" of codes and ciphers. It finds weakness in crypto systemes. The Cryptology studies the cryptography and the cryptanalysis.

3. Classical Cryptography The Atbash Cipher 500 - 600 BCE and Caesar Cipher 100 - 4 4 BCE are ones of first simple monoalphabetic substitution ciphers The frequency analysis is a technique for breaking monoalphabetic substitution ciphers around 1000 CE

4. Medieval cryptography The Alberti’s Code ca 1465 and Vigenere Cipher 1586 are the most famous polyalphabetic ciphers The Verman Code, considered the theoretically perfect code, derives from Vigenere cipher.

5. Mechanical and electromechanical cipher machines Enigma machine Is one of the more notable machines employed by Germany during World War II, it was developed by Arthur Scherbius.

6. Plaintext ciphertext Plaintext ciphertext Secret key message transmission Secret key Sender recipient Symmetric or secret key All useful modern encryption algorithms had been symmetric key algorithms, in which the same cryptographic key is used with the underlying algorithm by both the sender and the recipient who must both keep it secret.

7. Plaintext Ciphertext Plaintext Asymmetric or public key cryptography (1976) In contrast, with asymmetric key encryption, there is a pair of mathematically related keys for the algorithm, one of which is used for encryption and the other for decryption. One of the keys may be made public since the other cannot be deduced from the 'public' key. The other key must be kept secret and is usually called the 'private' key. Ciphertext Public key Message trasmission Private key Sender Recipient

8. RSA ALGORITHM The RSA algorithm was invented in 1978 by Ron Rivest, Adi Shamir, and Leonard Adleman. It is an asymmetric or public key algorithm This algorithm uses large (e.g., 1024-bit) prime numbers and Mathematical functions that it’s hardly impossible to reverse It is considered sure because there are no known easy methods of calculating P or Q given only P x Q, if P and Q are large prime number. If P and Q are each 1024 bits long, the sun will burn out before the most powerful computers presently in existence can factor your modulus into P and Q.

9. RSA Encryption It's relatively easy to understand math behind RSA encryption. Find P and Q, two large (e.g., 1024-bit) prime numbers. Choose E such that E is greater than 1, E is less than PQ, and E and Z=(P-1)(Q-1) are relatively prime, which means they have no prime factors in common. E does not have to be prime, but it must be odd. Z can't be prime because it's an even number.

10. Compute D such that (DE - 1) is evenly divisible by Z. Mathematicians write this as DE = 1 (mod Z). This is easy to do, simply find an integer X which causes D = (XZ+ 1)/E to be an integer, then use that value of D. The encryption function is C = (T^E) mod Z, where C is the ciphertext (a positive integer), T is the plaintext (a positive integer), and ^ indicates exponentiation. The message being encrypted, T, must be less than the modulus, Z. The decryption function is T = (C^D) mod Z, where C is the ciphertext (a positive integer), T is the plaintext (a positive integer), and ^ indicates exponentiation.

11. Your public key is the pair (Z, E). Your private key is the number D (reveal it to no one). Z is the modulus, E is the public exponent, D is the secret exponent. You can publish your public key freely, because there are no known easy methods of calculating D, P, or Q given only (Z, E) (your public key). If P and Q are each 1024 bits long, the sun will burn out before the most powerful computers presently in existence can factor your modulus into P and Q.

12. EXAMPLE EXAMPLE (with not large prime numbers) P=11 Q=17 N=PxQ=187 Z=(P-1)(Q-1)=160 1<E<PQ E=3 (odd and relatively prime with 160 ) I find D such that DE-1 is divisible by 160, D=107 in fact 107*3=2*160 T (plaintext) = 123 C (ciphertext) = 123^3 mod187 = 1860867 mod 187 = 30 T ( plaintext ) = 30^107 mod 187 = 123 Public key = 187, 3 Private key = 107