Private Key Cryptography traditional private secret single key cryptography uses one key shared by both sender and receiver if this key is disclosed communications are compromised also is symmetric parties are equal hence does not protect sender from receiver forging a message claiming is sent by sender Public Key Cryptography the most significant advance in the 3000 year history of cryptography uses two keys a public a private key asymmetric since parties are not equal uses clever application of number theoretic concepts to function complements rather than replaces private key crypto Public Key Cryptography public key two key asymmetric cryptography involves the use of two keys a public key which may be known by anybody and can be used to encrypt messages and verify signatures a private key known only to the recipient used to decrypt messages and sign create signatures is asymmetric because those who encrypt messages or verify signatures cannot decrypt messages or create signatures Public Key Cryptography Why Public Key Cryptography developed to address two key issues key distribution how to have secure communications in general without having to trust a KDC with your key digital signatures how to verify a message comes intact from the claimed sender public invention due to Whitfield Diffie Martin Hellman at Stanford Uni in 1976 known earlier in classified community Public Key Characteristics Public Key algorithms rely on two keys with the characteristics that it is computationally infeasible to find decryption key knowing only algorithm encryption key computationally easy to en decrypt messages when the relevant en decrypt key is known either of the two related keys can be used for encryption with the other used for decryption in some schemes Encryption using a Public Key System 7 Authentication using a Public Key System 8 Public Key Cryptosystems Public Key Applications can classify uses into 3 categories encryption decryption provide secrecy digital signatures provide authentication key exchange of session keys some algorithms are suitable for all uses others are specific to one Security of Public Key Schemes like private key schemes brute force exhaustive search attack is always theoretically possible but keys used are too large 512bits The public key algorithms are based on a known hard problem The its just made too hard to do in practise RSA Problem Given n pq with p and q primes Find p and q requires the use of very large numbers hence is slow compared to private key schemes RSA Rivest Shamir Adleman MIT 1977 Each user generates a public private key pair by 1 selecting two large primes at random p q 2 computing their system modulus N pq note N p 1 q 1 3 selecting at random the encryption key e where 1 e N gcd e N 1 4 solve following equation to find decryption key d ed 1 mod N and 0 d N 5 6 publish their public encryption key KU e N keep secret private decryption key KR d p q RSA Use to encrypt a message M the sender obtains public key of recipient KU e N computes C Me mod N where 0 M N to decrypt the ciphertext C the owner uses their private key KR d p q computes M Cd mod N note that the message M must be smaller than the modulus N block if needed Why RSA Works because of Euler s Theorem a n mod N 1 where gcd a N 1 in RSA have N p q N p 1 q 1 carefully chosen e d to be inverses mod N hence e d 1 k N for some k hence Cd Me d M1 k N M1 M N q M1 1 q M1 M mod N RSA Example 1 2 3 4 5 Select primes p 17 q 11 Compute n pq 17 11 187 Compute n p 1 q 1 16 10 160 Select e gcd e 160 1 choose e 7 Determine d de 1 mod 160 and d 160 Value is d 23 since 23 7 161 10 160 1 6 Publish public key KU 7 187 7 Keep secret private key KR 23 17 11 RSA Example cont sample RSA encryption decryption is given message M 88 nb 88 187 encryption C 887 mod 187 11 decryption M 1123 mod 187 88 RSA Key Generation users of RSA must determine two primes at random p q select either e or d and compute the other primes p q must not be easily derived from modulus N pq means must be sufficiently large typically guess and use probabilistic test RSA Security three approaches to attacking RSA brute force key search infeasible given size of numbers mathematical attacks based on difficulty of computing N by factoring modulus N timing attacks on running of decryption Factoring Problem mathematical approach takes 3 forms factor N pq hence find N and then d determine N directly and find d find d directly currently believe all equivalent to factoring Long range factoring prediction from Applied Cryptography by B Schneier Year Key length in bits 1995 1024 2005 2048 2015 4096 2025 8192 2035 16 384 2045 32 768 RSA recommended Key sizes from Applied Cryptography by B Schneier Year Individual Corporation Government 1995 768 1280 1536 2000 1024 1280 1536 2005 1280 1536 2048 2010 1280 1536 2048 2015 1536 2048 2048