Cryptography and Network Security Third Edition by William Stallings Lecture slides by Lawrie Brown Chapter 9 Public Key Cryptography and RSA Every Egyptian received two names which were known respectively as the true name and the good name or the great name and the little name and while the good or little name was made public the true or great name appears to have been carefully concealed The Golden Bough Sir James George Frazer 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 probably 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 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 security relies on a large enough difference in difficulty between easy en decrypt and hard cryptanalyse problems more generally the hard problem is known its just made too hard to do in practise requires the use of very large numbers hence is slow compared to private key schemes RSA by Rivest Shamir Adleman of MIT in 1977 best known widely used public key scheme based on exponentiation in a finite Galois field over integers modulo a prime nb exponentiation takes O log n 3 operations easy uses large integers eg 1024 bits security due to cost of factoring large numbers nb factorization takes O e log n log log n operations hard RSA Key Setup each user generates a public private key pair by selecting two large primes at random p q computing their system modulus N p q note N p 1 q 1 selecting at random the encryption key e where 1 e N gcd e N 1 solve following equation to find decryption key d e d 1 mod N and 0 d N 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 Exponentiation can use the Square and Multiply Algorithm a fast efficient algorithm for exponentiation concept is based on repeatedly squaring base and multiplying in the ones that are needed to compute the result look at binary representation of exponent only takes O log2 n multiples for number n eg 75 74 71 3 7 10 mod 11 eg 3129 3128 31 5 3 4 mod 11 Exponentiation 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 p q means must be sufficiently large typically guess and use probabilistic test exponents e d are inverses so use Inverse algorithm to compute the other 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 p q hence find N and then d determine N directly and find d find d directly currently believe all equivalent to factoring have seen slow improvements over the years as of Aug 99 best is 130 decimal digits 512 bit with GNFS biggest improvement comes from improved algorithm cf Quadratic Sieve to Generalized Number Field Sieve barring dramatic breakthrough 1024 bit RSA secure ensure p q of similar size and matching other constraints Timing Attacks developed in mid 1990 s exploit timing variations in operations eg multiplying by small vs large number or IF s varying which instructions executed infer operand size based on time taken RSA exploits time taken in exponentiation countermeasures use constant exponentiation time add random delays blind values used in calculations Summary have considered principles of public key cryptography RSA algorithm implementation security