Cryptography and Network Security Chapter 9 Fifth 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 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 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 related private key known only to the recipient used to decrypt messages and sign create signatures infeasible to determine private key from public is asymmetric because those who encrypt messages or verify signatures cannot decrypt messages or create signatures Public Key Cryptography Symmetric vs Public Key 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 Public Key Requirements Public Key algorithms rely on two keys where it is computationally infeasible to find decryption key knowing only algorithm encryption key it is 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 for some algorithms these are formidable requirements which only a few algorithms have satisfied Public Key Requirements need a trapdoor one way function one way function has Y f X easy X f 1 Y infeasible a trap door one way function has Y fk X easy if k and X are known X fk 1 Y easy if k and Y are known X fk 1 Y infeasible if Y known but k not known a practical public key scheme depends on a suitable trap door one way function 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 but is made hard enough to be impractical to break 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 En decryption to encrypt a message M the sender obtains public key of recipient PU e n computes C Me mod n where 0 M n to decrypt the ciphertext C the owner uses their private key PR d n computes M Cd mod n note that the message M must be smaller than the modulus n block if needed 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 PU e n keep secret private decryption key PR d n Why RSA Works because of Euler s Theorem in RSA have a n mod n 1 where gcd a n 1 n p q n p 1 q 1 carefully chose 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 k M1 1 k M1 M mod n RSA Example Key Setup 1 2 3 4 5 6 7 Select primes p 17 q 11 Calculate n pq 17 x 11 187 Calculate n p 1 q 1 16x10 160 Select e gcd e 160 1 choose e 7 Determine d de 1 mod 160 and d 160 Value is d 23 since 23x7 161 10x160 1 Publish public key PU 7 187 Keep secret private key PR 23 187 RSA Example En Decryption 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 c for 0 f 1 i k downto 0 do c 2 x c f f x f mod n if bi 1 then c c 1 f f x a mod n return f Efficient Encryption encryption uses exponentiation to power e hence if e small this will be faster often choose e 65537 216 1 also see choices of e 3 or e 17 but if e too small eg e 3 can attack using Chinese remainder theorem 3 messages with different modulii if e fixed must ensure gcd e n 1 ie reject any p or q not relatively prime to e Efficient Decryption decryption uses exponentiation to power d this is likely large insecure if not can use the Chinese Remainder Theorem CRT to compute mod p q separately then combine to get desired answer approx 4 times faster than doing directly only owner of private key who knows values of p q can use this technique 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 …