# chapter5-1 - The RSA Cryptosystem and Factoring Integers...

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1 The RSA Cryptosystem and Factoring Integers • Introduction to Public-Key Cryptography • More Number Theory • The RSA Cryptosystem • Primality Testing • Square Roots Modulo n • Factoring Algorithm • Other Attack on RSA • The Rabin Cryptosystem • Semantic Security of RSA

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2 Public-Key Cryptography • Symmetric-key (or Shared-key) cryptosystems – A secret key is chosen by the sender and the receiver before messages are sent – The decryption rule d K is either the same as the encryption rule e K , or easily derived from e K • Drawback of such cryptosystem – Problem: how to distribute K to Alice & Bob securely – It might be impossible for the sender and the receiver to setup a secret key before communication
3 Public-Key Cryptography Basic idea of Asymmetric-key (or public-key) cryptosystems – The receiver makes his/her encryption key K e (called public key ) public. The sender can encrypt the message easily. But the decryption key K d (called private-key ) is very hard to compute from K e . Only the receiver knows K d

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4 Trapdoor Functions Definition: • A function f: {0,1}* {0,1}* is a trapdoor function iff f(x) is a one-way function such that given some extra information it becomes feasible to find for any y in Img(f) x in X, s.t. y = f(x) • Public key cryptography relies on trapdoor functions
5 Trapdoor Functions Factoring integers: f(p,q) = n = pq • Easy: pq • Hard: factoring pq into p and q Root-extraction: f(p,q,e,y)= y e mod pq • Easy: y e mod pq • Hard: given pq, e, and y e mod pq, compute a y' such that y' e = y e mod pq Discrete log problem: f(g,p,x) = g x mod p • Easy: g x mod p • Hard: determine x from p, g, and g x mod p

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6 RSA Algorithm Invented in 1978 by Ron Rivest, Adi Shamir and Leonard A dleman Security relies on the difficulty of factoring large composite numbers Published as R L Rivest, A Shamir, L Adleman, " On Digital Signatures and Public Key Cryptosystems ", Communications of the ACM, vol 21 no 2, pp. 120- 126, Feb 1978
7 Public-key system: how it works • Everybody selects its own public key P and private key S , and publicizes P • Therefore Alice has ( P a , S a ), and Bob has ( P b , S b ) • Everybody knows P a , P b , … • Suppose Alice wants to send a message to Bob. – Alice encrypts the message with Bob‘s public key P b and sends out – (only) Bob can decrypt the message using his private key S b . Nobody else can

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8 RSA cryptosystem • Suppose n=p q , where p and q are big primes • Select a and b , such that a b≡ 1 mod φ ( n ) K =( n,p,q,a,b ), publicize n,b , but keep p,q,a secret • For any x,y Z n , define e K ( x )= x b mod n d K ( y )= y a mod n • Of course, from n,b , it is very difficult to get a (as well as p,q,φ ( n ))
9 More number theory For any positive n Z n is a ring φ ( n )= i=1 m ( p i e i - p i e i - 1 ) where n = i=1 m p i e i b Z n has a multiplicative inverse iff gcd( b , n ) = 1 Z n * = { b | b is coprime to n }, then ( Z n * , ) is an abelian group 1. (modulo n ) is associative and commutative 2. 1 is the multiplicative identity 3. Any element b

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chapter5-1 - The RSA Cryptosystem and Factoring Integers...

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