of Outline implementation
RSA algorithm for key generation select two prime numbers p, q compute n = p q v = (p-1) (q-1) select small odd integer k such that gcd(k, v) = 1 compute d such that (d k)%v = 1 RSA algorithm for encryption/decryption encryption: compute E(M) = (Mk)%n decryption: compute D(M) = (E(M)d)%n
RSA algorithm for key generation
Input: none Computation: select two prime integers p, q compute integers n = p q v = (p-1) (q-1) select small odd integer k such that gcd(k, v) = 1 compute integer d such that (d k)%v = 1 Output: n, k, and d
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RSA algorithm for encryption
Input: integers k, n, M M is integer representation of plaintext message Computation: let C be integer representation of ciphertext C = (Mk)%n Output: integer C ciphertext or encrypted message
RSA algorithm for decryption
Input: integers d, n, C C is integer representation of ciphertext message Computation: let D be integer representation of decrypted ciphertext D = (Cd)%n Output: integer D decrypted message
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This seems hard
How to find big primes? How to find mod inverse? How to compute greatest common divisor? How to translate text input to numeric values? Most importantly: RSA manipulates big numbers Java integers are of limited size how can we handle this? Two key items make the implementation easier understanding the math Javas BigInteger class
What is a BigInteger?
Java class to represent and perform operations on integers of arbitrary precision Provides analogues to Javas primitive integer operations, e.g. addition and subtraction multiplication and division Along with operations for modular arithmetic gcd calculation generation of primes http://java.sun.com/j2se/1.5.0/docs/api/
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Using BigInteger
If we understand what mathematical computations are involved in the RSA algorithm, we can use Javas BigInteger methods to perform them To declare a BigInteger named B
BigInteger B;
Randomly generated primes
BigInteger probablePrime(int b, Random rng) Returns random positive BigInteger of bit length b that is probably prime probability that BigInteger is not prime < 2-100 Random is Javas class for random number generation The following statement Random rng = new Random(); creates a new random number generator named rng What about randomized algorithms in general?
Predefined constants
BigInteger.ZERO BigInteger.ONE
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probablePrime
Example: randomly generate two BigInteger primes named p and q of bit length 32 :
Integer operations
Suppose have declared and assigned values for p and q and now want to perform integer operations on them use methods add, subtract, multiply, divide result BigInteger of operations is a BigInteger Examples:
BigInteger BigInteger BigInteger BigInteger w x y z = = = = p.add(q); p.subtract(q); p.multiply(q); p.divide(q);
/* create a random number generator */ Random rng = new Random(); /* declare p and q as type BigInteger */ BigInteger p, q; /* assign values to p and q as required */ p = BigInteger.probablePrime(32, rng); 13.9 q CompSci 001 = BigInteger.probablePrime(32, rng);
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Greatest common divisor
The greatest common divisor of two numbers x and y is the largest number that divides both x and y this is usually written as gcd(x,y) Example: gcd(20,30) = 10 20 is divided by 1,2,4,5,10,20 30 is divided by 1,2,3,5,6,10,15,30 Example: gcd(13,15) = 1 13 is divided by 1,13 15 is divided by 1,3,5,15 When the gcd of two numbers is one, these numbers are said to be relatively prime
Eulers Phi Function
For a positive integer n, (n) is the number of positive integers less than n and relatively prime to n Examples: (3) = 2 1,2 (4) = 2 1,2,3 (but 2 is not relatively prime to 4) (5) = 4 1,2,3,4 For any prime number p, (p) = p-1 For any integer n that is the product of two distinct primes p and q, (n) = (p) (q) = (p-1)(q-1)
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Relative primes
Suppose we have an integer x and want to find an odd integer z such that 1 < z < x, and z is relatively prime to x We know that x and z are relatively prime if their greatest common divisor is one randomly generate prime values for z until gcd(x,z)=1 if x is a product of distinct primes, there is a value of z satisfying this equality
Relative BigInteger primes
Suppose we have declared a BigIn...