(Tuesday, 20 October, 7:30-9PM) Encryption-Decryption CS100A Lecture 13 15 Oct. 1998 Discussion of Prelim 2 (Tuesday, 20 October, 7:30-9PM) Rooms for Prelim 2 A-K: Hollister B14 L-Z: Kimball B11 Cryptography: Encryption-Decryption using arrays (assignment 6) CS100A, Lecture 13, 15 October 1998

Prelim 2 1. Everything that was on Prelim 1. In particular: Be able to define four kinds of variables: local variables, parameters, fields (instance variables) and static fields (class variables) Know the three kinds of methods (procedures, functions, constructors) Know what an argument is. Know precisely the steps in executing a method call. Know precisely the steps in evaluating new C(…) 2. Loops: Know what a loop invariant is and how it is used. Be able to write a loop, given an invariant. Write simple loops without be given an invariant. 3. Arrays: know how to declare an array, allocate an array, use an array. 4. Understand type char. CS100A, Lecture 13, 15 October 1998

Encryption-Decryption CS100A Lect. 13, 15 Oct. 1998 Cryptography: Encryption-Decryption using arrays (assignment 6) Cryptography even before Caesar’s time: Encode messages, with the hope that only “friends”, who have been given the decoding scheme, can decode them. “Have a nice day” replace every character by the next one: a --> b, etc. “Ibwf b ojdf ebz” CS100A, Lecture 13, 15 October 1998

During world war II, the Germans encoded messages using a kind of “computer” (not a real computer, as we know them today) they had built, called the Enigma. The British had a group that worked continu-ously to intercept and decode the messages. They succeeded in breaking the code, and this was one reason for the success of the Allies. At times, they couldn’t use what they had learned because they didn’t want the Germans to suspect that their codes had been broken. Alan Turing, a mathematician who did a lot for computing (about 1936) even before computers were developed, had a big part in this. You’ll learn about Turing’s contributions to the theory of computing --the Turing Machine-- in CS481. CS100A, Lecture 13, 15 October 1998

Two types of cryptosystems: Secret key: both the sender and the receiver have the key to encoding/decoding. Hopefully, no one else does. How to send the receiver the secret key (assuming it has to be changed) without others intercepting it? Public key-private key: Gries decides on a public key - private key pair. He makes the public key available to everyone. Anyone wanting to send Gries an encoded message encodes it using the public key. Only Gries, who knows the private key, can decode the message. Diffie and Hellman published the idea in 1976, but without a good implementation. RSA (by Ron Rivest, Adi Shamir, and Leonard Adelman) found a way to implement it, using number theory. This assignment concerns the RDA public key - private key method. CS100A, Lecture 13, 15 October 1998

long integers: -9223372036854775808.. 9223372036854775807 Public key (puk, m) Private key (prk, m) Examples: puk prk m 401 137 551 229 349 399 241 481 551 109 493 Send Gries/Cardie messages using public key (109, 493). Only they can decode them because only they know the value prk. With small numbers, it can be guessed, but remember that these can be long integers --or even larger integers if we use some other representation of itnegers in Java. We don’t show how to generate public key - private key pairs. Must be hard to guess the prk. For example, given two primes p1 and p2, it’s easy to calculate p1*p2. But, given p1*p2, it’s very hard to calculate p1 or p2! Easy to multiply two integers; hard to factor some integers. CS100A, Lecture 13, 15 October 1998

Arithmetic modulo m (for m>0) Numbers can get too big when encrypting and decrypting (bigger than the biggest number in type long). We need a way to keep integers small. Use arithmetic modulo m, in which all integers are kept in the range 0..m-1. For any integer i, mod(i,m), or i mod m , is the integer that satisfies i = q*m+r and 0<=r < m (for some q) 6 mod 5 = 1 1 mod 5 = 1 -4 mod 5 = 1 5 mod 5 = 0 0 mod 5 = 0 -5 mod 5 = 0 4 mod 5 = 4 -1 mod 5 = 4 3 mod 5 = 3 -2 mod 5 = 3 2 mod 5 = 2 -3 mod 5 = 2 To calculate (i mod m): If i >=0: Use i%m (remainder when i is divided by m) If i<0: Use (i%m) + m See method mod in class Crypto for an analysis. CS100A, Lecture 13, 15 October 1998

Use arithmetic modulo m When encrypting and decrypting, after EVERY opera-tion that might produce an integer r that is larger than m, reduce it modulo m, that is, use r mod m instead! RSA: To encrypt an integer i as an integer j, use j = i puk mod m To decrypt an integer j to yield i, use i = j prk mod m In RSA, puk. prk, and m are chosen to guarantee that i = ( i puk mod m)prk mod m CS100A, Lecture 13, 15 October 1998

Encrypt a String s of characters as a long array c[0..s.length()] Each element c[i] of c is the encryption of s[i]: ((int) s.charAt(i)) puk mod m To decrypt long array c and produce the String s: each character s[i] is (char) (c[i] prk mod m) Example: the String “CS100” with prk = 401 and m =551 is encrypted as the array {383, 277, 197, 98, 98} CS100A, Lecture 13, 15 October 1998