1 Applications of Number Theory CS 202 Epp section 10.4 Aaron Bloomfield

2 About this lecture set I want to introduce RSA –The most commonly used cryptographic algorithm today Much of the underlying theory we will not be able to get to –It’s beyond the scope of this course Much of why this all works won’t be taught –It’s just an introduction to how it works

3 Private key cryptography The function and/or key to encrypt/decrypt is a secret –(Hopefully) only known to the sender and recipient The same key encrypts and decrypts How do you get the key to the recipient?

4 Public key cryptography Everybody has a key that encrypts and a separate key that decrypts –They are not interchangable! The encryption key is made public The decryption key is kept private

5 Public key cryptography goals Key generation should be relatively easy Encryption should be easy Decryption should be easy –With the right key! Cracking should be very hard

6 Is that number prime? Use the Fermat primality test Given: –n: the number to test for primality –k: the number of times to test (the certainty) The algorithm is: repeat k times: pick a randomly in the range [1, n−1] if a n−1 mod n ≠ 1 then return composite return probably prime

7 Is that number prime? The algorithm is: repeat k times: pick a randomly in the range [1, n−1] if a n−1 mod n ≠ 1 then return composite return probably prime Let n = 105 –Iteration 1: a = 92: mod 105 = 1 –Iteration 2: a = 84: mod 105 = 21 –Therefore, 105 is composite

8 Is that number prime? The algorithm is: repeat k times: pick a randomly in the range [1, n−1] if a n−1 mod n ≠ 1 then return composite return probably prime Let n = 101 –Iteration 1: a = 55: mod 101 = 1 –Iteration 2: a = 60: mod 101 = 1 –Iteration 3: a = 14: mod 101 = 1 –Iteration 4: a = 73: mod 101 = 1 –At this point, 101 has a (½) 4 = 1/16 chance of still being composite

9 More on the Fermat primality test Each iteration halves the probability that the number is a composite –Probability = (½) k –If k = 100, probability it’s a composite is (½) 100 = 1 in 1.2  that the number is composite Greater chance of having a hardware error! –Thus, k = 100 is a good value However, this is not certain! –There are known numbers that are composite but will always report prime by this test Source:

10 Google’s recruitment campaign

11 RSA Stands for the inventors: Ron Rivest, Adi Shamir and Len Adleman Three parts: –Key generation –Encrypting a message –Decrypting a message

12 Key generation steps 1.Choose two random large prime numbers p ≠ q, and n = p*q 2.Choose an integer 1 < e < n which is relatively prime to (p-1)(q-1) 3.Compute d such that d * e ≡ 1 (mod (p-1)(q-1)) –Rephrased: d*e mod (p-1)(q-1) = 1 4.Destroy all records of p and q

13 Key generation, step 1 Choose two random large prime numbers p ≠ q –In reality, 2048 bit numbers are recommended That’s  617 digits –From last lecture: chance of a random odd 2048 bit number being prime is about 1/710 We can compute if a number is prime relatively quickly via the Fermat primality test We choose p = 107 and q = 97 Compute n = p*q –n = 10379

14 Key generation, step 1 Java code to find a big prime number: BigInteger prime = new BigInteger (numBits, certainty, random); The number of bits of the prime Certainty that the number is a prime The random number generator

15 Key generation, step 1 Java code to find a big prime number: import java.math.*; import java.util.*; class BigPrime { static int numDigits = 617; static int certainty = 100; static final double LOG_2 = Math.log(10)/Math.log(2); static int numBits = (int) (numDigits * LOG_2); public static void main (String args[]) { Random random = new Random(); BigInteger prime = new BigInteger (numBits, certainty, random); System.out.println (prime); } }

16 Key generation, step 1 How long does this take? –Keep in mind this is Java! –These tests done on a 850 Mhz Pentium machine –Average of 100 trials (certainty = 100) –200 digits (664 bits): about 1.5 seconds –617 digits (2048 bits): about 75 seconds

17 End of lecture on 23 March 2007  In the later class; the earlier class did the next 3 slides

18 Key generation, step 1 Practical considerations –p and q should not be too close together –(p-1) and (q-1) should not have small prime factors –Use a good random number generator

19 Key generation, step 2 Choose an integer 1 < e < n which is relatively prime to (p-1)(q-1) There are algorithms to do this efficiently –We aren’t going over them in this course Easy way to do this: make e be a prime number –It only has to be relatively prime to (p-1)(q-1), but can be fully prime

20 Key generation, step 2 Recall that p = 107 and q = 97 –(p-1)(q-1) = 106*96 = = 2 6 *3*53 We choose e = 85 –85 = 5*17 –gcd (85, 10176) = 1 –Thus, 85 and are relatively prime

21 Key generation, step 3 Compute d such that: d * e ≡ 1 (mod (p-1)(q-1)) –Rephrased: d*e mod (p-1)(q-1) = 1 There are algorithms to do this efficiently –We aren’t going over them in this course We choose d = 4669 –4669*85 mod = 1 Use the script at

22 Key generation, step 3 Java code to find d: import java.math.*; class FindD { public static void main (String args[]) { BigInteger pq = new BigInteger("10176"); BigInteger e = new BigInteger ("85"); System.out.println (e.modInverse(pq)); } } Result: 4669

23 Key generation, step 4 Destroy all records of p and q If we know p and q, then we can compute the private encryption key from the public decryption key d * e ≡ 1 (mod (p-1)(q-1))

24 The keys We have n = p*q = 10379, e = 85, and d = 4669 The public key is (n,e) = (10379, 85) The private key is (n,d) = (10379, 4669) Thus, n is not private –Only d is private In reality, d and e are 600 (or so) digit numbers –Thus n is a 1200 (or so) digit number

25 Encrypting messages To encode a message: 1.Encode the message m into a number 2.Split the number into smaller numbers m < n 3.Use the formula c = m e mod n c is the ciphertext, and m is the message Java code to do the last step: –m.modPow (e, n) –Where the object m is the BigInteger to encrypt

26 Encrypting messages example 1.Encode the message into a number –String is “Go Cavaliers!!” –Modified ASCII codes: Split the number into numbers < n –Recall that n = – Use the formula c = m e mod n – mod = 4501 – mod = 2867 – mod = 4894 –Etc… Encrypted message: –

27 Encrypting RSA messages Formula is c = m e mod n Formula is c = m e mod n

28 Decrypting messages 1.Use the formula m = c d mod n on each number 2.Split the number into individual ASCII character numbers 3.Decode the message into a string

29 Decrypting messages example Encrypted message: – Use the formula m = c d mod n on each number – mod = 4181 – mod = 0237 – mod = 6788 –Etc… 2.Split the numbers into individual characters – Decode the message into a string –Modified ASCII codes: –Retrieved String is “Go Cavaliers!!”

30 modPow computation 1.How to compute c = m e mod n or m = c d mod n? –Example: mod = 4181 Use the script at Other means: –Java: use the BigInteger.modPow() method –Perl: use the bmodpow function in the BigInt library –Etc…

31 Why this works m = c d mod n c = m e mod n c d ≡ (m e ) d ≡ m ed (mod n) Recall that: –ed ≡ 1 (mod p-1) –ed ≡ 1 (mod q-1) Thus, –m ed ≡ m (mod p) –m ed ≡ m (mod q) m ed ≡ m (mod pq) m ed ≡ m (mod n)

32 Cracking a message In order to decrypt a message, we must compute m = cd mod n –n is known (part of the public key) –c is known (the ciphertext) –e is known (the encryption key) Thus, we must compute d with no other information –Recall: choose an integer 1 < e < n which is relatively prime to (p-1)(q-1) –Recall: Compute d such that: d*e mod (p-1)(q-1) = 1 Thus, we must factor the composite n into it’s component primes –There is no efficient way to do this! –We can, very easily, tell that n is composite, but we can’t tell what its factors are Once n is factored into p and q, we compute d as above –Then we can decrypt c to obtain m

33 Cracking a message example In order to decrypt a message, we must compute m = c d mod n –n = –c is the ciphertext being cracked –e = 85 In order to determine d, we need to factor n –d*e mod (p-1)(q-1) = 1 –We factor n into p and q: 97 and 107 –This would not have been feasible with two large prime factors!!! –d * 85 (mod (96)(106)) = 1 We then compute d as above, and crack the message

34 Signing a message Recall that we computed: d*e mod (p-1)(q-1) = 1 Note that d and e are interchangable! –You can use either for the encryption key You can encrypt with either key! –Thus, you must use the other key to decrypt

35 Signing a message To “sign” a message: 1.Write a message, and determine the MD5 hash 2.Encrypt the hash with your private (encryption) key 3.Anybody can verify that you created the message because ONLY the public (encryption) key can decrypt the hash 4.The hash is then verified against the message

36 PGP and GnuPG Two applications which implement the RSA algorithm –GnuPG Is open-source (thus it’s free) –PGP was first, and written by Phil Zimmerman The US gov’t didn’t like PGP…

37 The US gov’t and war munitions

38 How to “crack” PGP Factoring n is not feasible Thus, “cracking” PGP is done by other means –Intercepting the private key “Hacking” into the computer, stealing the computer, etc. –Man-in-the-middle attack (next 2 slides) –Etc.

39 Man-in-the-middle attack: “Normal” RSA communication What is your public key? My public key is 12345… What is your public key? My public key is 67890… Here’s message encrypted with 12345… Here’s a response encrypted with 67890…

40 What is your public key? My public key is 12345… My public key is abcde… What is your public key? My public key is 67890… My public key is vwxyz… Here’s message encrypted w/ abcde… Decrypts message with corresponding private key to abcde…; re-encrypts message with blue’s public key (12345…) Here’s message encrypted w/ 12345… Here’s response encrypted w /vwxyz… Decrypts message with corresponding private key to vwxyz…; re-encrypts message with yellow’s public key (67890…) Here’s response encrypted w/ 67890… Black has the private decryption key for abcde… Black has the private decryption key for vwxyz…

41 Other public key encryption methods Modular logarithms –Developed by the US government, therefore not widely trusted Elliptic curves

42 Quantum computers A quantum computer could (in principle) factor n in reasonable time –This would make RSA obsolete! –Shown (in principle) by Peter Shor in 1993 –You would need a new (quantum) encryption algorithm to encrypt your messages This is like saying, “in principle, you could program a computer to correctly predict the weather” A few years ago, IBM created a quantum computer that successfully factored 15 into 3 and 5 I bet the NSA is working on such a computer, also

43 Sources Wikipedia article has a lot of info on RSA and the related algorithms –Those articles use different variable names –Link at