1. Creating Secret Messages

2. 2 Why do we need to keep things secret? Historically, secret messages were used in wars and battles For example, the Enigma cipher in WW2 Computers are nowadays used to store lots of information about us The recent “lost CDs in the post” contained data that was not kept secret Also need to make sure that online purchases are secure

3. 3 Terminology Cryptography: study of creating secret codes Plaintext: the original message Ciphertext: the message after encryption Encryption: the process of creating a ciphertext Decryption: reversing encryption Cipher: name often given to an encryption

4. 4 A tiny bit of history Here’s some encryption methods: Caesar cipher: apparently used in 2 nd century AD by Julius Caesar Substitution cipher: a simple version (using pairs of letters) was used in 4 th century AD Vignere cipher: devised in the 16 th century RSA: developed in 1977 by Rivest, Shamir and Adleman – still used today

5. 5 Keys Many encryption methods use a key The key is used to create a ciphertext Different keys give different ciphertexts Encryption Ciphertext Plaintext

6. 6 Keys again However for many encryptions, the same key is used for encryption and decryption These are called symmetric ciphers It is important to keep the key secret Decryption Ciphertext Plaintext

7. 7 The Caesar Cipher One of the oldest and simplest ciphers The key is a letter which tells you how much to “shift” the letters in the plaintext by ‘A’ means shift 0 places, ‘B’ shift 1 place, etc

8. 8 Problems with the Caesar Cipher There are only 25 possible keys so we can simply display all the possible decryptions We can reduce the number of possible keys further by using frequency analysis We can also look at one and two letter words to help us decode the text Methods for analysing the text can also be used with other ciphers

9. 9 Frequency Analysis The values below give the average frequencies of letters in English The most common letters are E,T,A and I. The values will differ for other languages ABCDEFGHI 6.4%1.4%2.7%3.5%10.0%2.0%1.4%4.2%6.3% JKLMNOPQR 0.3%0.6%3.5%2.0%5.6% 1.7%0.4%4.9% STUVWXYZOth 5.6%7.1%3.1%1.0%1.8%0.3%1.8%0.2%16.6%

10. 10 Substitution Cipher We replace (substitute) each letter in the plaintext with a different letter This time the key is a table of replacements To encrypt all texts, we need a key that contains 26 letters with no repeats

11. 11 An example THIS SENTENCE IS ENCRYPTED OAZH HLEOLEXL ZH LEXKMVOLJ

12. 12 Another example DECRYPTED JLXKMVOLJ

13. 13 Weaknesses We can still do frequency analysis to help to guess of the letters The key can be quite hard to remember but we can use a word or phrase at the beginning of the cipher key to help What is the problem with doing this?

14. 14 Vignère Cipher For this a key word is used. We repeat the key word and each letter of the key word acts a key for a Caesar cipher Cannot do a frequency analysis on the cipher text But it is possible to guess the length of the key and the key itself KEYKEYKEYK PlainHELLOWORLD CipherRIJVSUYVJN

15. 15 Weaknesses and Keys All the examples we have seen so far use the same key for encryption and decryption (called symmetric ciphers) If we can guess the key then we can break the cipher It is hard to keep the key secret An alternative method is to use different keys for encryption and decryption – this is called a asymmetric cipher

16. 16 RSA The RSA cipher uses two keys and is much more complicated then any of the ciphers we’ve seen so far The key used for encryption is called the public key and the key used for decryption is called the private key (which is kept secret) To make it hard to break, one of the keys needs to be around 600 digits (2048 bits)

17. 17 Modular Arithmetic RSA encryption uses many different Maths techniques. One is called modular arithmetic. We write 10 (mod 6) to mean “give me the remainder when 10 is divided by 6” So 10 (mod 6) = 4 (because 10 = 6 × 1 + 4) 75 (mod 9) = 3 (75 = 9 × 8 + 3) 56 (mod 12) = 8 (56 = 12 × 4 + 8) 89 (mod 10) = 9 (89 = 10 × 8 + 9)

18. 18 Primes and Factoring For RSA, we have to pick two prime numbers, let’s say p and q. Remember a prime number has exactly two factors (so 2 and 3 are prime, but not 1 or 4) We have to work out the product n of the two primes so n = p × q To find p and q from n is hard – we would have to check all the (prime) numbers up to p or q. This is what makes RSA secure.

19. 19 Keys We now pick a public key e We find a private key d such that: d £ e = 1 (mod (p – 1)(q – 1)) (There’s a method called the extended Euclidean algorithm which can find this) To encrypt a number (message) M we do M e To decrypt, we do C d

20. 20 Example Pick p = 7, q = 13 so n = 7 × 13 = 91 Pick e = 5 and then d = 29 Check 5 × 29 = 145 = (2 × 6 × 12) + 1 To encrypt the number 10, we do 10 5 (mod 91) = 10 × 10 × 10 × 10 × 10 (mod 91) = 82 To decrypt we do 82 29 (mod 91) = 10

21. 21 Comments on RSA To make RSA fast we need to be able to find powers quickly – there are methods to help us do this The strength on RSA relies on the fact that it is generally hard to factor a very large number By using two keys, we can keep the private key secret There are many other “public key” methods

22. 22 Conclusions We have seen that there are many ways in which we can keep information secret We need to keep the decryption keys secret – in a symmetric cipher this is very difficult It is quite easy to break some ciphers by analysing the text Good encryption methods rely on using lots of different Maths techniques