1. Cryptography: from substitution cipher to RSA
Alex Karassev

2. Description of the problem
Alice wants to send a secret message to Bob
Problem: this message can be captured
Possible ways to intercept a message:
steal a letter from mailbox
wiretap \ listen in a phone call
steal data from network cable
use a hacker attack

3. Possible solutions Conceal the fact of sending
Conceal the text of the message

4. Conceal the fact of sending
However, in all these examples
the message is send as a plaintext
(i.e. non-encrypted text)
Use “invisible” ink
“Hide” the message inside a larger text
Use very small font (also used as anti-counterfeiting measure: look at dollar bills for example)
Example from ancient time: shaving the head of a slave and tattooing the message on the slave's bald scalp. When his hair had grown enough to conceal the tattoo, the slave was sent to another person who shaved the slave's head to receive the message

5. Conceal the text of the message
Use cipher or, more precisely, encryption
Encryption is a process that transforms a plaintext of the message into a different text (which usually looks as a meaningless collection of symbols or numbers) according to certain algorithm
The resulting text is called ciphertext (or encrypted message)
Algorithms used are called ciphers
The study of ciphers is called cryptography

6. Alice sends a message to Bob…
How to make all this happen?
Alice sends a message to Bob…
Alice takes the message and replaces each letter in the message by some other letter of symbol (or by several symbols) according to some cipher
She sends it then to Bob
Nobody else except for Bob knows how to read the encrypted message
Bob receive the message and uses the process of deciphering (or decrypting) to read the message

7. The oldest algorithm: Substitution cipher
Each letter of alphabet is replaced by another letter or symbol, or several symbols
Example: A → 1, B → 2, C → 3 and so on
Less trivial example:
A → 26, B → 25, C → 24, …, Z → 1

8. Main Requirement
We must have a procedure allowing us to restore plaintext from encrypted text without ambiguities
Then we need…
One-to-one correspondence: different letter of alphabet are replaced by different symbols

9. Immediately, we have a problem: What is 262524? Is it ABC?
Substitution table:
Immediately, we have a problem: What is ?
Is it ABC?
Or is it YUYVYW?
Or maybe ABYW?
Also, we need to encode spaces between words A B C D E F G H I J K L M 26 25 24 23 22 21 20 19 18 17 16 15 14 N O P Q R S T U V W X Y Z 13 12 11 10 9 8 7 6 5 4 3 2 1

10. It would be better to use the following cipher: A B C D E F G H I J K L M 26 25 24 23 22 21 20 19 18 17 16 15 14 N O P Q R S T U V W X Y Z 13 12 11 10 09 08 07 06 05 04 03 02 01
It would be better to use the following cipher:
A → 26, …,X →03, Y →02, Z → 01 and space is 00
We know that every TWO symbols represent a letter
Thus
is…
MATH IS THE BEST

11. A historical example of substitution cipher – shift (or Caesar) cipher
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Choose k
Shift all letters by k
For example, if k = 5
A becomes F, B becomes G, C becomes H, and so on…
What will replace X?

12. Modular arithmetic
In the Caesar cipher, the following algorithm is used
If n is the number of a letter in the alphabet, this letter is replaced by another letter, whose number is (n+k) modulo 26 (shortly (n+k) mod 26)
This is a remainder of division of (n+k) by 26
For example, take k=5 and take letter X
Its number n = 24
(n+k) mod 26 = mod 26 = 3
So X is replaced with C

13. Exercise
Use shift cipher with k=7 to encrypt the text “TOP SECRET” (ignore space)

14. Is substitution cipher really good?
It seems it satisfies the main condition: if Alice and Bob agree on the table for substitution then they can exchange messages and keep them secret
Is it really the case?
Substitution cipher is completely ruined by
FREQUENCY ANALYSIS

15. Frequency analysis Discovered by Arabs (approx. 9th century)
Statistically, it is possible to determine how often each letter appears in an “average” text
Frequency table
Other useful observation:
ST, NG, TH, and QU are common pairs of letters (bigrams), while NZ and QJ are rare.
What is one the most common trigram?
THE
Letters that often appear at the beginnings of words

16. To analyze the text…
Count the number of appearance of each letter and divide it by the total number of words in the ciphertext
Compare the results with the frequency table
Note: this method applies effectively to sufficiently large texts

17. Weaknesses of the substitution cipher
Every letter is ALWAYS encoded by the same symbol, which makes frequency analysis a very effective tool
Another problem: knowing context of the message is very useful (for instance, if you know that the text is about the types and number of airplanes in the enemy’s army, you should expect words “airplane”, “aircraft”, “weapon”, and so on)
Guesses also help: for instance, if you deciphered a part of the sentence “A cat drinks …” you would guess that the last word is “milk” (although it could be “lemonade”)

18. Examples from Literature and History
the use of frequency analysis to attack simple substitution ciphers:
Arthur Conan Doyle “Sherlock Holmes tale: The Adventure of the Dancing Men”
Edgar Allan Poe “The Gold Bug”
Greeks: scytale cipher
Morse code
WWII: Enigma machine
ASCII (American Standard Code for Information Interchange): Each character is encoded by numbers from 0 to127 in binary format

19. Alternatives to substitution cipher
Symmetric key algorithms (private key)
Algorithms with two keys (private and public)

20. Symmetric key algorithm: simplified description
Block cipher: a plaintext is divided into blocks of equal length
Key: a sequence of symbols (or, equivalently, a number)

21. Encryption and Decryption with symmetric key: mathematical model
Plaintext PT
Cipher defines a function F that has two arguments: PT and a key k
Ciphertext CT = F(PT,k)
Decryption
PT = F-1 (CT,k)

22. Example: Vernam’s cipher
Represent each letter in binary form
Transform the plaintext into the sequence of 0’s and 1’s
Choose a length of the key (say, 7 digits)
Generate a random 7-digits key
Divide this sequence into blocks (7 digits in each block)
Use addition modulo 2 to encrypt the message

23. Vernam's cipher
It was mathematically proved unbreakable by Claude Shannon (in 1940s) assuming that the following conditions are satisfied:
Key is as long as the plaintext
Key is random
Used only once, and kept entirely secret
One-time pad

24. Main problem with symmetric cipher
If Alice wants to send encrypted messages to Bob she needs to send the key to Bob
This is not safe
It would be nice if
Bob lets Alice know the algorithm
Bob send Alice a key called PUBLIC KEY (more precisely, Bob’s public key) (so everybody knows it)
Alice uses this key to send an encrypted message to Bob
Bob uses another key (PRIVATE KEY) to decipher the message from Alice
Nobody else (who does not know Bob’s private key) can decipher this message

25. Is it at all possible? YES! RSA algorithm is one of such methods
The algorithm was described in 1977 by Ron Rivest, Adi Shamir and Len Adleman at MIT
RSA is an “unbreakable” algorithm

26. Why is RSA so good? Factorization into primes
A prime number is a positive integer greater than 1 that is divisible only by 1 and itself
so 2, 3, 5, 7, 11, … are prime numbers
while 84 = 3*4*7 is not
Fundamental theorem of arithmetic: every integer is a product of prime numbers. Such a representation is unique (up to permutation of factors)
Exercise: there are infinitely many primes

27. Why is RSA so good?
At the present time, no effective algorithm of finding prime factorization of numbers is known
Effective means an algorithm that works “reasonably” fast
Example: is 943 prime?

28. References and Further Reading
Undergraduate
“Introduction to Cryptography” by Johannes Buchmann (Springer)
“Cryptography: An Introduction” by V. V. Yaschenko (editor) (American Mathematical Society)
Graduate
“A Course in Number Theory and Cryptography” by Neal Koblitz (Springer)