JavaScript Algorithms: Cryptography

Katy Moe

Introduction

intro to crypto

the RSA algorithm

mathematical concepts

key generation

signing, encryption/decryption

Crypto is about keeping secrets

THIS IS A SECRET MESSAGE TO BE XORED WITH A SECRET KEY TO PRODUCE A CIPHERTEXT. THIS IS CALLED ENCRYPTION.

crypto in JS!

home-made crypto!

RSA

Rivest, Shamir, Adleman (1977)

Cocks (1973, declassified 1997)

Puzzle

Bob wants to send a box to Alice.

Alice and Bob each have a distinct padlock and key (i.e. Alice has key which fits only her padlock, and Bob has a key which only fits his).  Bob wants to lock a box and send it to Alice, but doesn't want Eve (who can intercept the package) to be able to open it. Eve has no padlocks or keys.

Alice and Bob can send messages to each other, as well as padlocks, keys, and boxes. A box is locked by attaching a padlock to it - the key is only required for unlocking. How can Bob send the box to Alice without Eve getting access to the contents?

Alice

Eve

Bob

H487SBFK593OSFH0N8BLSK4278SBTOB9P

Bob

Alice

+ Alice's public key

+ Alice's private key

Hello, Alice!

Hello, Alice!

Some mathematical concepts

Prime

A number only divisible by itself

2, 3, 5, 7, 11, 13, 17, 19, 23, 29...

Fundamental Theorem of Arithmetic

Every number has a unique set of prime factors

12 = 3 x 2 x 2

330 = 11 x 5 x 3 x 2

Coprime

Two numbers are coprime if they share no prime factors

12 = 3 x 2 x 2

35 = 7 x 5

Modular arithmetic

a is congruent to b mod n if they have the same remainder when divided by n

17 ≡ 5 (mod 12)

125 ≡ 5 (mod 12)

Euler's Totient

Euler's Totient ϕ(n) is the number of numbers less than or equal to n that are coprime to n

ϕ(4) = 2

ϕ(7) = 6

ϕ(2) = 1

Fermat's Little Theorem

a   ≡ 1 (mod p)

p - 1

If p is prime and is coprime to a then:

RSA key generation

1. Large Primes: Generate two large prime numbers, p and q.

2. Modulus: From the two primes, calculate the modulus n = p * q.
3. Totient: Calculate Euler's Totient of n, ϕ(n).
4. Public Key: A prime number is calculated from the range [3,ϕ(n)) that is coprime with ϕ(n).
5. Private Key: Because the prime in step 4 has a gcd of 1 with ϕ(n), we are able to determine its inverse with respect to modϕ(n).

RSA encryption/decryption

For message m and key k:

F(m,k)= m    (mod n)

k

RSA signing

1. Alice encrypts the ciphertext with her own private key and sends this along with the ciphertext to Bob.

2. Bob decrypts the signature with Alice's public key and checks whether it matches the ciphertext.

Advanced challenges

1. [JS] Delete the helper functions and write them yourself.
2. [JS] Expand the algorithm to deal with Unicode characters.
3. [crypto/JS] Design a brute-force algorithm against RSA.
4. [UX] How would you incorporate RSA into a messaging app?
5. [crypto] Have a go at symmetric key exchange.
6. [JS] How can signing be made more performant?

Let's code!

slides.com/katharinemoe/crypto-js

github.com/LondonAlgorithms/Cryptography

Resources

1. Prime Number Hide-and-Seek: How the RSA Cipher Works [article]
2. Public Key Cryptography: RSA Encryption Algorithm [video]
3. Gambling with Secrets: 8/8 (RSA Encryption) [video]
4. Public Key Cryptography and the RSA Cipher [book chapter]
5. Why RSA Works: Three Fundamental Questions Answered [article]

Encryption in JavaScript

1. What's wrong with in-browser cryptography? [article]
2. No I didn't use the Web Crypto API [article]
3. JavaScript cryptography considered harmful [article]
4. The anatomy of a bad idea [article]

RSA