1. What use are prime numbers? ?

2. o All the primes, apart from 2, are odd numbers. o 1 is not a prime number. Why? o There are infinitely many prime numbers! o The largest known prime number is 12,978,189 digits long. o Prime numbers can be used to keep private information safe. (in a similar way to how codes are used to send secret messages)

3. Internet security o Every time you enter a password or PIN number into a computer HUGE numbers are used to encrypt it and keep your information protected from attackers!

4. RSA o The type of encryption process used today on the internet is called the RSA public key cryptosystem, named after its inventors Ronald R ivest, Adi S hamir and Leonard A dleman.RSA o In order to send information securely a ‘padlock’ system is used. What is special about the RSA is that it is very easy to lock up but very hard to open - what is called a one-way operation in maths, something that is very easy to calculate in one direction but very hard in the other.

5. o Multiplying two prime numbers together is easy. However, working in the opposite direction, finding the two prime factors of a number, is much more tricky especially when the number you’re factorising is HUGE! o Find the 2 prime factors of 3799283 Answer: 2437 and 1559

6. Fermat’s Little Theorem o The method used in RSA cryptography has been developed from Fermat’s Little Theorem. Fermat’s Little Theorem If p is a prime number and m is any number which is not divisible by p then m (p-1) ≡ 1 (mod p) i.e. m (p-1) leaves remainder 1 when divided by p. The version used by the RSA requires p to be the product of two prime numbers.

7. Modular Arithmetic Fermat’s Little Theorem uses modular arithmetic (sometimes called clock arithmetic) which is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value— the modulus.

8. Example

9. Try these questions