1. Data encryption with big prime numbers DANIEL FREEMAN, SLU

2. Old school codes Full knowledge of the code is needed to both encrypt messages and to decrypt messages. The code can only be used between a small number of trusted people.

3. Public key encryption  If you buy something online, you need to send your credit card number to Amazon.  Your computer needs to be able to encrypt your credit card number.  Amazon does NOT want you to be able to decrypt other people’s credit card numbers.  Everyone needs to be able to encrypt but only Amazon should be able to decrypt.  We need a mathematical technique that is computationally very simple to evaluate, but is extremely computationally difficult to invert.

4. Multiplication is easy, factoring is hard 3568535685356853568535723 * 7564533681359827542555893 Typing in the following into Wolfram Alpha gives an output of 26994308385016394749558484505346578147056894665639 Typing in the following into Wolfram Alpha factor 26994308385016394749558484505346578147056894665639 gives an output of factor 26994308385016394749558484505346578147056894665639

5. Key points of modular exponentiation x (p-1)(q-1) x (p-1)(q-1)+1 More generally, if m ≡ 1 mod (p-1)(q-1) then x m mod n can be efficiently calculated by expressing the exponent m in binary. Euler’s theorem or Fermat’s little theorem : Let p be a prime number and let x be an integer that is not divisible by p. Then, x p-1 Euler’s theorem (a little more general) : Let p and q be distinct prime numbers and let x be an integer that is not divisible by p or q. Then,

6. RSA encryption Choose 2 large prime numbers p and q. Calculate n=pq. p and q should be so large that it is not computationally feasible to factor n. Choose a positive integer e which is relatively prime to (p-1)(q-1). e will be publicly shared. Choose a positive integer d such that ed ≡ 1 mod (p-1)(q-1) Suppose someone wants to encrypt the integer x such that 1<x<n-1. They encrypt x as the value y ≡ x e mod n n will be publicly shared, but p and q will be secret. d will be secret To decrypt, we exponentiate to the power d to get y d ≡ x ed ≡ x mod n

7. RSA example Choose p = 2498359 q = 5418341 Calculate n = pq = 2498359 * 5418341 = 13536961002419 Choose e = 234234239 Solve ed ≡ 1 mod (p-1)(q-1) You calculate and send y= 10021380275883 ≡ x e mod n Amazon then calculates x ≡ y d mod n p and q are just two big prime numbers e was picked to be a prime number big enough to most likely not a factor of (p-1)(q-1) Calculate φ(n) = (p-1)(q-1) = 2498358 * 5418340 = 13536953085720 d = 9846393595559 Suppose you want to send the number x=432564456 to Amazon.