Presentation is loading. Please wait.

Presentation is loading. Please wait.

Data encryption with big prime numbers DANIEL FREEMAN, SLU.

Similar presentations


Presentation on theme: "Data encryption with big prime numbers DANIEL FREEMAN, SLU."— Presentation transcript:

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 Mod n We think of x mod n as the remainder when x is divided by n. More generally, x≡y mod n means that x and y have the same remainder when divided by n, or that x-y is a multiple of n.

6 Modular exponentiation Fermat’s little theorem : Let p be a prime number and let x be an integer that is not divisible by p. Then,

7 Key points of modular exponentiation 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. Fermat’s little theorem : Let p be a prime number and let x be an integer that is not divisible by p. Then, Euler’s theorem : Let p and q be distinct prime numbers and let x be an integer that is not divisible by p or q. Then,

8 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 { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/4317659/14/slides/slide_7.jpg", "name": "RSA encryption Choose 2 large prime numbers p and q.", "description": "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

9 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.


Download ppt "Data encryption with big prime numbers DANIEL FREEMAN, SLU."

Similar presentations


Ads by Google