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Data encryption with big prime numbers DANIEL FREEMAN, SLU

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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.

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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.

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Multiplication is easy, factoring is hard * Typing in the following into Wolfram Alpha gives an output of Typing in the following into Wolfram Alpha factor gives an output of factor

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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.

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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,

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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,

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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

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RSA example Choose p = q = Calculate n = pq = * = Choose e = Solve ed ≡ 1 mod (p-1)(q-1) You calculate and send y= ≡ 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) = * = d = Suppose you want to send the number x= to Amazon.

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