Introduction to Modular Arithmetic and Public Key Cryptography

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• Introduction to Modular Arithmetic and Public Key
  Cryptography

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What is modular arithmetic?

• Modular arithmetic is arithmetic with the
  remainders upon division by a fixed number n.
• It is based upon the idea that the remainder of
  the sum/difference/product of two numbers is the
  remainder of the sum/difference/product of the
  remainders.
• For example, if n5,
• (317)5 385 3, and
• (31575)5 123

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So, what is arithmetic mod n?

• Our numbers are 0, 1, 2, ... (n-1).
• We add, subtract as usual, but subtract or add n
  as necesary to get an answer between 0 and n-1.
• For multiplication, the process is similar
  multiply the two numbers together, and then take
  the remainder dividing by n.

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Some examples, mod n 6.

• 4 3 7 - 6 1
• 3 5 -2 6 4
• 4 5 20 6 2
• WHAT ABOUT DIVISION?????
• Let us say there is an x such that x 2 1.
• Let us also say there is a y such that y 3 1

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Some examples, mod n 6.

• 4 3 7 - 6 1
• 3 5 -2 6 4
• 4 5 20 6 2
• WHAT ABOUT DIVISION?????
• Let us say there is an x such that x 2 1.
• Let us also say there is a y such that y 3 1
• Then x y 2 3 6 xy 1.

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Some examples, mod n 6.

• 4 3 7 - 6 1
• 3 5 -2 6 4
• 4 5 20 6 2
• WHAT ABOUT DIVISION?????
• Let us say there is an x such that x 2 1.
• Let us also say there is a y such that y 3 1
• Then x y 2 3 6 xy 1.
• But 6anything 0!!!

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Some examples, mod n 6.

• 4 3 7 - 6 1
• 3 5 -2 6 4
• 4 5 20 6 2
• WHAT ABOUT DIVISION?????
• Let us say there is an x such that x 2 1.
• Let us also say there is a y such that y 3 1
• Then x y 2 3 6 xy 1.
• But 6anything 0!!!
• So 1 0 ?!?!?!?!?!?!?!?!?

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Can we divide if n is a prime? Yes, but......

• From now on, our modulus will be a prime p.
• We will show how to divide in arithmetic mod p.
• Devious method!
• We will need a result, called the extended
  euclidean algorithm to pull this off.
• But first, we need the euclidean algorithm to
  understand what is going on.
• The euclidean algorithm computes the greatest
  common divisor of two positive integers.

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Elementary Euclidean Algorithm
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Extended Euclidean Algorithm
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What can we do with the egcd?

• Given two numbers a,b, the extended euclidean
  algorithm finds their gcd g and two numbers s and
  t such that as bt g.
• In particular, if a and b have no common factors
  (aside from 1) (i.e. they are relatively
  prime), we can find two numbers s,t such that as
  bt 1
• For modular division, if p is prime, given a, we
  can find s and t such that as tp 1. s is then
  the multiplicative inverse of a (suitably
  reduced, if necesary).

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Some more, strange, results.
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Another result

• Chinese remainder theorem
• Given m1, m2, m3, .... mk and a1, a2 a3, ak,
  where
• The mi, mj are positive, pairwise relatively
  prime
• The ai are positive integers less than mi
  respectively.
• Then, there exists a b such that mi divides b-ai
  for each i.
• If we require that b be less than the product of
  the mi, then this b is unique.
• (Proof in next slide)?

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Proof of Chinese Remainder Theorem

• Suffices to take k2 by induction. Thus, need to
  prove that, for 0 lt a lt m and 0 lt b lt n if m and
  n are relatively prime, there exists a unique u
  between 0 and mn such that u m a u n b
• Since m and n are relatively prime, there exist p
  ,q such that pm qn 1.
• Then bpm aqn mn u satisfies all the
  conditions.

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The RSA Theorem
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Proof of the RSA Theorem
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How RSA works

• Take two primes, p, q, let npq
• Chose an e, relatively prime to (p-1)(q-1).
• Find a d such that de k(p-1)(q-1) 1 with the
  extended euclidean algorithm then
• de 1k(p-1)(q-1)?
• Publish, n, e as public key.
• Encryption raise a to the e-th power
• Decryption raise result to the d-th power.

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Efficient powering to compute an
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Another Crypto-system Diffie-Hellman key exchange

• Let p be a large prime, s a number between 2 and
  p-2 p and s are publicly known.
• Each person has a private key a.
• Whenever two people want to exchange messages,
  they send each other sa mod p
• They raise the number they receive to their
  private key power mod p, and have an exchange key
  for a symmetric crypto-system.

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Another Crypto System El-Gamal

• As before, let p be a large (publicly known)
  prime number, s some number between 2 and p-2.
• Each person chooses a private key e and
  publishes E s raised to the e-th power mod p.
• To send message x, we first generate a session
  key k, and send t sk and y Ek x mod p
• We decrypt by computing t(-e) y x mod p