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Cryptography

Cryptography. Part 5. Where is this headed?. We are developing the math we need to understand RSA encryption , a remarkably powerful cipher. The system is based on simple modular arithmetic… …But with very large numbers. Inventors of RSA, 1970’s (more on this later).

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Cryptography

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  1. Cryptography Part 5

  2. Where is this headed? • We are developing the math we need to understand RSA encryption, a remarkably powerful cipher. • The system is based on simple modular arithmetic… • …But with very large numbers. Inventors of RSA, 1970’s (more on this later)

  3. The system is so simple it fits on a T-shirt:

  4. Prime numbers • RSA encryption relies on prime numbers • A positive integer p>1 is prime if its only positive integer factors are 1 and itself. • Examples: 17 is prime; it’s only factors are 1,1733 is not prime since 33 = (3)(11).

  5. Is 14 prime? • Yes • No

  6. Is 41 prime? • Yes • No

  7. Is 11000000000000037 prime? • No • Yes

  8. Is 3141618245328721 Prime? • Who Knows? • Can figure out with a fancy program on a computer but how can we tell?

  9. How can you tell if a number is prime? • It is very hard to determine whether large numbers are prime. • Euclid (4th century BC): There are infinitely many prime numbers. • The largest known one has 12,978,189 digits. • How big is this? Replace each letter in the Bible with one of the digits; you would need 3 Bibles to contain the entire number. • In August 2008, a team won a $100,000 prize for the first known prime with at least 10,000,000 digits. • Of course it’s easy to find bigger numbers; the hard part is telling whether they are prime!

  10. Sieve of Erasthones • Discovered by Erasthones (Greek, 300 BC) • Works great for finding small primes, but not big ones. • Creates a sieve (or filter) through which non-primes fall out, and all that remains are primes. • It is a method for finding primes less than a given number. (See board or wikipedia.)

  11. Use the Sieve of Erasthones to find all the primes between 1 and 100. What is the 23rd prime? • 83

  12. Prime factorization • Euclid (4rd century BC): Every integer is a product of prime numbers. • Examples: 24 = 3 * 8 = 3 * 23150 = 52 * 3 * 2

  13. Infinitely Many Primes 2,3,5,7,11,13,17,19,23,29,. . . • Euclid gave the first known proof that there are infinitely many primes. First of all what does it mean to have infinitely many primes? One way to think about this is that if you write down any list of primes, no matter how many you list, there are more primes that are not on your list. For example, the list at the top of the page is incomplete because 31 is prime but is not on the list. • Write down a list of primes. Multiply them together and add 1. Factor this number completely into primes. Any prime that arises is not on the original list. We can repeat this process as much as we want in order to produce new primes. • For example, if you start with 2 and 3, multiplying and adding 1 yields 7, which is prime. Now, starting with 2, 3, 7, multiplying and adding 1 yields 43, which is also prime. To do this one more time, if we multiply 2, 3, 7, and 43 and add 1, we get 1807 = 13 × 139. So, we have the list 2, 3, 7, 13, 37, 139. We could continue.

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