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Cryptography

Cryptography. Audrey Jones Michael Healy Lillian Coletta. http://spectrum.ieee.org/computing/software/cryptographers-take-on-quantum-computers. What is it?. Cryptography is from the Greek for “ secret writing ” Altering a message to appear incomprehensible to an outside party.

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Cryptography

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  1. Cryptography Audrey Jones Michael Healy Lillian Coletta http://spectrum.ieee.org/computing/software/cryptographers-take-on-quantum-computers

  2. What is it? • Cryptography is from the Greek for “secret writing” • Altering a message to appear incomprehensible to an outside party http://paralleluniversestory.blogspot.com/2009/09/german-enigma-cipher-machine.html

  3. RSA CRYPTOGRAPHY Rivest, Shamir and Adleman http://www.cse.iitb.ac.in/~aditiwari/cs296.htm

  4. Examples of rsa http://www.loansafe.org/?attachment_id=3441

  5. Public Key: (n, e) Private Key: (p,q,d) http://www.iis.ee.ethz.ch/~kgf/acacia/c3.html 0

  6. Steps • Bob chooses p and q • Compute n = pq, and (p - 1)(q - 1) • Find an e that is relatively prime to (p-1)(q-1) p = 13 and q = 17 n = (p • q) = 221 (p -1) = 12 and (q -1) =16 (p -1)(q -1) = 192 e =11

  7. Find d such that: de ≡ 1(mod (p-1)(q-1)) • Euclidean Algorithm d = 35

  8. Alice picks a number to be message • Alice encrypts message and sends to Bob • E(m) = me (mod pq) • Using the private key Bob will decrypt message and understand Alice’s message • D(c) = cd (mod pq) n = 221 (p -1)(q -1) = 192 e = 11 d = 35 m = 12

  9. Calculating with mathematica MATHEMATICAL!!!!

  10. Why does Rsa work? • It all goes back to… Fermat’s Little Theorem

  11. Outline of proof • Fundamental Theorem of Modular Arithmetic • Fermat’s Little Theorem • Putting it all together

  12. Strength of RSA Facebook BIG NUMBERS!!!

  13. The strength of RSA • Factoring n is currently the most efficient method for breaking RSA. • One problem with this: factoring enormous numbers takes an enormous amount of time. • As of today, the largest RSA number to be factored was232digits long (768 bits). • It took that team almost 3 YEARS to do this. • Compare that to Facebook's n, which is309digits long (1024 bits).

  14. References • Melanie Brown (Group mentor) • Voight, John. "Prime Factorization Algorithm." Message to the author. 18 Apr. 2012. E-mail. • Stankova-Frenkel, Zvezdelina. "RSA Encryption." Berkeley Math Circle. University of California at Berkeley, 22 Dec. 2000. Web. 23 Apr. 2012. http://mathcircle.berkeley.edu/BMC3/rsa/node4.html • Perez, Pascal and Weisstein, Eric W. "Successive Square Method." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SuccessiveSquareMethod.html • Kleinjung, Aoki, and Franke. "Factorization of a 768-bit RSA Modulus." 18 Feb. 2010. Web. 23 Apr. 2012. <http://eprint.iacr.org/2010/006.pdf>.

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