RSA Cryptography

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21 Νοε 2013 (πριν από 3 χρόνια και 6 μήνες)

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CRYPTOGRAPHY

Audrey Jones

Michael Healy

Lillian Coletta

http://spectrum.ieee.org/computing/software/c
ryptographers
-
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

http://paralleluniversestory.blogspot.com/2009/09/
german
-
enigma
-
cipher
-
machine.html


RSA CRYPTOGRAPHY

http://www.cse.iitb.ac.in/~aditiwari/cs296.htm


Rivest
, Shamir and
Adleman

EXAMPLES OF
RSA

http://www.loansafe.org/?attachment_id=3441


Public Key: (n, e)

Private Key: (p,q,d)


http://www.iis.ee.ethz.ch/~kgf/acacia/c3.html





0

STEPS

1.
Bob chooses p and q


Compute n =
pq
, and (p
-

1)(q
-

1)

2.
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

3.
Find d such that:


de


1(mod (p
-
1)(q
-
1))


Euclidean Algorithm

d = 35

4.
Alice picks a number to be message

5.
Alice encrypts message and sends to Bob


E(m) = m
e

(mod pq)

6.
Using the private key Bob will decrypt message
and understand Alice

s message


D(c) = c
d

(mod pq)



n = 221

(p
-
1)(q
-
1) = 192

e = 11

d = 35


m = 12

CALCULATING WITH
MATHEMATICA

MATHEMATICAL!!!!

WHY DOES
RSA

WORK?


It all goes back to…


Fermat

s Little Theorem


OUTLINE OF PROOF

I.
Fundamental Theorem of Modular
Arithmetic

II.
Fermat

s Little Theorem

III.
Putting it all together

STRENGTH OF RSA

Facebook


BIG NUMBERS!!!


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 was

232

digits long (768 bits).


It took that team almost 3 YEARS to do this.


Compare that to Facebook's n, which is

309

digits long
(1024 bits).


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>.