Unformatted text preview: rk on efficient specialpurpose
The
implementation (e.g. special circuit board, and then the “RSA chip”,
which did RSA in 0.4 seconds) to prove practicality of RSA.
which
IBM PC debuts in 1981 and Moore’s Law to the rescuesoftware
IBM
now runs 2000x faster…
now
also, software and the Web rule…now ;
also,
Speed differs on types of operations, (i.e. encryption, decryption,
Speed
digital signing and signature verification), as well as relatively size
of e and d ; e.g. with e = 3, encryption and signature verification are
e.g.
typically much (510 times) faster than decryption and digital
much
signing respectively ; Why not make d = 3 instead ? DiffieHellman Key Exchange
DiffieHellman keyexchange enables two users to establish
a shared secret key securely using an open/ public
communications channel. YA
XA Public channel:
anyone can listen to YB
XB = (YB)XA mod q = aXBXA mod q =Secret = aXAXB mod q = (YA) XB mod q DiffieHellman Key Exchange enables two users to establish a shared secret key via
enables
an open/ public communications channel.
open/ Choose a prime number q, and α ( < q and is a primitive
Choose
root of p ); both made public
); Alice randomly chooses x in {2, …, q1} as her secret;
Alice send Bob αx (mod q)
(mod
Bob randomly chooses y in {2, …, q1} as his secret;
Bob
send Alice αy (mod q)
(mod Shared key KAB = (αy)x = (αx)y
Shared DiffieHellman Example users Alice & Bob who wish to swap keys:
agree on prime q=353 and α=3
agree
q=353
select random secret keys: A chooses xA=97, B chooses xB=233
chooses =97,
compute respective public keys:
97
97 mod 353 = 40 233 mod 353 = 248 (Bob) yA=3
yB=3 (Alice) compute shared session key as:
x 97 KAB= yB A mod 353 = 248 mod 353 = 160
248 KAB= yA B mod 353 = 40 mod 353 = 160
40 x 233 (Alice)
(Bob) How secure is DiffieHellman Key
Exchange ? It relies on the fact that “Discrete Logarithm” is a computationally
It
difficult problem, i.e.:
difficult Knowing that YA = aXA mod q and the values of a, q and YA
that
It is still computationally difficult to find XA But still...
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 Fall '13
 CHOWSzeMing,Sherman
 Cryptography, Publickey cryptography

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