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G special circuit board and then the rsa chip which

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Unformatted text preview: rk on efficient special-purpose 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 rescue---software 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 (5-10 times) faster than decryption and digital much signing respectively ; Why not make d = 3 instead ? Diffie-Hellman Key Exchange Diffie-Hellman key-exchange 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 Diffie-Hellman 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, …, q-1} as her secret; Alice send Bob αx (mod q) (mod Bob randomly chooses y in {2, …, q-1} as his secret; Bob send Alice αy (mod q) (mod Shared key KAB = (αy)x = (αx)y Shared Diffie-Hellman 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 Diffie-Hellman 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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