ICS 268, Fall'04
Lecture Summaries, Homeworks, Solutions, Handouts
[+ a tentative schedule for what's to come]
[back to course main page]
Lectures 12
(lect1.pdf)
Lectures 34
(h1primes.pdf)
,
(h2composites.pdf)
,
(Dana Angluin's notes on
computation and number theory.pdf
)
.
We covered some basic modular arithmetic in the "primes" handout, and the extended Euclidean
algorithm for computing gcd and modular inverses from chapter 4 of Dana's notes.
Lecture 5
We showed that modular exponentation can be done efficiently (polynomial time), but we posed
the inverse of the exponentiation, namely the discrete logarithm problem, as a problem for which
no known efficient algorithm is known. We looked at two trivial attacks against discrete
logarithm: exhaustive search and guessing, and concluded that the first runs in exponential time
while the second one has a negligible probability of success. We saw Shank's discrete logarithm
running in time O(\sqrt(q)) and the index calculus methods which run in time about O(2^{p
^{1/3}), and we translated these two algorithms into bounds on the size of p and q needed to
achieve security for the discrete logarithm in practice. Finally, we stated the discrete logarihtm
assumption.
Reading: Most of this material is in Stinson, chapter 6, sections 6.1, 6.2 (esp 6.2.1, the other
attacks are an optional reading), and 6.6.
In the next lecture we'll abstract the assumption that discrete logarithm is hard into an
assumption that "exponentiation is a oneway function". The best lecture notes which introduce
oneway functions is
Yevgeni Dodis's lecture notes #2.pdf
. For now read up sections 17.
[If you are curious why we are skipping Stinson 6.35, here is a quick overview of that
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 Fall '04
 Jarecki
 Cryptography, Discrete logarithm, discrete logarithm assumption, discrete logarithm problem, Stinson, schnorr signature scheme

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