MATH 343, REVIEW SHEET FOR FIRST HOUR EXAM
ANDREW J. BLUMBERG
1. Things to know
(1) The Euclidean algorithm for computing the gcd.
(2) The fact that gcd(x, y) = m is equivalent to the existence of a solution to
the equation ax + by = m.
(3) Modular arithmetic, Z/m, and (Z/m)× .
(4) Fast exponentiation.
(5) Primitive roots.
(6) Fermat’s little theorem.
(7) The discrete logarithm problem.
(8) Diffie-Hellman key exchange.
(9) Elgamal public key encryption protocol.
(10) Shank’s algorithm (Babystep-Giantstep) for solving discrete log.
(11) The Chinese remainder theorem. (You must know the proof.)
(12) Pohlig-Hellman. (But only at a high level.)
(13) Euler’s formula.
(14) Computing roots mod N .
(15) RSA.
(16) Primality testing.
(17) Pollard’s p − 1 algorithm.
(18) Smooth numbers and the quadratic sieve.
(19) (You are not responsible for quadratic residues or quadratic reciprocity.)
(20) RSA signatures.
(21) Elgamal signatures.
(22) Collision algorithms for discrete log. (You must understand the probabilistic analysis, at least roughly.)
(23) Pollard’s ρ algorithm. (But only at a high level.)
(24) Elliptic curves (including the idea of the Fp points).
(25) The addition law on an elliptic curve.
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