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1 Multiple Precision Arithmetic
Some Lectures on Number Theory, Elliptic Curves and Cryptology
Kapil Hari Paranjape
1 Multiple Precision Arithmetic
1.1 Addition and Subtraction Algorithm
1.2 Multiplication Algorithm
1.3 Long Division Algorithm
1.4 Using the shift operations
1.5 Faster Algorithms
1.6 GMP and other implementations
2 Greatest common divisor
2.1 Euclid's algorithm
2.2 Binary GCD algorithm
2.3 Lehmer's Algorithm
2.4 Extended GCD
3 Arithmetic modulo
N
3.1 Faster arithmetic
3.2 Encrypting and Decrypting RSA messages
4 Primes and Composites
4.1 Eratosthenes' sieve
4.2 Trial division
4.3 Combinations of the methods
4.4 Compositeness Tests
4.5 Hensel's lemma
5 Factorisation and Certificates
5.1 Pollard's
5.2 Group theoretic method
5.3 Primality Certificates
6 Algebraic Number Fields
6.1 Algebraic Numbers
6.2 Algebraic Number Fields as Matrix Algebras
6.3 Orders and Maximal orders
6.4 Lattices and ideals
6.5 Groups of invertible fractional ideals
6.6 Minkowski's Geometry of Numbers
6.7 Prime ideals
7 Quadratic fields
7.1 Prime ideals
7.2 Naive computation of the class group
7.3 Binary Quadratic Forms
8 Algebraic Schemes for Cryptosystems
8.1 Finite rings
8.2 Functors of points
8.3 Morphisms of schemes
8.4 Relativisation and categorical constructions
8.5 The category of vector space schemes
8.6 Vector Bundles and regular schemes
8.7 Action of correspondences
8.8 Cryptosystems
9 Hyperelliptic Cryptosystems
9.1 Hyperelliptic curves
9.2 Closed points
9.3 Divisors
9.4 Computing with the divisor class group
9.5 Frobenius Endomorphism
10 Symmetric Cryptosystems
10.1 Types of operations
10.2 Modes of enciphering
10.3 Cryptanalytic methods
11 Algorithms for groups
11.1 Pollard's
11.2 Shanks' Baby step-Giant step
11.3 Pohlig-Helman factor method
11.4 Other problems
11.5 Applicability and efficiency
11.6 Index calculus
A. A missing Lemma
A..1 Counterexample
B. Comparison with ``classical'' definition
Bibliography
Kapil Hari Paranjape 2002-10-20
