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Covers fundamental algorithms for integer arithmetic, greatest common divisor calculation, modular arithmetic, and other number-theoretic computations. Algorithms are derived, implemented, and analyzed for primality testing and integer factorization. Applications to cryptography are explored including symmetric and public-key cryptosystems. A cryptosystem will be implemented and methods of attack investigated. To be able to implement and analyze algorithms for integer factorization and primality testing. To be able to use a system like Maple to explore concepts and theorems from number theory. To understand fundamental algorithms from symmetric-key and public-key cryptography.
MODULE (I):
NUMBER THEORY: Introduction-Divisibility-Greatest common divisor -Prime numbers - Fundamental theorem of arithmetic - Mersenne primes -Fermat numbers - Euclidean algorithm -
Fermat’s theorem - Euler totient function - Euler’s theorem. Congruences: Definition - Basic properties of congruences - Residue classes - Chinese remainder theorem.
MODULE (II):
ALGEBRAIC STRUCTURES: Groups - Cyclic groups, Cosets, Modulo groups -Primitive roots - Discrete logarithms. Rings – Sub rings, ideals and quotient rings, Integral domains. Fields - Finite fields – GF(Pn), GF(2n) - Classification -Structure of finite fields. Lattice, Lattice as Algebraic system, sub lattices, some special lattices.
MODULE (III):
PROBABILITY THEORY: Introduction – Concepts of Probability – Conditional Probability - Baye’s Theorem - Random Variables – discrete and continuous central Limit Theorem-Stochastic Process Markov Chain.
MODULE (IV):
CODING THEORY: Introduction - Basic concepts: codes, minimum distance, equivalence of codes, Linear codes - Linear codes - Generator matrices and parity check matrices - Syndrome decoding – Hamming codes - Hadamard Code –Goppa codes.
PSEUDORANDOM NUMBER GENERATION: Introduction and examples -
Indistinguishability of Probability Distributions - Next Bit Predictors - The Blum
Blum-Shub Generator – Security of the BBS Generator.
Fermat numbers, Fermat’s theorem
https://www.youtube.com/watch?v=5NmSvKtaDAs&t=88s
https://www.youtube.com/watch?v=TtCT5Yv3_kI
Euler totient function - Euler’s theorem
https://www.youtube.com/watch?v=gLCXaxL7DFs
https://www.youtube.com/watch?v=lgJFRPsbgCQ
Congruences: Definition - Basic properties of congruences
https://www.youtube.com/watch?v=MNj_e-t9tIs
https://www.youtube.com/watch?v=boSgvgmkLvw
Fields - Finite fields – GF(Pn), GF(2n)
https://www.youtube.com/watch?v=KCSZ4QhOw0I
https://www.youtube.com/watch?v=z9bTzjy4SCg
https://www.youtube.com/watch?v=FUdEpUoA4mc
Classification -Structure of finite fields
Lattice, Lattice as Algebraic system, sublattices, some special lattices
https://www.youtube.com/watch?v=3EAYX-wZH0g
https://www.youtube.com/watch?v=6otfMyGRd58
https://www.youtube.com/watch?v=l93lewDdorA
https://www.youtube.com/watch?v=7lYk-5otTV4
discrete and continuous central Limit Theorem
https://www.youtube.com/watch?v=k_zZ4IgI7_g
https://www.youtube.com/watch?v=b5xQmk9veZ4
Hamming codes - Hadamard Code
https://www.youtube.com/watch?v=1A_NcXxdoCc
https://www.youtube.com/watch?v=EBMrX9Z3cGg
Indistinguishability of Probability Distributions
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