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Essential Concepts
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Module 1
Number-Theoretic Foundations
1
Extended Euclidean Algorithm and Modular Inverses
Prove the correctness of the Extended Euclidean Algorithm and use it to compute modular inverses.
2
Euler's and Fermat's Little Theorems
State and prove Euler's theorem and Fermat's Little Theorem.
3
Euler's Totient Function and Prime Factorization
Compute Euler's totient function for arbitrary integers using prime factorization.
4
Solving Systems of Linear Congruences with the Chinese Remainder Theorem
Solve systems of linear congruences using the Chinese Remainder Theorem.
Analyzing Square-and-Multiply for Modular Exponentiation
Analyze the computational complexity of modular exponentiation using the square-and-multiply method.
6
Quadratic Residuosity with Legendre and Jacobi Symbols
Determine quadratic residuosity using the Legendre and Jacobi symbols.
7
Quadratic Reciprocity: Statement and Application
State and apply the Law of Quadratic Reciprocity.
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