Number Theory

1. Computational Number Theory
   1. Primality Testing
      1. Deterministic Tests
         1. AKS primality test
            1. Polynomial-time algorithm
            2. Importance in theoretical computer science
         2. Miller test
            1. Use of complex numbers and cyclotomic fields
            2. Theoretical framework and proofs
      2. Probabilistic Tests
         1. Miller-Rabin test
            1. Strong pseudoprimes
            2. Complexity and reliability analysis
         2. Solovay-Strassen test
            1. Use of Euler's criterion
            2. Statistical significance of test results
         3. Fermat primality test
            1. Fermat's Little Theorem foundation
            2. Carmichael numbers as exceptions
   2. Factorization Algorithms
      1. Trial Division
         1. Basic algorithm efficiency
         2. Impact of small primes in trial division
      2. Pollard's Rho Algorithm
         1. Floyd's cycle-finding algorithm
         2. Use of pseudo-random polynomial sequences
         3. Practical applications and limitations
      3. Quadratic Sieve
         1. Smooth numbers and their role
         2. Sieving process
         3. Comparison to more complex algorithms
      4. General Number Field Sieve
         1. Most efficient known classical algorithm for large integers
         2. Concept of number fields and algebraic integers
         3. Importance for cryptographic security
   3. Cryptography Applications
      1. RSA Encryption
         1. RSA algorithm steps
         2. Role of large prime numbers in security
         3. Key generation, encryption, and decryption processes
      2. Diffie-Hellman Key Exchange
         1. Concept of shared secrets
         2. Discrete logarithm problem in finite group
         3. Application in secure communications
      3. Elliptic Curve Cryptography (ECC)
         1. Use of elliptic curves over finite fields
         2. Benefits over RSA in terms of key size and efficiency
         3. Implementation challenges and security concerns
   4. Integer Arithmetic in Computation
      1. Fast algorithms for integer multiplication
         1. Karatsuba algorithm
         2. Schönhage-Strassen algorithm
         3. Recent advances in multiplication techniques
      2. Modular arithmetic in algorithms
         1. Modular exponentiation
         2. Applications in digital signatures and cryptographic protocols
   5. Computational Techniques
      1. Use of symbolic computation
         1. Simplifying complex number theoretic expressions
         2. Software tools aiding in computational analysis
      2. Parallel computing in number theory
         1. Techniques to distribute computational tasks
         2. Impact on factorization and primality testing
   6. Applications in Algorithms and Complexity Theory
      1. Importance in algorithmic analysis
      2. Impact on P vs NP problem
      3. Implications for theoretical computer science