Algorithms for integer arithmetic: 

Divisibility, GCD, modular arithmetic, modular exponentiation, Montgomery arithmetic, congruence, Chinese remainder theorem, orders and primitive roots, quadratic residues, integer and modular square roots, prime number theorem, continued fractions and rational approximations. 

Prime and extension fields, representation of extension fields, polynomial basis, primitive elements, normal basis, optimal normal basis, irreducible polynomials, Root-finding and factorization algorithm, Lenstra-Lenstra- Lovasz algorithm. 

Elliptic curves: The elliptic curve group, elliptic curves over finite fields, Schoof’s point counting algorithm. 

Primality testing algorithms: Fermat Basic Tests , Miller–Rabin Test , AKS Test.

Integer factoring algorithms: Trial division, Pollard rho method, p-1 method, CFRAC method, quadratic sieve method, elliptic curve method. 

Computing discrete logarithms over finite fields: Baby-step-giant-step method, Pollard rho method, Pohlig-Hellman method, index calculus methods, linear sieve method, Coppersmith’s algorithm. 

Quantum Computational Number Theory : Grover’s algorithm, Shor’s algorithm Applications in Algebraic coding theory and cryptography.

1. Yan, Song Y. Computational Number Theory and Modern Cryptography. John Wiley & Sons,2012. 
2. Meijer, Alko R. Algebra for Cryptologists. Springer, 2016 
3. Lidl, Rudolf, and Harald Niederreiter. Introduction to finite fields and their applications. Cambridge university press, 1994. 
4. Apostol, Tom M. Introduction to analytic number theory. Springer Science & Business Media,2013.