How is Math used in Cryptology/ Cryptography

              Summary:

My topic is about how math is used in cryptography. Math is used in many cryptosystems to protect data like making public keys. Prime factorizing is one of the main factors of cryptography.There are so many branches of math that we use in cryptography (matrices, primes, ellipses, modular arithmetic and many more).

Abstract math is very Important in cryptography as well.

From Analytical number theory, tools like factorization and computing logarithms in a finite field. 

Combinatorial problems, like knapsack and subset-sum has been used in cryptosystem. You can find a very nice connection between subset-sum and Lattice based cryptography

Coding theory and many combinatorial designs (BIBDs, Orthogonal arrays) have been used in the constructing universal hash function families and thereby randomness extractor and pseudorandom number generators. They are mostly used in the unconditional setting.

Algebraic geometry have been used in elliptic curve cryptography.

Group theory and in general Algebraic number theory has been used (for example, hidden subgroup problem) to construct cryptographic primitives secure against quantum attack.

Discrete Fourier Analysis has been used to prove and construct hard-core predicates, something of great use in the theoretical cryptography