Abstract mathematics has played an important role in the development of cryptography.

1. From Analytical number theory, tools like factorization and computing logarithms in a finite field. Enough is said and known about these techniques!

2. 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. Try working it out yourself, if you can't do it, then look for a neat result in [Public-key cryptographic primitives provably as secure as subset sum ][1].

3. Game theory has been used in constructing protocols in rational setting, mainly for a weaker notion of fairness in Secure multiparty computation. Recall that fair computation is impossible because of Cleve's seminal work in STOC 1986.

4. 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.

5. Algebraic geometry have been used in elliptic curve cryptography. Enough has already been said by other people here.

6. Group theory and in general Algebraic number theory has been used (for example, hidden subgroup problem) to construct cryptographic primitives secure against quantum attack. Recall that quantum computers are not known to solve hidden subgroup problems. More so, Algebraic number theory gives rise to ideals and rings on which all the FHE are based and most of the lattice based cryptographic assumptions that have worst case to average case reduction are defined.

7. Analytical tools like exponential sums has been used in proving uniformity of certain distribution. Mostly, they use Weil's critereon and prove that the exponential sum corresponding to a particular distribution has a non-trivial bound and from discrete analog of Weil's critereon, it is uniformly distributed. This has been used to give an evidence that certain form of DH problems have uniform distribution over a group of prime order.

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

9. Additive combinatorics has been used in few cryptosystems indirectly (they are used in complexity theory and from there find application in cryptography), especially the famous BKT03 result. You can find more about these results on Jean Bourgain's and Igor Shparlinski's webpage.

At the moment, I can't remember any more. [1]: http://www.di.ens.fr/~lyubash/papers/subsetsumcrypto.pdf

Abstract mathematics has played an important role in the development of cryptography.

1. From Analytical number theory, tools like factorization and computing logarithms in a finite field. Enough is said and known about these techniques!

2. 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. Try working it out yourself, if you can't do it, then look for a neat result in Public-key cryptographic primitives provably as secure as subset sum .

3. Game theory has been used in constructing protocols in rational setting, mainly for a weaker notion of fairness in Secure multiparty computation. Recall that fair computation is impossible because of Cleve's seminal work in STOC 1986.

4. 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.

5. Algebraic geometry have been used in elliptic curve cryptography. Enough has already been said by other people here.

6. Group theory and in general Algebraic number theory has been used (for example, hidden subgroup problem) to construct cryptographic primitives secure against quantum attack. Recall that quantum computers are not known to solve hidden subgroup problems. More so, Algebraic number theory gives rise to ideals and rings on which all the FHE are based and most of the lattice based cryptographic assumptions that have worst case to average case reduction are defined.

7. Analytical tools like exponential sums has been used in proving uniformity of certain distribution. Mostly, they use Weil's critereon and prove that the exponential sum corresponding to a particular distribution has a non-trivial bound and from discrete analog of Weil's critereon, it is uniformly distributed. This has been used to give an evidence that certain form of DH problems have uniform distribution over a group of prime order.

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

9. Additive combinatorics has been used in few cryptosystems indirectly (they are used in complexity theory and from there find application in cryptography), especially the famous BKT03 result. You can find more about these results on Jean Bourgain's and Igor Shparlinski's webpage.

At the moment, I can't remember any more.