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I would like to know which area of Mathematics would be most beneficial to cryptography. Surely Algebraic Number Theory and maybe to a lesser extend, Elliptic Curves, are closely linked to Cryptography; but are there more fields that would apply neatly to Cryptography?

What topics would you recommend to a Maths graduate who is looking for a career in Cryptography?

(I wanted to tag this as a "soft-question" as they do in math.stackexchange)

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    $\begingroup$ Discrete mathematics and group theory can be a start. You could also find this (somewhat different) question useful. A soft question (one that does not admit a definite answer) is a tad against the policy (see this section) $\endgroup$ – rath Jul 10 '13 at 22:42
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  1. Abstract Algebra
  2. Number theory
  3. Statistics
  4. Computer/Network security (so you can see the application)
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This seems to me a situation of the tail wagging the dog. Personally I have always found something I want to do such as zero knowledge proofs, faster primality testing, shorter signatures and then learned the relevant maths - bilinear pairings, elementary number theory and elliptic curves respectively. Cryptography is now a big subject and you can't hope to know enough maths to be comfortable in all areas - for instance the maths needed for differential/linear cryptanalysis is completely separate from the maths for asymmetric cryptography.

If you really are a maths graduate, what would a career in cryptography look like? Who would be your supervisor? Which university? Similar constraints are relevant in a commercial setting too.

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I would add Probabilities Theory and Linear Algebra as it turns to become the de facto source for intractable problems for the unreal yet fully homomorphic encryption schemes

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