An Intensive Introduction to Cryptography — Boaz Barak

Index

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Course recap

It might be worthwhile to recall what we learned in this course:

Some things we did not cover

  • We did not cover what is arguably the other “fundamental theorem of cryptography”, namely the equivalence of one-way functions and pseudorandom generators. A one-way function is en efficient map \(F:\{0,1\}^*\rightarrow\{0,1\}^*\) that is hard to invert on a random input. That is, for any afficient algorithm \(A\) if \(A\) is given \(y=F(x)\) for uniformly chosen \(x\leftarrow_R\{0,1\}^n\), then the probability that \(A\) outputs \(x'\) with \(F(x')=y\) is negligible. It can be shown that one-way functions are minimal in the sense that they are necessary for a great many cryptographic applications including pseudorandom generators and functions, encryption with key shorter than the message, hash functions, message authentication codes, and many more. (Most of these results are obtained via the work of Impagliazzo and Luby who showed that if one-way functions do not exist then there is a universal posterior sampler in the sense that for every probabilistic process \(F\) that maps \(x\) to \(y\), there is an efficient algorithm that given \(y\) can sample \(x'\) from a distribution close to the posterior distribution of \(x\) conditioned on \(F(x)=y\). This result is typically known as the equivalence of standard one-way functions and distributional one-way functions.) The fundamental result of Hastad, Impagliazzo, Levin and Luby is that one-way functions are also sufficient for much of private key cryptography since they imply the existence of pseudorandom generators.
  • Related to this, although we mentioned this briefly, we did not go in depth into “Impagliazzo’s Worlds” of algorithmica, heuristica, pessiland, minicrypt, cryptomania (and the new one of “obfustopia”). If this piques your curiosity, please read this 1995 survey.
  • We did not go in detail into the design of private key cryptosystems such as the AES. Though we discussed modes of operation of block ciphers we did not go into a full description of all modes that are used in practice. We also did not discuss cryptanalytic techniques such as linear and differential cryptanalysis. We also not discuss all technical issues that arise with length extention and padding of encryptions in practice. In particular we did not talk
  • While we talked about bitcoin, the TLS protocol, two factor authentication systems, and some aspects of pretty good privacy, we restricted ourselves to abstractions of these systems and did not attempt a full “end to end” analysis of a complete system. I do hope you have learned the tools that you’d be able to understand the full operation of such a system if you need to.
  • While we talked about Shor’s algorithm, the algorithm people actually use today to factor numbers is the number field sieve. It and its predecessor, the quadratic sieve, are well worth studying. The (freely available online) book of Shoup is an excellent source not just for these algorithms but general algorithmic group/number theory.
  • We talked about some attacks on practical systems, but there many other attacks that teach us important lessons, not just about these particular systems, but also about security and cryptography in general (as well some human tendencies to repeat certain types of mistakes).

What I hope you learned

I hope you got an appreciation for cryptography, and an understanding of how it can surprise you both in the amazing security properties it can deliver, as well in the subtle, but often devastating ways, that it can fail. Beyond cryptography, I hope you got out of this course the ability to think a little differently- to be paranoid enough to see the world from the point of view of an adversary, but also the lesson that sometimes if something sounds crazy but is not downright impossible it might just be feasible.
But if these philosophical ramblings don’t speak to you, as long as you know the difference between CPA and CCA and I won’t catch you reusing a one-time pad, you should be in good shape :)

I did not intend this course to teach you how to implement cryptographic algorithms, but I do hope that if you need to use cryptography at any point, you now have the skills to read up what’s needed, and be able to argue intelligently about the security of real-world systems. I also hope that you have now sufficient background to not be scared by the technical jargon and the abundance of adjectives in cryptography research papers, and be able to read up on what you need to follow any paper that is interesting to you.

Mostly, I just hope you enjoyed this last term and felt like this course was a good use of your time. I certainly did.