This course will introduce you to the foundations of modern cryptography, with an eye toward practical applications.

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From the course by University of Maryland, College Park

Cryptography

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This course will introduce you to the foundations of modern cryptography, with an eye toward practical applications.

From the lesson

Week 7

Digital Signatures

- Jonathan KatzProfessor, University of Maryland, and Director, Maryland Cybersecurity Center

Maryland Cybersecurity Center

[MUSIC]

Â Well this brings us to the end of the course.

Â I really hope you enjoyed the course, as much as I enjoyed teaching it, and

Â as much as I enjoyed interacting with, many of you on the discussion boards.

Â In fact, I hope you enjoyed the course so

Â much, that you're motivated to learn more, about the field of cryptography.

Â And what I wanted to just briefly talk about here is, where to go next?

Â What more is there to learn, and where can you turn to learn more about it?

Â So, first of all,

Â I want to point out that, even though I emphasize proof-able security.

Â Back when we talked about the principles of modern cryptography.

Â And even though we did see examples of proofs, and

Â I did talk about formal theorem statements expressing what kind of security we could

Â prove, I really didn't give very many proofs in the second half of the course.

Â And the reason simply is that the proofs become, a bit more difficult.

Â They become more time consuming.

Â They require a bit more background.

Â And also it's a little bit difficult to present them, on a PowerPoint slide and

Â I much prefer doing them on a,

Â on a whiteboard and being more active and interactive.

Â But if you're interested in the field,

Â it's really important to understand, how these proofs of security work.

Â And understand how to evaluate them, as well.

Â One thing in particular that I gave very short script to in this course,

Â is the random oracle model.

Â Even though I did mention a handful of times, this assumption of treating

Â a cryptographic hash function as if it's a random function.

Â I didn't go into any detail about it, and

Â we didn't really see any proof, based on that assumption.

Â This is a technique that's become more and

Â more widely used, in the analysis of cryptographic schemes today.

Â And if you're interested in the field, again,

Â it is important to go and see some examples of these proofs and

Â really understand what it is the random-oracle model entails.

Â As well as what it's limitations are.

Â Another topic that we really didn't have time to cover in this course was

Â design principals.

Â For modern stream ciphers, block ciphers, and hash functions.

Â We defined what a stream cipher is.

Â A stream cipher is supposed to act as a pseudo random generator.

Â We defined, what a block cipher should be.

Â A block cipher is supposed to be behave like a random function.

Â We talked about the notion of collision resistant hash functions.

Â I gave examples by name, of modern day cryptographic primitive,

Â that are assumed to realize these different functionalities.

Â But, we didn't go into any detail at all,

Â about how these modern day primitives are actually constructed.

Â And that's another very important area to learn about,

Â a very interesting one as well.

Â To really understand and get a sense,

Â of how these things can be constructed in practice and why we have any

Â belief that these things really do achieve the properties that we claim they do.

Â Another very interesting topic, is to look at developing cryptographic primitives and

Â cryptographic schemes.

Â Based on minimal assumptions.

Â So, in our discussion of for example private key encryption,

Â we took as our basic building block, pseudo random functions i.e block ciphers.

Â And we showed how to construct encryption schemes satisfying strong definitions of

Â security, based on any broad cipher.

Â The previous bullet talks about how block ciphers are actually

Â constructed and practiced.

Â But the practical constructions we have, are ultimately heuristic,

Â in the sense that we can't really prove anything about them.

Â The best we can do, is to analyze them.

Â And use the fact that there has not been any successful attack on them after

Â years of analysis, to therefore give us the belief, that they are indeed secure.

Â But, what's very interesting is that you can actually take a more

Â foundational approach and start with a very, very weak assumption,

Â namely the assumption that what's called one way functions exist.

Â A one way function, is roughly speaking, a function that is easy to compute, but hard

Â invert, such that i.e that it's difficult to compute the inverse, of that function.

Â That's a very basic, minimal assumption, but it turns out that that

Â assumption suffices, for constructing all of private key cryptography.

Â That is, you can build block ciphers, based on the assumption that one way

Â functions exist and then, as we've seen in this course, you can build, private key

Â encryption and message authentication codes, based on block ciphers.

Â I specifically did not cover this topic,

Â because it's really only a theoretical interest.

Â It doesn't have any practical significance today.

Â But for those of you with a more mathematical orientation or

Â a more theoretical orientation, I would advise you to

Â look into this topic because it really contains some very interesting results.

Â We also did not talk very much, about modern day algorithms for factoring and

Â computing discrete logarithms.

Â We introduced the problems.

Â We said that they're considered to be hard.

Â Namely, that there's no polynomial time algorithm, for

Â solving these problems, but that doesn't mean that the best algorithms we have for

Â solving them, are the trivial, brute force, exponential time ones.

Â And in fact it's very important to understand,

Â what the best algorithms are for factoring and computing discrete logarithms,

Â when determining the key length, of public key schemes.

Â For those of you who again are more mathematically oriented,

Â there's also a lot of very interesting mathematics, and

Â group theory, involved in designing and analyzing these algorithms.

Â Finally, we only gave very relatively little attention to public key

Â encryption and signature schemes.

Â And there's much more to learn about in that area as well.

Â I'm very happy to announce, that the second edition of my textbook,

Â Introduction to MODERN CRYPTOGRAPHY, was published and actually this happened,

Â I wasn't sure exactly when this was going to happen.

Â It happened sometime after I began recording week one but

Â before I'm ending the course now.

Â And so all of these topics,

Â all the topics listed on this slide are in fact covered in that book, and

Â I think it's a great place to turn to next if you're interested in learning more,

Â about cryptography beyond what we had time for in this course.

Â Now beyond that, you can start looking at more advanced material.

Â And here you begin getting into the current Cryptographic research and or

Â things that you might learn if you go to graduate school, to study Cryptography.

Â I'll just mention these very briefly.

Â In this course, we've talked primarily about a two party setting, where we have

Â say a sender and receiver communicating in the presence of an attacker.

Â Who's trying to eavesdrop or otherwise interfere with their communication.

Â But cryptography can also study the setting, where you

Â have a network of many parties all interacting and running some protocol and

Â where it's not even clear, which parties trust other parties.

Â Or which parties can be trusted or which might be compromised.

Â And this leads to a general area of the cryptographic design of protocols for

Â various tasks, with security even in the face of compromise of

Â some number of the participants in the protocol.

Â I mentioned in the last slide, design principles for

Â stream ciphers, block ciphers, and hash functions.

Â But of course, the counterpart to that is modern-day cryptanalysis, of

Â the constructions stream ciphers and block ciphers and hash functions that we have.

Â This is again a very active area of research today.

Â And one where you can really get very deep,

Â deeply involved in that as you try to understand these practical constructions.

Â There's also a lot of interesting work surrounding.

Â Number-theoretic algorithms.

Â And here I'm talking about things beyond necessarily algorithms for factoring and

Â computing discrete logarithms.

Â But algorithms for other aspects as relevant to cryptography as well.

Â Another very interesting area, of modern day cryptographic research,

Â is the investigation of what is sometimes called post-quantum cryptography.

Â So in this class, when it came to public key cryptography,

Â we looked exclusively at systems that were based on, really only two assumptions.

Â The assumption that factoring was hard and the assumption that

Â computing discrete logarithms in certain classes of groups, was also hard.

Â It turns out, that if quantum computers are ever built.

Â Both of these problems could then be solved in polynomial time on

Â a quantum computer.

Â And so for that reason, people who are thinking ten, 20,

Â 30 years ahead are already worried about, what will replace modern day

Â public key cryptosystems, in case a quantum computer is ever built.

Â And so people are investigating various other assumptions.

Â On which to base public key cryptography.

Â And there's much, much more beyond that.

Â And what I would encourage you to do if you're really interested,

Â is to take a look at the web page for

Â the International Association for Cryptologic Research, IACR.org.

Â And take a look at both their flag ship conferences, Crypto, Eurocrypt,

Â and Asiacrypt.

Â Held annually, as well as the journal,

Â journal of cryptology that they are in charge of publishing.

Â And there you can get a sense of what kind of problems,

Â researchers in cryptography are working on nowadays and

Â see what interests you and develop your tastes that way.

Â With that, I only have left to wish you luck on the final exam.

Â And to encourage you to check out the capstone course as part of

Â the cyber security specialization being offered by the University of Maryland.

Â Again, I really do hope you've enjoyed the class, and

Â I look forward to meeting some of you in the future.

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