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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University of Maryland, College Park

345 ratings

Course 3 of 5 in the Specialization Cybersecurity

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

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

Maryland Cybersecurity Center

[SOUND].

Â In this lecture, our final lecture on number theory,

Â I want to just to talk very briefly about appropriate choice of parameters.

Â Now we've discussed two classes of cryptographic assumptions.

Â The first, was based on the underlying assumption that factoring was hard, and

Â the problem that we spoke about here was the RSA assumption.

Â The other set that we considered, was in the setting of cyclic groups, and

Â we looked there at the discrete logarithm, the confrontational diffie hunnam and

Â the dicisional diffie hunnam assumptions.

Â And we talked very briefly about two classes of

Â groups in which those assumptions could be studied.

Â Now all of these problems are indeed believed to be hard in the sense that we

Â don't believe that they have any polynomial time algorithms for

Â solving these problems.

Â But this is not enough for using cryptography in practice.

Â In practice, it's not sufficient to know that there's no poly-time algorithm for

Â solving some problem.

Â In practice, we need to set some concrete value for the parameter.

Â For example, for the length of the modulus.

Â And to properly set that, we need to have a more fine grain understanding of how

Â hard these problems are.

Â And it's useful here to review what we saw for

Â the case of symmetric-key cryptography.

Â So for symmetric-key cryptography,

Â we said that we could hope for a block cipher with an n-bit key

Â being secure against an attacker running in time roughly two to the n.

Â And so if we want to achieve security against an attacker running for

Â a particular amount of time, we can easily set the length of our key appropriately,

Â to ensure security against such an attacker.

Â And, similarly, for the case of hash functions, we said that if

Â we have a hash function with an n-bit output, then we could hope for

Â security against an attacker running in time two to the n over two.

Â So here the security is worse than it is for the case of block ciphers.

Â But nevertheless, because we can characterize the security very

Â exactly we know how to set the length of

Â the output in order to achieve some desired level of security.

Â And to do analogous calculations for

Â the public key setting, we need there too to have a better understanding of

Â the exact difficulty of factoring in computing discrete logarithms.

Â So, for example, is it the case that factoring a modulus of

Â length n requires two to the n time or maybe two to the n over two time?

Â What is it?

Â And similarly, does computing discreet logarithms in a group with on the order of

Â two to the n elements?

Â Take two to the n time, two to the n over two time, something else?

Â Or does it depend on the group in which we're working?

Â These are the questions we're going to look at here.

Â Now I do want to give a little bit of a disclaimer.

Â And just say that the goal of this lecture is not to actually give you

Â parameters that you can then use in practice,

Â although you could do that from the slide I give you at the end.

Â But the goal instead is to give you an idea as to how these

Â parameters can be calculated.

Â And if you're serious about this, then there are many other

Â important considerations that you need to consider before setting a value for

Â the parameters you're going to use in your scheme.

Â And this is really just meant as an introduction.

Â And just to give you a broad brush, brush strokes.

Â An idea of how of the parameters are calculated.

Â Rather than to prescribe any particular setting of the parameters.

Â So with that in mind lets forge ahead.

Â In terms of factoring it turns out that there do

Â exist factoring algorithms that run in much less than two to the n time.

Â If I let n say denote the length of modulus that we're factoring.

Â In fact, there are algorithms that run much better than exponential time.

Â And the best known algorithm asymptotically, is the the general number

Â field sieve with a heuristic running time of approximately two to the, length

Â of n to the one third times some lower outer parameters, which are very important

Â in practice, but not relevant to the high level discussion that I want to have here.

Â So the point is that we might expect or we might hope for

Â exponential security which would be two to the n, where I

Â mean here again the length of the modulus, or maybe two the constant times n.

Â But instead we get something lower we get actually something a running time which is

Â sub exponential in the length of the modulus.

Â And this means in turn that the problem is much easier than say attacking

Â a symmetric key algorithm whose key length is equal to the modulith length, right?

Â So if you have a, a 128 bit modulith, that's trivial to factor.

Â Whereas the 128 bit key is sufficient enough to give good security in practice.

Â Now as far as the discrete logarithm problem goes

Â it's kind of interesting here because we have two classes of algorithms and

Â they both need to be taken into account.

Â The first class of algorithms are those that work for

Â arbitrary groups they're sometimes called generic group algorithms.

Â They work regardless of the group,

Â they don't care the details of the group you're working in.

Â All they actually care about, is the order of the group.

Â The other class of algorithms,

Â are those that target the discrete logarithm problem in specific groups.

Â As far as generic algorithms go, the best generic algorithms we have run in

Â time two to the n over two in a group who's order is about two to the n.

Â Or to say it differently if we have a group of several order q and the length of

Â q, the bit length of q is n bits then we get about two to the n over two security.

Â And it's kind of interesting because in this setting we know that

Â these algorithms are in fact optimal in some sense.

Â However, we have this other class of algorithms to consider which looks at

Â the particular group in which we're performing the discreet log

Â calculation and tries to tailor the algorithm for that group.

Â The best algorithm, that's known for the case of discreet logarithms in z,

Â p star or in subgroups of z, p star, is the so called number field sieve.

Â This algorithm has running time which is heuristically about two to the bit

Â length of p, to the one third.

Â So again, as in the case of factoring, this,

Â the running time of this algorithm is sub exponential in the length of p.

Â Which again means that the discrete logarithm problem in z p

Â star is easier than the corresponding, then

Â attacking a corresponding symmetric key algorithm with an equivalent key length.

Â Now what's particularly interesting in what makes elliptic curve cryptography

Â so exciting.

Â Is that if you choose your elliptic curve groups appropriately,

Â then there's no algorithms known that are better than the generic algorithms.

Â So this means that in some sense, elliptic curves are currently,

Â elliptic curve groups are currently optimal with respect to what we

Â could hope for as far as security is concerned.

Â Now, there is this caveat that I did put in parenthesis appropriately chosen.

Â Some care does need to be taken you can't just willy nilly pick any elliptical curve

Â group you want, there's some that's specifically need to be avoided.

Â However because we didn't go into any detail about ecliptic curve I'm not

Â going to go into detail about that either.

Â But you can find suitable references online.

Â Now this, in turn has a big impact on the efficiency of cryptographic algorithms.

Â So just as an example, NIST has recommended different security levels, or

Â rather, different parameters corresponding to different security levels.

Â And I've listed, here, the recommendations of NIST.

Â If you want to achieve 112 bit security, that is,

Â roughly speaking, security against attacks running in time two to the 1/12th, or

Â maybe a better way to put it would be security equivalent to

Â what you would get with a 112 bit, for, for a 112 bit symmetric key cypher.

Â For factoring the recommendation is to use a 2048 bit modulus.

Â So of course this is about a factor of 20 larger than the 112-bit key that

Â you would need to obtain equivalent security for a blog cypher.

Â And this reflects the fact that we do know sub exponential time

Â algorithm for factoring.

Â For the discreet logarithm problem,

Â if we look at the discreet log problem in order q sub group of z p star.

Â Then remember we have to be concerned both about the generic algorithms,

Â as well as for the specific algorithm tailored to z p star.

Â So to ensure 112 bit security against a generic algorithm, we need to make sure

Â that the order of the group is on the order, is roughly two to the 224.

Â Right? Because a group of order,

Â roughly two to the n, gives you about two to the n over two security.

Â So that means that the bit length of q,

Â if q is the order of the group, should be, should be 224.

Â But to protect against the Taylor algorithm,

Â that specifically targets the discrete log problem in z p star.

Â We need to set p much larger.

Â Because we have sub exponential time algorithms working in z p star.

Â And in particular, NIST recommends taking the bit length of p equal to 2048.

Â So, it's a little bit difficult to think about what this means exactly.

Â But wh, what it means is that you take your prime p of length 2048.

Â But then you're working in a much, much smaller subgroup of that larger group,

Â and this balances the security that you get against generic algorithms on

Â the one hand and against tailored algorithms on the other hand.

Â If we come to the case of elliptic curve groups or

Â elliptic curve subgroups, well as we said a moment ago for elliptic curve groups.

Â No algorithms known no, no algorithms are known which are better than what you get

Â by using generic algorithms.

Â And this means that it's sufficient here to take the group order equal to something

Â that would give you the requisite security against a generic algorithm.

Â Meaning that you could take the order to be something approximately two to

Â the 224 or equivalently you set your group order q, such that q has bit length 224.

Â And again this is why elliptic curves are so exciting because you can use

Â smaller parameters, I really haven't defined what elliptic curve groups are so

Â it's a little bit hard for me to say what those parameters are, but

Â you can use smaller smaller integer calculations and get equivalent security

Â to what you would get in using a much larger p in the case of z p star.

Â So, roughly speaking, elliptic-curve,

Â using elliptic curve groups will give you the same security at better efficiency.

Â In any case, one takeaway point here is that these parameters are much

Â larger than what you have for symmetric-key algorithms.

Â Perhaps with the exception of hash functions.

Â And this explains in part why public ecryto in a particular factoring based

Â crypto or crypto based on discreet log in some groups of z p star is so

Â much less, so much less efficient than symmetric ecryto.

Â This concludes our discussion of number theory.

Â And in the following weeks we're going to see some application of

Â all the number theory we learned to public-key cryptography.

Â And I look forward to seeing you all there.

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