The Joy of Cryptography
joyofcryptography.com
joyofcryptography.com
Like a lot of people (I imagine) I made it through a CS bachelors program not really ‘getting’ the discrete math combinatorics part. Crypto is an area where those concepts really really matter. It’s great to see this resource available!
(Though, what do you mean by calculus? I assume you mean integration and differentiation and real numbers and stuff? Or something else?)
I took extra calculus classes from the math department. Those are way harder than what you need for introductory physics.
(I recall one day in 3rd year calculus course realizing "wait a second... I don't need to be here!". I just kept signing up for calculus every year since 11th grade and suddenly realized I don't need, don't want, and don't like the class nor the prof I was in, and that was one pain less I could instill upon myself :)
At least to my knowledge there is no 'theory of calculus' which allows one to solve generic integrals or differentials, calc is as much an unrelated art to comp sci as learning to play chess will help with accounting.
It's a shame that match curriculum for non-math-majors is an all-roads-lead-to-calculus affair. I think we scare a lot of potentially talented people away from math with that approach.
Discrete math is very important to computer science. But in the rest of the world, calculus (and differential equations) dominate.
Anything to do with numbers besides the reals. Anything to to with finite sets. Anything to do with groups, rings, polynomials, trees, graphs, ordinals, lattices, compass-straightedge-constructions, polygons, knots, categories, sheaves, topological spaces, vector spaces, and anything to do with oddities like map coloring or plane tiling, and a lot else too.
It's a course in "everything else" for students that are being pigeonholed into a mathematical specialization by the fact that it's been fashionable to use real numbers to describe the world since Newton.
I'd have to ask around but I know a few algebraic topologists and I'd bet that if pressed to describe their work as more-continuous or more-discrete they'd first tell you that this is a silly way to classify subfields in math and then they'd says it's probably more on the discrete side since it's all about categorizing topological spaces based on whether they have certain discrete properties:
- separable - countable - metric - compact
Sure the spaces themselves might be continuous, but topology is for telling those spaces apart based on where and how continuity fails. It's a discretization of things formerly suspected to be the the same.
There's something about real analysis that makes me uncomfortable. I've been struggling for years to put my finger on it. Whatever it is, topology doesn't have it.
I had mistakenly decided that it was an overappreciation of continuity, but I think it must be something else.
Something about the homework in Real Analysis left me feeling angry. Not because it was difficult or presented poorly, but because it was somehow... untrustworthy? As if my betters had decided which ideas were the good ones and the only thing left for me to do was optimize along the one dimension that they had assigned me. I realize that this is nonsense, but I can't seem to shake it.
I'll start with sets and you can start with the real numbers, and we'll see whose list is longer.
To a certain degree the whole game is nonsense, there's a countable infinity of axioms and a countable infinity of theorems that follow therefrom, so we're never going to get anywhere rigorous with this, but just like it's not unreasonable to say that there are more multiples of 2 than there are of 2000, I think it's fair to say that continuity games on the reals represents a relatively small share when compared to mathematics in general.
book.pdf
Really? I'm saving the file for a later review and this name will guarantee I either completely lose the file or will spend more time than necessary to locate it. The other option is to rename the file on saving, which is some work which needs to be performed by each user, instead of being done just once on the source. I think this is just impolite.
Here's a proper name you can copy-paste:
Mike_Rosulek_-_The_Joy_Of_Cryptography_(draft_2021-01-03).pdf
PS: The short-link-for-verbal-sharing problem is solved via a symbolic link.
On the second thought, one part of my job is preparing products for the final consumption, so this attention to minutiae is probably a professional deformation.
I'd like to see material around modern stuff (Keccak and ECC) with only a cursory discussion of pitfalls of the past standards in contrast with the current stuff. And more time on asymmetric, discrete log and fiat-shamir protocols that are so much fun and require really a 7th grade algebra to grasp.
I've also made a couple of videos on ECC basics with that approach: https://www.youtube.com/channel/UCk0r9S76wngNPJb_GlhJYdw
EDIT: what i mean, is focus first on how to use primitives and what they promise, rather than how they actually work. Once you feel like you can play with these legos, it's more fun to dig deeper and see why ECDLP is hard and how ciphers are made. But please don't start from within - it's an annoying boring mess that distracts from the true beauty: the cool things you can build with these tools.
These sorts of "provable security" textbooks (like the one in the article) try and do that. They emphasise less the exact inner workings of algorithms, and more on what they supposedly achieve. Various kinds of security proofs are possible. This school of thought in cryptography does also allow you to argue fairly convincingly that some concrete ciphers are secure. So it can talk about both the inner workings of ciphers as well as what security properties they achieve.
The "provable security" approach seems to be in line with your suggestion. In fact, it was the framework under which Zero Knowledge Proofs were developed. Not every cryptographer knows about that though, so I'm not sure if you were aware...
This makes it language agnostic. Code can be very opinionated in my opinion.
In some sense, programming languages are a mathematical notation, too.
Thank you for posting this book, I'll give it a go! Crypto is one of those things I have on my shameful "how-come-you-don't-know-it-yet" to-learn list :).
[1] https://www.youtube.com/channel/UC1usFRN4LCMcfIV7UjHNuQg
Compared to this book you seem right, the glimps I had was easily digestable and still concise.
For inquiries related to this book, use x@joyofcryptography.com, where x is the author's first name.
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Nearly everyone with a high school education can “crack” that puzzle and figure out his email and yet most bots will fail.
Mail to the literal 'x@joyofcryptography.com' address would just be dropped. There's no expectation that mail to a nonexistent address would be placed in the postmaster mailbox.
i always thought the hotness would be to encrypt the address in source text and render/decrypt it client side in javascript, but i suppose scrapers these days use full blown headless browsers complete with javascript runtimes.
That said, depending on the construction you use, nonrandom IVs can be a problem (they can leak that there's a relationship between different plaintexts if they line up with a difference in the first block), so you're better off at least making it a random function of the key and said blob/etc ID.
Someone please correct me if I'm wrong.
[1] https://github.com/signalapp/libsignal-protocol-java/blob/ma...
[2] https://signal.org/docs/specifications/doubleratchet/Set0_1....
Or more specifically how could I create a book with formatting that looks as nice as this.
So should I see primes as a weakness in cryptography?
First of all, primes are only used to arrive at a session key, and once you have a session key you're in the land of symmetric algorithms, which provide security by permutations rather than vectoring into prime spaces. The content of a web page does not matter at all in terms of the security being provided. A 404 is just as secure as a valid home page, in terms of cryptography. (Not in terms of application security, but that's a whole different thing.)
Second, the supply of prime numbers is countable but also infinite, and the relation between a number space and the number of primes within it is well established within the workable sizes. We have upper and lower bounds on the number of primes within certain ranges. This is partly why we end up with certain key sizes as being secure and other key sizes as being insufficient. Secure key sizes (in asymmetric algorithms) partly are secure because there are so many primes that can be fodder for key generation.
A fun way to improve one's intuition for the huge space of primes in the regions that are used in cryptography is to run something like
openssl prime -generate -bits 1024
(This would give you one of two secret parameters for a 2048-bit RSA modulus.)
Here is an example output:
171512683213144641422165911053946068613736141364433791764418217818329805042681574174826781297743340609766353737741548521280185783254625590756752560896455821248403501924867464886010664489035275551178636464876794754383911750620888923620343965444339734253757757361106691647372710935609902417522529107877592063581
Not very easy to guess just because the space in question is limited to "1024-bit primes"!
And indeed, that's part of why we use primes that are this big when we still use RSA. They're big enough that the number of possibilities makes them not easy to guess (even given the extra hint of "the secret number you're looking for is one of the factors of this 2048-bit semiprime").
Exponentials are always hard for human intuition. We start with some kind of pattern and it feels like there just aren't that many numbers that would satisfy it. But when you get out to large numbers (even numbers as large as the one above), there just are that many numbers that would satisfy the pattern!
The point about the webserver was using methods to work out what the underlying webserver and version was and potentially the OS, after all not much point trying to decrypt an algo that is not used on the webserver, but metadata is given out by many webservers to suggest what type of webserver it is.
And then there is education, we see people believe many things because this is what's taught to them, we see this cognitive dissonance with religious people but also in "educated" people like some GP's or academics because its their life's work and I'm sure cryptography has consumed many hours of people thinking things through, without much time spent on other areas of weakness like the limitations of a cpu and memory registers.
After all how many crypto experts are fully conversant with the inner workings of a CPU design? If they were conversant with CPU designs would Spectre and Meltdown have been found years earlier?
Admittedly side channel attacks but I did say in my original post "Whilst theory is different in practice due to machine limitations" and this is the thing is the current implementation of cryptography on devices suitable when considering the design of CPU's and devices?
http://sunnyday.mit.edu/accidents/Ariane5accidentreport.html and yet again another OpenSSL bug , another flaw in the open source fallacy of many eyes to spot bugs... https://nakedsecurity.sophos.com/2021/03/28/serious-security...
plus when considering in maths there are ways to rearrange formulas including new discoveries like this one (a new way to do quadratic equations) https://news.ycombinator.com/item?id=29342161 https://www.sciencealert.com/math-genius-finally-discovers-e...
I wonder if discovering a new way to do things could be done to cryptographic algorithms.
Look into post quantum cryptography if you're interested to know about one of the big trains heading our way.
Build time into an algo and then you need a quantum computer which can also roll back or forward time.
You see the problem is, how do you build time into an algorithm like a particle decay? And if you could, could you then make it like a time delayed bank vault which only opens for certain windows during the course of the particle decay?
I also wonder if a place, some co-ordinates could also be built into an algo if that is even possible. There are medieval examples, namely trusted couriers but then you are not looking so much at an algo but more a self contained device to defeat quantum computing.
Anyway for now, you've got the NSA data centre at Bluffdale Utah which can sort of replicate quantum computing by running multiple instances with each using one pwd from all the possible pwd combination to look for the magic bytes in a file https://en.wikipedia.org/wiki/List_of_file_signatures in order to decrypt files. A variation of that Could be used for replay attacks on network communication.
At least thats how I would do it quickly and easily.
(Not because of any weakness, but because there are better, more efficient algorithms instead.)
Elliptic curves are faster and use smaller keys, but they're also a more fragmented ecosystem; plus, in particular for the older NIST curves, there is the unshakable fear that they're backdoored by the NSA (since they use magic unexplainable numbers, which RSA does not).
Thankfully ed25519 gave us an alternative free of magic numbers, and it's seeing a lot of adoption (in particular in open source software), but it's nowhere near taking over RSA.
This is all 100% pedantry. But the belief that RSA is risky because "prime numbers" is false, and worth pushing back on. There are reasons not to use RSA, but they're not as simple as "we don't trust prime field cryptography".
More info: https://safecurves.cr.yp.to/rigid.html
Ed25519 is a replacement for RSA in the x.509 WebPKI, which is what I was trying to refer to when I said TLS. Classic TLS (as in non-PFS, the one that also used RSA to encrypt the session secret) is dead and nobody cares about replacing it with anything. There is no public key encryption involved in modern TLS; instead all you need is a signature scheme (for the certificate and for the final server to authenticate itself) and a key exchange scheme. The former can be ed25519. The latter can be curve25519 (specifically, the retroactively named X25519 ECDH key exchange).
My point is precisely that there's no inherent distrust in RSA (and some concerns with NIST EC, both the magic numbers and secure implementation difficulty), which is why we haven't abandoned it yet. There is certainly no inherent issue with prime field cryptography.
New designs shouldn't use the P-curves, because it's too easy to implement them vulnerably (for all intents and purposes any random string is a workable Curve25519 point, and that's not the case for the P-curves --- you have to do fussy input validation). But that has nothing to do with conspiracy theories about how the curve was generated.
You don't have to take my word for this; you can just read the Koblitz and Menezes paper, which takes this question on in detail.
So, no, really, none of this is true.
Yes, curves in the Web PKI are largely P curves right now, but ed25519 is also standardized.
You're certainly free to trust the NIST curves, but my point still stands: there is no good reason not to pick such constants using a nothing up my sleeve algorithm, and the fact they didn't do that means they either know something we don't, they backdoored it, they wanted people to suspect they backdoored it, or they're idiots and decided to ignore established best practice for no reason. It's entirely possible the answer is the latter, of course :)
It's almost impossible to prove they were backdoored, but considering the sensitivity of the subject, I understand why many consider this unknown a reason to distrust NIST P-curves.
[1] https://cr.yp.to/talks/2013.05.31/slides-dan%2Btanja-2013053...
The most well-known failure has to be the PS3 jailbreak [2]. Sony just failed to implement their RNG (or alternatively copied their RNG code from xkcd #221), which rendered their ECDSA-based crypto completely worthless.
Another famous case is the long list of JWT/JWE libraries which were vulnerable to invalid curve attacks, again completely destroying the security of their NIST p-curves (when used for encryption) [3].
Really, I don't think nobody should be using NIST P-curves if they have any choice, unless you verified your implementation yourself. And I don't even want to claim to be able to do it.
(I don't think tptacek ever said you should use the NIST curves[4], so there's no controversy there)
[1] https://safecurves.cr.yp.to/
[2] https://www.youtube.com/watch?v=LP1t_pzxKyE
[3] https://auth0.com/blog/critical-vulnerability-in-json-web-en...
[4] https://latacora.micro.blog/2018/04/03/cryptographic-right-a...
Again: Koblitz and Menezes take up this topic in detail.
All these libraries listed here are bad in some way for RSA?
* https://en.wikipedia.org/wiki/Comparison_of_cryptography_lib...
[1] https://www.ijser.org/researchpaper/Performance-Based-Compar...
>We noted that the encryption by the RSA algorithm is faster than the one with ECC algorithm.
While we are here, I will point out that the performance was almost the same for the same key lengths in the encryption case.
So ECC seems to be faster for key generation but overall slower for everything else.
Look at the ed25519 benchmarks:
71000 signature verifications per second and 109000 signature generations per second on a quad-core machine. That is much, much faster than RSA.
It's well established that ECC is much faster than RSA in general. I suspect I know what happened. DSA signature schemes require randomness to generate signatures, unlike RSA. They probably were using an entropy-constrained random number generator. That bottlenecked ECDSA through no fault of the algorithm. Ed25519 does not suffer from this issue, since it is constructed in a way that requires no external source of randomness (the random part is substituted with a deterministic hash of the key and input).
RSA 2048 bits: 1001 signatures per second ECDSA P-256: 9516 signatures per second
Note that 256-bit RSA provides a higher security level than 2048-bit RSA, so this is biased in RSA's favor.
https://connect2id.com/blog/nimbus-jose-jwt-6
That one has ed25519 signing 60x faster than RSA-2048. Verification is a bit slower with ed25519 (RSA is very asymmetric there due to the low hamming weight public exponents), but considering the difference in security level, it basically works out to about the same.
So signature verification is about the same for EC vs RSA, while EC is much faster than RSA for signature generation, and ridiculously faster for key generation. EC certainly isn't significantly slower than RSA for any equivalent operation, and definitely not by 40x like the paper you linked claims. That is just wrong.
This is all in response to another user who said:
> There is a massive performance difference—not just size, but computation as well.
... and we don't know exactly what they meant by that, but I was pointing out that it was not that simple...
Asymmetrical performance is normally not very important in most systems due to the use of symmetric stuff to do all the heavy lifting. The exception would be if you were doing some sort of forward secrecy scheme that involved forgetting the private key for each and every message. There the speed of ECDHE is helpful vs the slow key generation of RSA. However, the Signal Protocol shows us we can use a hash ratchet for such short cycle forward secrecy schemes so that key generation speed does not have to be a significant issue.