There are a lot of people who learn and teach the RSA algorithm superficially without a sufficient grasp of the number theory to really understand what is going on. I know because I've been one of them (on both sides). The Carmichael vs. Euler totient issue confused me for a long time.
Needless to say, those people should not be implementing RSA for a system that needs actual security. I'm looking for a better way to teach "real" RSA without needing the students to be math majors or to spend a whole semester on it. Does anybody have any suggestions?
Given how much more favored ECDSA and ECDH is these days, i recommend teaching elliptic curves. They're actually quite simple to understand mathematically if you want a shallow comprehension.
The undergrad lectures I took placed almost equal emphasis on crypto over Zp and crypto over EC. For most students without deep abstract algebra backgrounds, introduction to operations and principles are more friendly and tractable over Zp.
During my 2-year CS degree (in France ) we learned the whole modular algebra with groups, and stuff (don't know the terminology in English sorry) and finally, we learned about RSA using all of this stuff and it really was a wow moment for the whole class!
I don't know how it's taught elsewhere but I feel like I both have "a sufficient grasp of the number theory to really understand what is going on" but also I "should not be implementing RSA for a system that needs actual security" !
I also added plenty of inline python code blocks students can change and run on the fly.
The reason i wrote this is the hand waving around group theory i saw in other explanations. Namely you shouldn't just say x^y always = x mod m for certain values of y (eg. x^13=x mod 35, even for factors of 35). You should give a detailed, intuitive understanding of why this occurs.
Yeah, learning maths for 1-2 months and then applying it to RSA in python at the very end was how I learned it and I think it was a great way. Even though it was a CS diploma we learned it with the maths teacher and spent the right amount of time on it.
Although there are continuums of teaching delivery from muddled to clear explanations of concepts, there are no student shortcuts to escape the irreducible mental exertion to acquire familiarity towards mastery. Uncurious people shouldn't be in the field (no pun intended).
> I'm looking for a better way to teach "real" RSA without needing the students to be math majors or to spend a whole semester on it. Does anybody have any suggestions?
Start and end with a reminder to use padding.
Actually if you want to make it not-so-mathy, talking about about how to be compatible with other programs could be nice. How do you import/export public key in pem or der? How do you (de)serialize ciphertext?
Hm, never encountered the Carmichael function before, but I have had a cursory understanding of Carmichael number.
Given a standard 2048-bit RSA modulus, the totient is still ~2048 bits. I'm not sure and haven't done or seen analysis given the reduction in size (and search space) when replaced with a Carmichael function.
This isn't used in practice because if you care about efficiency you're not calculating M^d mod N; instead you compute exponents mod p and mod q and use the CRT to combine (as mentioned in the author's link to "Garner's algorithm").
BTW the Carmichael function and Carmichael numbers have little in common aside from their author and the fact they concern whether x^b = 1 mod N for x relatively prime to N.
Thanks, I thought about this a bit more. Would the security argument for using the Carmichael function essentially be the same as RSA with totient function, as the adversary can always find d that satisfies either function (Carmichael or Euler totient) regardless of which function is used?
Correct. You could construct a weird scenario with a buggy side channel attack where using a different value for d would matter, but generally speaking the attacker doesn't know and doesn't care what value (out of the infinitely large number of options!) you're using.
Annoyingly, while that d = e^-1 usually isn't used in practice (except in cases where you care about side-channel / fault resistance more than the 4x speedup), the Carmichael totient itself still is used in practice. At least if you want to conform to FIPS 186-5 / SP800-56B, which says that the private key includes d = e^-1 mod the Carmichael totient LCM(p-1,q-1), even if you're going to use the CRT. And that means you have to compute LCM(p-1,q-1), which also has side-channel considerations.
Another similar one is that we don't care for strong primes anymore and even though the standards for RSA specifically require it, it's not actually helpful at all, see https://eprint.iacr.org/2001/007
Strong primes are ones where the totient (both carmichael and euler totients) have large primes in them. This happens naturally for 2048 bit and above RSA keys in any-case, they'll statistically absolutely have primes that are larger than the bits needed to factor using elliptic curve methods (>256 bits). In general it's just not that helpful, similar to trying to require carmichael rather than Euler totient. Ok you've made the 2048 bit key 3 bits stronger, great, but let's not bother right?
Do the standards require strong primes for RSA? I think FIPS doesn't ... it gives you that option, either for the legacy reasons or to get a proof with Pocklington's theorem that (p,q) really are prime, but just choosing a random (p,q) and running enough rounds of Miller-Rabin on them is considered acceptable IIRC.
There are a lot of people who learn and teach the RSA algorithm superficially without a sufficient grasp of the number theory to really understand what is going on. I know because I've been one of them (on both sides). The Carmichael vs. Euler totient issue confused me for a long time.
Needless to say, those people should not be implementing RSA for a system that needs actual security. I'm looking for a better way to teach "real" RSA without needing the students to be math majors or to spend a whole semester on it. Does anybody have any suggestions?
Given how much more favored ECDSA and ECDH is these days, i recommend teaching elliptic curves. They're actually quite simple to understand mathematically if you want a shallow comprehension.
The undergrad lectures I took placed almost equal emphasis on crypto over Zp and crypto over EC. For most students without deep abstract algebra backgrounds, introduction to operations and principles are more friendly and tractable over Zp.
The task for teaching is much harder now as these need to be combined into hybrid PQC protocols
Sure, but teaching the original as a fundamental building block would still be just that.
During my 2-year CS degree (in France ) we learned the whole modular algebra with groups, and stuff (don't know the terminology in English sorry) and finally, we learned about RSA using all of this stuff and it really was a wow moment for the whole class!
I don't know how it's taught elsewhere but I feel like I both have "a sufficient grasp of the number theory to really understand what is going on" but also I "should not be implementing RSA for a system that needs actual security" !
I wrote an article trying to give a simple overview for teaching. https://rubberduckmaths.com/eulers_theorem
I also added plenty of inline python code blocks students can change and run on the fly.
The reason i wrote this is the hand waving around group theory i saw in other explanations. Namely you shouldn't just say x^y always = x mod m for certain values of y (eg. x^13=x mod 35, even for factors of 35). You should give a detailed, intuitive understanding of why this occurs.
I use this as a teaching aid: https://github.com/jcalvinowens/toy-rsa
It's an ugly naive implementation, but it's much simpler and more accessible than any real one I've ever seen, and depends on nothing but libc.
Depending on what you're trying to teach, I would think something like these would be nicer to read (but with minimal dependencies): https://github.com/jackkolb/TinyRSA or https://github.com/i404788/tiny-rsa
> I'm looking for a better way to teach "real" RSA without needing the students to be math majors or to spend a whole semester on it.
RSA is math so it seems like you're trying to shove a square peg into a round hole here.
Yeah, learning maths for 1-2 months and then applying it to RSA in python at the very end was how I learned it and I think it was a great way. Even though it was a CS diploma we learned it with the maths teacher and spent the right amount of time on it.
Anyone with an undergraduate CS background should be able to handle Dan Boneh's course:
https://www.coursera.org/learn/crypto
Although there are continuums of teaching delivery from muddled to clear explanations of concepts, there are no student shortcuts to escape the irreducible mental exertion to acquire familiarity towards mastery. Uncurious people shouldn't be in the field (no pun intended).
> I'm looking for a better way to teach "real" RSA without needing the students to be math majors or to spend a whole semester on it. Does anybody have any suggestions?
Start and end with a reminder to use padding.
Actually if you want to make it not-so-mathy, talking about about how to be compatible with other programs could be nice. How do you import/export public key in pem or der? How do you (de)serialize ciphertext?
Hm, never encountered the Carmichael function before, but I have had a cursory understanding of Carmichael number.
Given a standard 2048-bit RSA modulus, the totient is still ~2048 bits. I'm not sure and haven't done or seen analysis given the reduction in size (and search space) when replaced with a Carmichael function.
I know, I'll attempt to summon cperciva.
This isn't used in practice because if you care about efficiency you're not calculating M^d mod N; instead you compute exponents mod p and mod q and use the CRT to combine (as mentioned in the author's link to "Garner's algorithm").
BTW the Carmichael function and Carmichael numbers have little in common aside from their author and the fact they concern whether x^b = 1 mod N for x relatively prime to N.
Thanks, I thought about this a bit more. Would the security argument for using the Carmichael function essentially be the same as RSA with totient function, as the adversary can always find d that satisfies either function (Carmichael or Euler totient) regardless of which function is used?
Correct. You could construct a weird scenario with a buggy side channel attack where using a different value for d would matter, but generally speaking the attacker doesn't know and doesn't care what value (out of the infinitely large number of options!) you're using.
Annoyingly, while that d = e^-1 usually isn't used in practice (except in cases where you care about side-channel / fault resistance more than the 4x speedup), the Carmichael totient itself still is used in practice. At least if you want to conform to FIPS 186-5 / SP800-56B, which says that the private key includes d = e^-1 mod the Carmichael totient LCM(p-1,q-1), even if you're going to use the CRT. And that means you have to compute LCM(p-1,q-1), which also has side-channel considerations.
The summoning worked!
"The efficiency gained from using Carmichael’s totient is minimal. More efficiency can be gained by using Garner’s algorithm."
The proof of which is left to the reader?
stares at the board for ten minutes
disappears into the back room for fifteen minutes
"Yes, it's trivial."
Another similar one is that we don't care for strong primes anymore and even though the standards for RSA specifically require it, it's not actually helpful at all, see https://eprint.iacr.org/2001/007
Strong primes are ones where the totient (both carmichael and euler totients) have large primes in them. This happens naturally for 2048 bit and above RSA keys in any-case, they'll statistically absolutely have primes that are larger than the bits needed to factor using elliptic curve methods (>256 bits). In general it's just not that helpful, similar to trying to require carmichael rather than Euler totient. Ok you've made the 2048 bit key 3 bits stronger, great, but let's not bother right?
Do the standards require strong primes for RSA? I think FIPS doesn't ... it gives you that option, either for the legacy reasons or to get a proof with Pocklington's theorem that (p,q) really are prime, but just choosing a random (p,q) and running enough rounds of Miller-Rabin on them is considered acceptable IIRC.