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A New Attack Easily Knocked Out a Potential Encryption Algorithm
SIKE was a contender for post-quantum-computing encryption. It took researchers an hour and a single PC to break it.
Photograph: Tuomas A. Lehtinen/Getty Images
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In the U.S. government's ongoing campaign to protect data in the age of quantum computers , a new and powerful attack that used a single traditional computer to completely break a fourth-round candidate highlights the risks involved in standardizing the next generation of encryption algorithms.
This story originally appeared on Ars Technica , a trusted source for technology news, tech policy analysis, reviews, and more. Ars is owned by WIRED's parent company, Condé Nast.
Last month, the US National Institute of Standards and Technology (NIST), selected four post-quantum-computing encryption algorithms to replace algorithms like RSA, Diffie-Hellman, and elliptic curve Diffie-Hellman, which are unable to withstand attacks from a quantum computer.
In the same move, NIST advanced four additional algorithms as potential replacements pending further testing, in hopes one or more of them may also be suitable encryption alternatives in a post-quantum world. The new attack breaks SIKE, which is one of the latter four additional algorithms. The attack has no impact on the four PQC algorithms selected by NIST as approved standards, all of which rely on completely different mathematical techniques than SIKE.
SIKE—short for supersingular isogeny key encapsulation —is now likely out of the running, thanks to research that was published over the weekend by researchers from the Computer Security and Industrial Cryptography group at KU Leuven. The paper, titled “ An Efficient Key Recovery Attack on SIDH (Preliminary Version) ,” described a technique that uses complex mathematics and a single traditional PC to recover the encryption keys protecting the SIKE-protected transactions. The entire process requires only about an hour’s time. The feat makes the researchers, Wouter Castryck and Thomas Decru, eligible for a $50,000 reward from NIST.
“The newly uncovered weakness is clearly a major blow to SIKE,” David Jao, a professor at the University of Waterloo and co-inventor of SIKE, wrote in an email. “The attack is really unexpected.”
The advent of public-key encryption in the 1970s was a major breakthrough, because it allowed parties who had never met to securely trade encrypted material that couldn’t be broken by an adversary. Public-key encryption relies on asymmetric keys, with one private key used to decrypt messages and a separate public key for encrypting. Users make their public key widely available. As long as their private key remains secret, the scheme remains secure.
In practice, public-key cryptography can often be unwieldy, so many systems rely on key encapsulation mechanisms, which allow parties who have never met before to jointly agree on a symmetric key over a public medium such as the internet. In contrast to symmetric-key algorithms, key encapsulation mechanisms in use today are easily broken by quantum computers. SIKE, before the new attack, was thought to avoid such vulnerabilities by using a complex mathematical construction known as a supersingular isogeny graph.
The cornerstone of SIKE is a protocol called SIDH, short for supersingular isogeny Diffie-Hellman. The research paper published over the weekend shows how SIDH is vulnerable to a theorem known as “glue-and-split” developed by mathematician Ernst Kani in 1997, as well as tools devised by fellow mathematicians Everett W. Howe, Franck Leprévost, and Bjorn Poonen in 2000. The new technique builds on what’s known as the GPST adaptive attack, described in a 2016 paper . The math behind the latest attack is guaranteed to be impenetrable to most non-mathematicians. Here’s about as close as you’re going to get:
“The attack exploits the fact that SIDH has auxiliary points and that the degree of the secret isogeny is known,” Steven Galbraith , a University of Auckland mathematics professor and the “G” in the GPST adaptive attack, explained in a short write-up on the new attack. “The auxiliary points in SIDH have always been an annoyance and a potential weakness, and they have been exploited for fault attacks, the GPST adaptive attack, torsion point attacks, etc.”
More important than understanding the math, Jonathan Katz, an IEEE member and professor in the Department of Computer Science at the University of Maryland, wrote in an email: “The attack is entirely classical, and does not require quantum computers at all.”
SIKE is the second NIST-designated PQC candidate to be invalidated this year. In February, IBM postdoc researcher Ward Beullens published research that broke Rainbow , a cryptographic signature scheme with its security, according to Cryptomathic , “relying on the hardness of the problem of solving a large system of multivariate quadratic equations over a finite field.”
NIST’s PQC replacement campaign has been running for five years. Here’s a brief history:
Rainbow fell during round 3. SIKE had made it until round 4.
It is perhaps a bit concerning that this is the second example in the past six months of a scheme that made it to the 3rd round of the NIST review process before being completely broken using a classical algorithm. (The earlier example was Rainbow, which was broken in February.) Three of the four PQC schemes rely on relatively new assumptions whose exact difficulty is not well understood, so what the latest attack indicates is that we perhaps still need to be cautious/conservative with the standardization process going forward.
I asked Jao, the SIKE co-inventor, why the weakness had come to light only now, in a relatively later stage of its development. His answer was insightful. He said:
It's true that the attack uses mathematics which was published in the 1990s and 2000s. In a sense, the attack doesn't require new mathematics; it could have been noticed at any time. One unexpected facet of the attack is that it uses genus 2 curves to attack elliptic curves (which are genus 1 curves). A connection between the two types of curves is quite unexpected. To give an example illustrating what I mean, for decades people have been trying to attack regular elliptic curve cryptography, including some who have tried using approaches based on genus 2 curves. None of these attempts has succeeded. So for this attempt to succeed in the realm of isogenies is an unexpected development.
In general there is a lot of deep mathematics which has been published in the mathematical literature but which is not well understood by cryptographers. I lump myself into the category of those many researchers who work in cryptography but do not understand as much mathematics as we really should. So sometimes all it takes is someone who recognizes the applicability of existing theoretical math to these new cryptosystems. That is what happened here.
The version of SIKE submitted to NIST used a single step to generate the key. A possible variant of SIKE could be constructed to take two steps. Jao says it’s possible that this latter variant might not be susceptible to the math causing this breakage. For now, though, SIKE is dead, at least in the current running. The schedule for the remaining three candidates is currently unknown.
This story originally appeared on Ars Technica .
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SIKE is one of the new algorithms that NIST recently added to the post-quantum cryptography competition.
We present an efficient key recovery attack on the Supersingular Isogeny Diffie­-Hellman protocol (SIDH), based on a “glue-and-split” theorem due to Kani. Our attack exploits the existence of a small non-scalar endomorphism on the starting curve, and it also relies on the auxiliary torsion point information that Alice and Bob share during the protocol. Our Magma implementation breaks the instantiation SIKEp434, which aims at security level 1 of the Post-Quantum Cryptography standardization process currently ran by NIST, in about one hour on a single core.
I read the article on The Register yesterday and while I haven’t the first clue about the underlying math it read to me like someone who’s installed an amazingly fancy and highly secure (oxymoron alert) digital door lock but forgot to fir the door itself into the frame.
Btw, is codecrypt utility still secure enough to use it?
codecrypt is a GnuPG-like Unix program for encryption and signing that only uses quantum-resistant algorithms:
Please, feel free to express your thoughts.
“while I haven’t the first clue about the underlying math it read to me like someone who’s installed an amazingly fancy and highly secure”
The fundamental protocol for SIKE is “Supersingular Isogeny Diffie-Hellman”(SIDH). Which is a tads difficult to fully explain even by mathmaticians…
The problem is that even mathmaticians can be unaware of the more referified parts of their art…
So it turns out it was not “secure”, and it had been known by some mathmetitians how to attack it since the late 1990’s…
Which is kind of a decade or two and some before anyone decided to use SIKE as a one way function for crypto. Which kind of makes it realy “face palm” embarrassing…
“The research paper published over the weekend shows how SIDH is vulnerable to a theorem known as “glue-and-split” developed by mathematician Ernst Kani in 1997, as well as tools devised by fellow mathematicians Everett W. Howe, Franck Leprévost, and Bjorn Poonen in 2000. The new technique builds on what’s known as the “GPST adaptive attack,” described in a 2016 paper. The math behind the latest attack is guaranteed to be impenetrable to most non-mathematicians.”
But as I said the other day this is not the real issue…
The issue is “One Way Functions”(OWFs) that have “Secret trap doors” to make them appropriate for Key Exchange” and “signing”.
We do not realy know if OWFs realy exist in a form that is “Quantum Computing”(QC) proof.
Firstly because there is no proof that the supset of OWFs suitable are actually secure. Secondly if we do get QC’s up and running, what new algorithms might pop up to ruin a cryptographas day…
Perhaps rather than chase “Post Quantumn Algorithms” that we have no idea how long before they get broken, we should actually take a different direction and look at how to do the “keye exchange” and “signing” functions in entirely different ways…
The statement “The math behind the latest attack is guaranteed to be impenetrable to most non-mathematicians” is a massive understatement. A bit like the dentist saying, this will be a bit uncomfortable.
I know several people who, while not career level mathematicians, have used and explored advanced maths throughout their careers and have opined on the rather significant effort to understand the math.
I await further insight from people more capable than I. 🙂
@Clive – good questions. Of course the next question is what might a new way to do key exchanges even look like. The old ways of exchanging keys scales poorly (to say the least).
The main problem is Shor’s not Grover’s. Double all key lengths and symmetric algorithms will continue to work. Asymmetric crypto on the other hand …
Notably a couple of weeks ago when NIST announced some of their PQC picks, it left open the asymmetric algorithms which is what we most need. A candidate algorithm face-planting is one thing, having an announced algorithm fail is quite another. The last time this happened was with Format Preserving Encryption which is a much more limited use case. Maybe this needs more time?
Btw, is codecrypt utility still secure enough to use it?
codecrypt is a GnuPG-like Unix program for encryption and signing that only uses quantum-resistant algorithms
I wouldn’t recommend using anything with only post-quantum algorithms. It’s more common to combine them with better-tested methods like Elliptic Curve Cryptography, such that both algorithms need to be broken to break the security. While ECC etc. are theoretically vulnerable to quantum computers, they’re believed to be safe from all existing computers—which, as you’ve seen, is harder to say about these newer post-quantum algorithms. The general belief is that no quantum computer exists, nor will in the next decade or so, that can crack any strong cryptography much faster than classical computers.
Additionally, codecrypt is unmaintained.
I’m not familiar with fmtseq, but took a quick look at the paper and some references. It looks to be based on the Merkle signature scheme, for which it’s important not to reuse keys (although the papers are not explicit about this, or about what happens if we do). That is, it requires the signer to maintain state and update it after every signature, which is something we don’t really want: we’ve seen severe cryptosystem failures in the past from improper entropy or state-maintenance. I wonder whether such methods could be (or already have been?) made deterministic in some way, or random-but-stateless based on a very low probability of leaks.
That said, Merkle’s scheme is well understood and believed to be secure when used correctly. I don’t know if that generalizes to fmtseq. I also don’t know enough about MDPC McEliece to comment, except to note that codecrypt’s original McEliece algorithm was broken.
Most people should probably stick to something more traditional, and move to a smartcard or hardware security module (HSM)—or multiple of these with secret-sharing—if they want to improve security; and, if possible, keep them offline. Under realistic threat models, attacks based on improper patching or zero-day exploits, or idiocy like checking private keys into git(hub), tend to be much more worrying than quantum attacks.
Kolmogorov complexity and One-way Functions
‘https://www.quantamagazine.org/researchers-identify-master-problem-underlying-all-cryptography-20220406/
Conversely, if calculating the approximate time-bounded Kolmogorov complexity is too hard to solve efficiently for many strings, then Liu and Pass showed that true one-way functions must exist.
“…how to do the “key exchange” and “signing” functions in entirely different ways…”
Key exchange without a recognizable middle man or process? Sounds like number stations from various indistinguishable sources. Internet white-noise, plus less secrets: wagingpeace.org

Kolmogorov complexity and One-way Functions

I believe someone posted a link to the original paper here some time ago. It is still sitting in my reading list. I think I will move it to the top of the list.
There are reasons why there are players that want you to chase the PQC Ghost.
It is to distract and waste your time.
We need Asymmetric PKI of course, but we need to know that it is not MITM-ed.
There are alternative methods for Symmetric, but if the TLS Handshake is MITM-ed, then that will fail also.
We definitely need better internet cryptography protocols. We need to be able to detect a MITM and treat it as a double bit error.
You must assume a MITM at all times.
Err the maths behind this attack on SIKE is also being looked at for breaking ECC (read ARS article for more details).
So far the search for an ECC attack of use is “ongoing”. This attack on SIKE is almost certainly going to renew attempts to break ECC. Doing so would give the successful person(s) a “Golden Ticket” C.V.[1]
Let’s put it this way I suspect ECC now has a very short shelf life… Maybe half a decade at most life left would be my advice to the cautious. So looking for a replacment should begin right away.
It could be a heck of a sight less, as there is a chance a successful attack may already have been discovered but not yet recognised. So it might come fast very fast.
Hency my earlier comments about thinking on how to replace the curent asymetric “Key Exchange” and “Signing” systems.
Because if we loose them before we replace them to a non QC algorithm then it’s going to be brutal very fast…
1, Banking / Finance.
2, Online Shoping/Commerce.
3, Software Patching.
4, Se
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