Why haven't quantum computers factored 21 yet?

by ingve • 11 months ago

340 197 11 months ago

And these are the same quantum computers that will eventually break ecliptic curve cryptography? Now I’m very confused.

Hasn't classical already severely crippled ECC because of some mathematical Assumptions that somebody came back in 2022 and Proved were wrong?

markusde• 11 months ago

Could you link to any more information about this?

kevindamm• 11 months ago

Which assumptions? ECDLP is still considered computationally hard, and ECC considered secure. There are invalid curve attacks and small subgroup attacks but that's a problem with key selection, not a fundamental problem with ECC.

Do you have a citation?

bitexploder• 11 months ago

Not in general, no. It is still secure and used. There are of course attacks but those were not completely breaking

cwmma• 11 months ago

I believe you are thinking of "Supersingular isogeny Diffie–Hellman key exchange" or SIKE which is a post quantum encryption algorithm that was spectacularly broken a couple years ago. The math involves elliptical curves but it's different from the elliptical curve cryptography used in your browser.

bArray• 11 months ago

I think the idea is that it doesn’t matter if it takes billions of gates to achieve, the point is that it can do it very fast. If we thought a table sized FPGA could do it, a state actor would most definitely build one.

lazide2• 11 months ago

theoretically

The practical problem is that ‘noise’ between gates seems to increase exponentially, so practically it may actually be impossible to construct anything with more than a handful of gates for the foreseeable (possibly indefinite?) future.

It’s essentially the crypto version of Fusion.

santiagobasulto• 11 months ago

The potential is there, we haven't made it yet. It's the same with AI, AGI and all that. Imagine if you'd read a response from GPT-2 back in 2019, you'd also be like "and these are the same models that will eventually give us AGI".

heyjamesknight2• 11 months ago

Not a great analogy, since there’s zero chance the kinds of model involved in GPT-2 will give us AGI.

fmbb• 11 months ago

Imagine if you'd read a response from GPT-5 in 2025, you'd also be like "and these are the same models that will eventually give us AGI".

z3phyr• 11 months ago

"Is this what you can conjure Saruman?"

I have a strong belief that new mathematical tools and methods can be developed that can make it "easy" to break a lot of modern cryptography primitives without ever using a quantum computer.

griffzhowl• 11 months ago

If we can build a machine with enough coherent qubits, then it'll be able to break ECC.

As it turns out, that's a big if, but the bigness of the if is about hardware implementation. The theory behind it is just basic quantum mechanics

oh_my_goodness8• 11 months ago

Article: it takes 2405 entangling gates to factor the number 21.

wongarsu3• 11 months ago

In many applications you are concerned about data you encrypt today still being secure 20 years from now. Especially if your threat model includes a malicious actor holding on to data in hopes of decrypting it later.

From the article it sounds like we will still be safe for 20+ years. On the other hand 15 was just extraordinarily easy, progress after 21 will be much quicker. And we never know which breakthroughs might come in the next decades that speed up progress.

oh_my_goodness8• 11 months ago

"progress after 21 will be much quicker"

Can you provide a quick verification for that?

Mistletoe• 11 months ago

This is the reality a million clickbait fearmongering articles won’t tell you. And pr machines for quantum computing companies won’t either.

taway789aaa62• 11 months ago

Third, notice that the only remaining multiplication is a multiplication by 4. Because 15 is one less than a power of 2, multiplying by 2 modulo 15 can be implemented using a circular shift. A multiplication by 4 is just two multiplications by 2, so it can also be implemented by a circular shift. This is a very rare property for a modular multiplication to have, and here it reduces what should be an expensive operation into a pair of conditional swaps.
Aside: multiplication by 16 mod 21 is the inverse of multiplying by 4 mod 21, and the circuits are reversible, so multiplying by 16 uses the same number of Toffolis as multiplying by 4.

I couldn't really find anything explaining the significance of this. The only info I found said that "4 mod 21 = 4" (but I don't know if it was AI slop or not).

Is "multiplying by 4 mod 21" something distinct to quantum computing?

oh_my_goodness8• 11 months ago

I mean 4 is equivalent to 4 mod 21. That part's not AI slop.

JohnKemeny• 11 months ago

I think, in fact, for every prime number p at least 5, 4 mod p is (congruent to) 4.

shiandow• 11 months ago

It is phrasing mathematicians sometimes use. In this sentence 'mod 4' does not indicate an operator but denotes what number system you are in.

For instance the following are equivalent:

2 = 6 mod 4

6 = 2 mod 4

This 'mod 4' can also appear in parentheses or in some other way, but it must appear at the end. Like I said it is not an operator rather it denotes that the entire preceding statement takes place in the appropriate quotient space.

So it is not (multiplying by (4 mod 21)) but ((multiplying by 4) mod 21)

layer8• 11 months ago

See https://en.wikipedia.org/wiki/Modular_arithmetic. It means that the multiplication is performed modulo 21.

taway789aaa62• 11 months ago

Thank you!

AnotherGoodName• 11 months ago

Fractions are under any modulus are actually representable as whole numbers (provided they don’t share factors with the modulus).

For example under mod 21 a half can actually be represented by 11. Try it. Times any even number by 11 and you’ll see you halved it.

Take any number that’s a multiple of 4 and times it by 16 under mod 21. You now have that number divided by 4.

Etc.

Absolutely nothing to do with quantum computers.

owlbite• 11 months ago

So how many gates are we talking to factor some "cryptographically useful" number? Is there some pathway that makes quantum computers useful this century?

dsclough• 11 months ago

Saw this earlier: https://x.com/adamscochran/status/1962148452072124879?s=46

As a layman the pathway seems to exist behind multiple massive materials science breakthroughs

andrewflnr• 11 months ago

How is it that we need to build logical qubits out of physical qubits for error correction purposes, but then still need to blow out our logic qubit numbers for error correction purposes, again? It seems like there's something missing from this explanation, at least.

tripplyons4• 11 months ago

Not sure about the gate count, but if you look at the number of logical qubits required, we are still very far away from factoring numbers that traditional computing has already factored like the 829-bit RSA-250 number.

Legend2440• 11 months ago

Realistically, you want millions to billions of qubits to compete with classical computers that already have trillions of transistors.

jameshart• 11 months ago

Ah - this helped me understand the numbers in quantum computing a little more clearly. I had been under the impression (based on my naive interpretation of the naming) that the number of qubits in a quantum processor might be something analogous to the number of bits of register state in a regular CPU; that qubits should be thought of more as analogous to transistors or maybe even gates makes it a little clearer why you need so many more qubits to perform more complex operations.

adgjlsfhk19• 11 months ago

the difference is that you need millions of 1 qbits to factor rsa 4096, but you only need 10s of millions to factor rsa 32k. qbits and quantum time scale almost linearly with factor size, but super-polynomially for regular computers

lisper9• 11 months ago

So how many gates are we talking to factor some "cryptographically useful" number?

That is a hard question to answer for two reasons. First, there is no bright line that delineates "cryptographically useful". And second, the exact design of a QC that could do such a calculation is not yet known. It's kind of like trying to estimate how many traditional gates would be needed to build a "semantically useful" neural network back in 1985.

But the answer is almost certainly in the millions.

[UPDATE] There is a third reason this is hard to predict: for quantum error correction, there is a tradeoff between the error rate in the raw qbit and the number of gates needed to build a reliable error-corrected virtual qbit. The lower the error rate in the raw qbit, the fewer gates are needed. And there is no way to know at this point what kind of raw error rates can be achieved.

Is there some pathway that makes quantum computers useful this century?

This century has 75 years left in it, and that is an eternity in tech-time. 75 years ago the state of the art in classical computers was (I'll be generous here) the Univac [1]. Figuring out how much less powerful it was than a modern computer makes an interesting exercise, especially if you do it in terms of ops/watt. I haven't done the math, but it's many, many, many orders of magnitude. If the same progress can be achieved in quantum computing, then pre-quantum encryption is definitely toast by 2100. And it pretty much took only one breakthrough, the transistor, to achieve the improvement in classical computing that we enjoy today. We still don't have the equivalent of that for QC, but who knows when or if it will happen. Everything seems impossible until someone figures it out for the first time.

---

[1] https://en.wikipedia.org/wiki/UNIVAC_I#Technical_description

fhdkweig• 11 months ago

> Is there some pathway that makes quantum computers useful this century?
This century has 75 years left in it, and that is an eternity in tech-time.

As a comparison, we went from first heavier than air flight to man walking on the moon in only 66 years.

ted_dunning• 11 months ago

Good reminder on the time scale.

On the other hand, the Univac could do more useful work than current quantum computers.

TheOtherHobbes• 11 months ago

It's not an eternity because QC is a low-headroom tech which is already pushing its limits.

What made computing-at-scale possible wasn't the transistor, it was the precursor technologies that made transistor manufacturing possible - precise control of semiconductor doping, and precision optical lithography.

Without those the transistor would have remained a lab curiosity.

QC has no hint of any equivalent breakthrough tech waiting to kick start a revolution. There are plenty of maybe-perhaps technologies like Diamond Defects and Photonics, but packing density and connectivity are always going to be huge problems, in addition to noise and error rate issues.

Basically you need high densities to do anything truly useful, but error rates have to go down as packing densities go up - which is stretching optimism a little.

Silicon is a very forgiving technology in comparison. As long as your logic levels have a decent headroom over the noise floor, and you allow for switching transients (...the hard part) your circuit will be deterministic and you can keep packing more and more circuitry into smaller and smaller spaces. (Subject to lithography precision.)

Of course it's not that simple, but it is basically just extremely complex and sophisticated plumbing of electron flows.

Current takes on QC are the opposite. There's a lot more noise than signal, and adding more complexity makes the problem worse in non-linear ways.

nabla92• 11 months ago

For RSA 4096 10^7 qubits with 10^-4 error rate (order of magnitude).

You can do useful and valuable quantum chemistry calculations already with few 100s of qubits with that low error rates, while post-quantum algorithms are becoming more common everyday removing incentives to build crypto cracking quantum computers.

I think the quantum computing will advance fastest in directions that are not easy to use in cryptography.

HappyPanacea• 11 months ago

Which valuable quantum chemistry calculations you can do with few 100s of qubits with that low error rates?

smj-edison• 11 months ago

Honestly, if all quantum computers manage to pull off is quantum chemistry, I feel like that'll be enough. It would be a massive boon to the field of material sciences at any rate, which underlies so much of current infrastructure.

gjrq• 11 months ago

Latest numbers are about 1e6 qubits with 1e-4 error rate: https://arxiv.org/abs/2505.15917. Gates (in the sense the OP means) is harder to quantify in the error corrected context once you compile to the operations that are native to your code. Total compute time of about a week assuming a 1MHz "clock" (code cycle time, for the experts). In some ways this is the harder metric to meet than the qubit numbers.

Note that the magic of quantum error correction (exponential improvement in the error rate goes both ways): if you could get another 9 in qubit fidelity, you get a much larger improvement in qubit numbers. On the other hand, if you need to split your computation over several systems, things get much worse.

adgjlsfhk19• 11 months ago

given the correct state of gate noise progress, it seems likely that we might get an extra order of magnitude of error before we get the 3 orders of magnitude in gates.

Strilanc6• 11 months ago

So how many gates are we talking to factor some "cryptographically useful" number?

Table 5 of [1] estimates 7 billion Toffoli gates to factor 2048 bit RSA integers.

Is there some pathway that makes quantum computers useful this century?

The pathway to doing billions of gates is quantum error correction. [1] estimates distance 25 surface codes would be sufficient for those 7 billion gates (given the physical assumptions it lists). This amplifies the qubit count from 1400 logical qubits to a million physical noisy qubits.

Samuel Jacques had a pretty good talk at PQCrypto this year, and he speculates about timelines in it [2].

(I'm the author of this blog post and of [1].)

[1] https://arxiv.org/pdf/2505.15917

[2] https://www.youtube.com/watch?v=nJxENYdsB6c

ktallett2• 11 months ago

It's not just quantum error correction that is required, it's also hard to make devices small enough due to cooling, to allow thousands of qubits let alone billions.

sllabres• 11 months ago

From the talk of Samuel Jacques: Timeline for RSA-2048 at about 2088 (conservative extrapolation) or ~2052 (Moore’s‑law‑style growth)

cubefox3• 11 months ago

How do you go from 7 billion Toffoli gates (which consist of 42 billion "two-qubit entangling gates", per the blog post) to merely 1400 logical qubits?

rendaw• 11 months ago

Is there a qubit-speed tradeoff, where you could factor 2048 bit RSA integers with fewer qubits but it would take more time? Or is it all or nothing?

DebtDeflation2• 11 months ago

Since most QC researchers are saying that the real use cases are not Shor's or Grover's but rather molecular simulation (calculating ground state energies), do we have a timeline for doing anything useful in this space with quantum computers? A quick Googling indicates it take about 12 logical qubits to simulate extremely simple molecules like LiH and H4. Does the qubit requirement just scale linearly with the total number of electrons or is there an exponential function in there somewhere? I'm trying to get a sense of whetherf material science and biological applications are similarly 30+ years out or might come substantially sooner.

AceJohnny2• 11 months ago

What does this mean about the size (and thus feasibility) of a circuit required to factor a cryptographically interesting number, say, to be generous, RSA1024?

add-sub-mul-div• 11 months ago

"(Quick aside: the amount of optimization that has gone into this factoring-21 circuit is probably unrepresentative of what would be possible when factoring big numbers. I think a more plausible amount of optimization would produce a circuit with 500x the cost of the factoring-15 circuit… but a 100x overhead is sufficient to make my point. Regardless, special thanks to Noah Shutty for running expensive computer searches to find the conditional-multiplication-by-4-mod-21 subroutine used by this circuit.)"

Davidzheng2• 11 months ago

Off topic, but are cryptographers convinced that on the new gigawatt data centers RSA1024 is infeasible to factor? I gather that the fastest known algorithms are still too slow to factor it in reasonable time. But is consensus that there will not be improvements to these algorithms in near future?

rwmj• 11 months ago

Number Field Sieves are still the best method, and the techniques are three or more decades old with only incremental improvements. (Of course there might be an incredible breakthrough tomorrow.)

ginko• 11 months ago

It recently occurred to me that now would be the best time ever for state actors to build out massive data centers without anyone noticing.

tripplyons4• 11 months ago

I've seen pretty credible evidence that factoring large semiprime numbers is effectively a solved problem, even without considering quantum computing or gigawatt-scale computing. I'm not able to share specifics, but I would personally not trust RSA.

mcswell• 11 months ago

Fiction, but interesting nonetheless: the 1992 film "Sneakers", with Robert Redford among others. The MacGuffin is a mathematician who discovers another way to factor large numbers.

Davidzheng2• 11 months ago

This is wrong btw: a 1GW data center can indeed factor RSA1024 in around a month--I expect state actors to crack it faster in specialized hardware

Strilanc6• 11 months ago

Estimates of the cost of RSA1024 use explicit circuit constructions at the target size, rather than extrapolating from the 4 bit case. So they implicitly account for the discontinuity being pointed out in the post. So this post has no impact on those costs.

raverbashing• 11 months ago

My belief in achieving actual quantum computing is going down as noise in qbits goes up

Digital computers were much easier than that. Make it smaller, make a larger number of it, and you're set.

Quantum computers complexity goes up with ~ n^2 (or possibly ~ e^n) where n is the number of qbits

At the same time, things like d-wave might be the most 'quantum' we might get in the practical sense

analog313• 11 months ago

It turns out that error correction was easy on digital computers, and was essentially a solved problem early in their development. In fact, "noise immunity" is arguably the defining feature of a digital system. And error correction can happen at each gate, since there's no reason to propagate an indeterminate number.

Der_Einzige2• 11 months ago

Except quantum error correction algorithms that are good don’t exist and probably theoretically never can exist: https://spectrum.ieee.org/the-case-against-quantum-computing

tiahura2• 11 months ago

What? I thought quantum computing was going to be factoring billion digit prime numbers in 5 years?

sincerely• 11 months ago

Sounds like a you problem

api2• 11 months ago

It’s worth noting that the reason we are deploying PQ crypto is not that we are 100% convinced QC is coming soon. It may or may not depending on how development goes.

The goal of cryptography is to make something as close to theoretically unbreakable as possible. That means even theoretical vulnerabilities are taken seriously.

For ECC and RSA and related algorithms we have a theoretical and physically plausible pathway toward a practical machine that could break them. That means many cryptographers consider them theoretically broken even if such a machine does not exist and may not exist for a long time. The math works even if we can’t build it yet.

So it’s considered prudent to go ahead and upgrade now while no QC exists. That way if some major advance does arrive we are ready.

Nobody’s talking seriously about replacing SHA2, AES, ChaCha, etc because there is no physically plausible theoretically valid path to a machine that can break these in, say, less than many millions of years. AFAIK there is no proof that such a path does not exist but nobody has found one, hence they are considered unbroken.

Note that cryptography is not the only or even the most useful application of QC. Things like physical stimulation of quantum systems, protein folding, machine learning, etc. could be more useful. Like digital computers there’s probably a ton of uses we don’t know about because we need to tinker with the machine to figure them out.

leeoniya• 11 months ago

Things like physical stimulation of quantum systems, protein folding, machine learning, etc. could be more useful

is there still more to do in protein folding after AlphaFold?

https://www.isomorphiclabs.com/articles/alphafold-3-predicts...

api2• 11 months ago

There’s a difference between good AI predictions and theoretically perfect QC computations. The AI estimates while the QC will give you the answer, full stop. The latter could be relied upon more strongly. It could also generate infinite training data to make much better models.

QC might be directly applicable to AI training too. It may be possible to compute the optimal model over a data set in linear time. It could allow training that is faster and consumes a tiny fraction of the energy current brute force methods need.

dwroberts• 11 months ago

The predictions don't tell us anything about why the answer is what it is. There is probably important (useful) fundamental scientific knowledge in being able to know that vs. just being able to predict the result.

tialaramex• 11 months ago

For the symmetric cryptography (so obviously AES and ChaCha, but also in effect the SHA-2 family) we can hand wave the quantum attacks as halving key length by enabling a sort of meet-in-the-middle attack (this attack is why it was 3DES not 2DES when they strengthened DES). There's a lot of hand waving involved. Your real Quantum Computer won't in fact be equivalent cost to the non-quantum computer, or as fast, or as small, the speed-ups aren't quite halving, and so on. But it's enough to say OK, what if AES-128 was as weak as a hypothetical AES-64, and that's fine because we have AES-256 anyway.

However, the main focus is on Key Exchange. Why? Well, Key Exchange is the clever bit where we don't say our secrets out loud. Using a KEX two parties Alice and Bob agree a secret but neither of them utters it. Break that and you can learn the secret, which was used to encrypt everything else - for any conversation, including conversations which you recorded any time in the past, such as today.

If future bad guys did have a Quantum Computer the Key Exchange lets them read existing conversations they've tapped but today can't read, whereas breaking say the signing algorithm wouldn't let them somehow go back in time and sign things now because that's not how time works. So that's why the focus on KEX. Once such a thing exists or clearly is soon to deliver it's important to solve a lot of other problems such as signing, but for KEX that's already too late.

d_silin• 11 months ago

Obligatory reference: https://scottlocklin.wordpress.com/2019/01/15/quantum-comput...

n4r9• 11 months ago

Obligatory how? It's sort of funny but comes across as uninformed, over the top, and lacking substance.

NooneAtAll3• 11 months ago

I think a more plausible amount of optimization would produce a circuit with 500x the cost of the factoring-15 circuit

I don't get this part

If author already produced "115x", how can optimizations make it worse?

tromp• 11 months ago

The 115x was obtained by doing a (less plausable) large amount of optimization...

turtletontine• 11 months ago

I think the idea is the minimal implementation will be unstable and unreliable. I don’t know the details, but there’s much work and thought on quantum error correcting qubits - where you hook up N qubits in a clever way to function as one very stable qubit. Terms such as “decoherence time” make appearances. You can imagine this quickly expands into an awful lot of qubits.

tsimionescu2• 11 months ago

The point was that the amount of optimization work that went into making the circuit for factoring 21 only ~100x larger than the one for factoring 15 is not feasible for larger numbers. So, the general rule should be that you need ~500x more gates for a similar increase in the number to be factored, not just ~100x.

Strilanc6• 11 months ago

The more plausible amount of optimization is less optimization. Or, more accurately, the benefits of optimization at large sizes is expected to be less beneficial than it was for the N=21 circuit.

BitsAndObjects• 11 months ago

I enjoyed the typo in "error corection" (sic).

avhon1• 11 months ago

The linked paper "Replication of Quantum Factorisation Records with an 8-bit Home Computer, an Abacus, and a Dog" is a good read!

https://eprint.iacr.org/2025/1237.pdf

tim333• 11 months ago

That was entertaining.

m1013• 11 months ago

Quantum mechanics is "true" insofar as it's useful for some purpose. Until then it's a theory and the jury is still out.

Randomness is something which I feel is a pretty weird phenomenon. I am definitely one of those 'God doesn't play with dice' types.

Randomness is also something that we call things when actually it's random from a subjective perspective. If we knew more about a system the randomness just falls away. E.g. if we knew the exact physical properties of a dice roll we could probably predict it better than random.

What if it's the case that quantum mechanics is similar. I.e. that what we think of as randomness isn't really randomness but only appears that way to the best of what we can observe. If this is the case, and if our algorithms rely on some sort of genuine randomness inherent in the universe, then doesn't that suggest there's a problem? Perhaps part of the errors we see in quantum mechanics arise from just something fundamental to the universe being different to our model.

I don't think this is that far fetched given the large holes that our current understanding of physics have as to predicting the universe. It just seems that in the realm of quantum mechanics this isn't the case, apparently because experiments have verified things. However, I think there really is something in the proof being in the pudding (provide a practical use case).

karmakurtisaani• 11 months ago

This has been considered, see https://en.m.wikipedia.org/wiki/Hidden-variable_theory

m1013• 11 months ago

I've heard of it but I'm not very familiar, or have a deep understanding of it. I buy into things like Wolfram physics for explaining how the universe actually works, with equations/calculus being limited to approximations to an error in measurement. The hidden variable theory seems to mostly covers physics unlike what Wolfram covers, so it's useful for a purpose but it still seems to be going down a similar rabbit hole to where normal physics ends up.

withinboredom2• 11 months ago

Most people don’t want to go down this hole. In the end, it means we might not have free will if there isn’t any randomness…

But if you are up for an existential crisis, just google “hidden variable theories”

AlienRobot• 11 months ago

Turns out we only have "free as in free beer" will.

Sorry, I was destined to make this joke before I was even born.

ryankrage77• 11 months ago

the 'no randomness = no free will' argument is pretty common, but how does randomness ensure free will? It's still something out of our control, just it can't be predicited. Why is it any better to be a random automaton than a predictable one?

dhampi• 11 months ago

Yes, Schrodinger’s equation is entirely deterministic. There is no randomness inherent in quantum mechanics. The perceived randomness only arises when we have incomplete information. (It is another matter that quantum mechanics to some extent forbids us from having perfect information.)

I mean no disrespect, but I don’t think it’s a particularly useful activity to speculate on physics if you don’t know the basic equations.

dekken_2• 11 months ago

Quantum mechanics, is not "just one thing", so to say "it is true" is somewhat wrong I think.

You are probably talking about the Copenhagen interpretation, involving superposition.

Personally, I don't think this is the final theory.

Any theory using calculus, cannot be considered discrete, so is therefore not quantized, and not possibly "physical".

Gerard 't Hooft has more to say on this if you want to hear something from a nobel laureate on the subject.

m1013• 11 months ago

Yes, agree that suggesting "true" is unclear, and in fact in science doesn't really talk about true things but rather than ability to predict the way things behave. We are still in the dark about the fundamental nature of reality. Science is still useful of course, but only insofar as it has a useful purpose. It's more like an engineering subject.

I think what I've just said foots with your calculus comment, and also a Wolfram-like interpretation is closer to "truth" and your point on discretisation.

Why do you think discretisation/quantisation is necessary for the "physical"?

What can I search for to find his comments on this subject?

hulitu• 11 months ago

Why haven't quantum computers factored 21 yet?

They tried. But because they know exactly the question, they cannot give a precise answer. /s

amelius• 11 months ago

That's ok as long as the VCs don't know the answer either /s

djha-skin• 11 months ago

Does this essentially mean that the Big-O "circuitry requirements" of factoring integers using Schorr's is super polynomial?

cvoss• 11 months ago

The article explains the gate cost mostly comes from O(n) multiply-under-mod ops on O(n)-bit numbers. Each op could be naively spelled out as O(n^2) gates. So the whole thing looks cubic to me at worst.

hvenev• 11 months ago

I wonder when quantum computers will be able to target post-quantum RSA [1]. Normal RSA operations (key generation, encryption, decryption) have an asymptotic advantage over Shor's algorithm, so it is not unreasonable to just use large enough keys. The advantage is similar to Merkle's puzzles [2], with the added bonus that the attacker also needs to run their attack on a quantum computer.

A while ago I generated a gigabit RSA public key. It is available at [3]. From what I remember, the format is: 4-byte little-endian key size in bytes, then little-endian key, then little-endian inverse of key mod 256**bytes. The public exponent is 3.

[1] https://eprint.iacr.org/2017/351.pdf

[2] https://dl.acm.org/doi/pdf/10.1145/359460.359473

[3] https://hristo.venev.name/pqrsa.pub

BoppreH• 11 months ago

Post-Quantum RSA is clearly a joke from djb, to have a solid reply when people ask "can't we just use bigger keys"?. It has a 1-terabyte RSA key taking 100 hours to perform a single encryption. And by design it should be beyond the reach of quantum computers.

vrighter2• 11 months ago

Because they haven't actually factored any other smaller number yet.

If your program has a compilation process that requires you to already know the answer to the problem you're trying to solve, then what they did was not factorization, but "print 3" with extra steps.

patrakov• 11 months ago

The article, if examined by a non-specialist like me, seems to contain an answer to this concern at the end:

There are papers that claim to have factored 21 with a quantum computer. For example, here’s one from 2021 [1]. But, as far as I know, all such experiments are guilty of using optimizations that imply the code generating the circuit had access to information equivalent to knowing the factors.

[1] https://arxiv.org/abs/2103.13855

DebtDeflation2• 11 months ago

The article claims that 15 was done without precompilation (in the 2015 experiment but not the 2001 experiment), but only because 15 is a very unique number (after decomposition, only the last multiplication has to be performed because all the previous multiplications involve multiplying by 1).

That said, we are REALLY far off from having a useful quantum computer. Jensen was probably being conservative when he said 20-30 years away, hence the immediate pressure he received form the investor community to reverse his statement followed by the flood of ridiculous press releases from the usual companies claiming to be 2-3 years away.

fsckboy• 11 months ago

deriving arithmetic from lambda calculus is not "arithmetic with extra steps"... not sure what I mean by that, but seems like that implies there is arithmetic without extra steps, then you just print the answer

brazzy• 11 months ago

I recently talked to a guy who works on quantum computing research.

He said that factoring and cryptography applications are red herrings. It's not what most people in the field are working on.

The practical application lies in simulating processes where quantum effects are actually directly relevant, such as quantum chemistry: https://en.wikipedia.org/wiki/Quantum_chemistry

Nevermark4• 11 months ago

I wonder if the circuit could be modularized into a quantum ALU. Where many of the otherwise identical components in the original layout, couldn't just be arranged as operations time, over a reduced number of distinct components.

Most digital algorithms would explode in terms of hardware needed, for increasing N, if we didn't distribute that computation in time, as well as across elements.

cubefox3• 11 months ago

Is this the same idea as using universal Turing machines (CPUs that can execute software) rather than conventional fixed-function Turing machines (ASICs/FPGAs)?

Nevermark4• 11 months ago

Yes, exactly

adgjlsfhk19• 11 months ago

unfortunately this is kind of fundamentally impossible. the whole power of quantum computation is that big quantum computers can do computation on a massive state space "for free". that benefit only exists if you have enough qbits to hold the state space.

Nevermark4• 11 months ago

Holding the state space and iteratively transforming it (in a reversible way, so the state space is preserved), seem like a solution to that.

This would still require more hardware than digital circuits (which can be made reversible for energy efficiency, but that is rarely done).

While still reducing the number of operation components, and reusing them.

Strilanc6• 11 months ago

In most quantum computer designs, gates are signals generated on demand at runtime rather than material deposited at fabrication time. In this regime, the concept of an ALU makes no sense. Instead of just sending pulses doing the exact gates you know need, why would you instead expand that short sequence into long sequences that emulate potentially applying every arithmetic operation to every input and then mux out the result you already knew you needed. It's a lot of extra work for the same final result.

A quantum ALU would also be substantially harder to design, because of the need to maintain reversibility. For example, every operation would have to run as slow as the slowest operation (or else the timing side channel would measure which operation occurred).

JoachimS• 11 months ago

Does this mean in a general sense there are numbers that are harder to factor, or is it due to constraints? That some keys will be much harder to crack? If so, how can we know beforehand?

adgjlsfhk19• 11 months ago

it's more that there are numbers that are easier to factor (e.g. 2^n-1). almost all numbers once you get past the tiny numbers are about the same.

Why haven't quantum computers factored 21 yet?

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