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RSA-240 Factored

This just in:

We are pleased to announce the factorization of RSA-240, from RSA’s challenge list, and the computation of a discrete logarithm of the same size (795 bits):

RSA-240 = 12462036678171878406583504460810659043482037465167880575481878888328 966680118821085503603957027250874750986476843845862105486553797025393057189121 768431828636284694840530161441643046806687569941524699318570418303051254959437 1372159029236099 = 509435952285839914555051023580843714132648382024111473186660296521821206469746 700620316443478873837606252372049619334517 * 244624208838318150567813139024002896653802092578931401452041221336558477095178 155258218897735030590669041302045908071447

[…]

The previous records were RSA-768 (768 bits) in December 2009 [2], and a 768-bit prime discrete logarithm in June 2016 [3].

It is the first time that two records for integer factorization and discrete logarithm are broken together, moreover with the same hardware and software.

Both computations were performed with the Number Field Sieve algorithm, using the open-source CADO-NFS software [4].

The sum of the computation time for both records is roughly 4000 core-years, using Intel Xeon Gold 6130 CPUs as a reference (2.1GHz). A rough breakdown of the time spent in the main computation steps is as follows.

RSA-240 sieving: 800 physical core-years
RSA-240 matrix: 100 physical core-years
DLP-240 sieving: 2400 physical core-years
DLP-240 matrix: 700 physical core-years

The computation times above are well below the time that was spent with the previous 768-bit records. To measure how much of this can be attributed to Moore’s law, we ran our software on machines that are identical to those cited in the 768-bit DLP computation [3], and reach the conclusion that sieving for our new record size on these old machines would have taken 25% less time than the reported sieving time of the 768-bit DLP computation.

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TPM-Fail Attacks Against Cryptographic Coprocessors

Really interesting research: TPM-FAIL: TPM meets Timing and Lattice Attacks, by Daniel Moghimi, Berk Sunar, Thomas Eisenbarth, and Nadia Heninger.

Abstract: Trusted Platform Module (TPM) serves as a hardware-based root of trust that protects cryptographic keys from privileged system and physical adversaries. In this work, we per-form a black-box timing analysis of TPM 2.0 devices deployed on commodity computers. Our analysis reveals that some of these devices feature secret-dependent execution times during signature generation based on elliptic curves. In particular, we discovered timing leakage on an Intel firmware-based TPM as well as a hardware TPM. We show how this information allows an attacker to apply lattice techniques to recover 256-bit private keys for ECDSA and ECSchnorr signatures. On Intel fTPM, our key recovery succeeds after about1,300 observations and in less than two minutes. Similarly, we extract the private ECDSA key from a hardware TPM manufactured by STMicroelectronics, which is certified at CommonCriteria (CC) EAL 4+, after fewer than 40,000 observations. We further highlight the impact of these vulnerabilities by demonstrating a remote attack against a StrongSwan IPsecVPN that uses a TPM to generate the digital signatures for authentication. In this attack, the remote client recovers the server’s private authentication key by timing only 45,000 authentication handshakes via a network connection.

The vulnerabilities we have uncovered emphasize the difficulty of correctly implementing known constant-time techniques, and show the importance of evolutionary testing and transparent evaluation of cryptographic implementations.Even certified devices that claim resistance against attacks require additional scrutiny by the community and industry, as we learn more about these attacks.

These are real attacks, and take between 4-20 minutes to extract the key. Intel has a firmware update.

Attack website. News articles. Boing Boing post. Slashdot thread.

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Former FBI General Counsel Jim Baker Chooses Encryption Over Backdoors

In an extraordinary essay, the former FBI general counsel Jim Baker makes the case for strong encryption over government-mandated backdoors:

In the face of congressional inaction, and in light of the magnitude of the threat, it is time for governmental authorities­ — including law enforcement­ — to embrace encryption because it is one of the few mechanisms that the United States and its allies can use to more effectively protect themselves from existential cybersecurity threats, particularly from China. This is true even though encryption will impose costs on society, especially victims of other types of crime.

[…]

I am unaware of a technical solution that will effectively and simultaneously reconcile all of the societal interests at stake in the encryption debate, such as public safety, cybersecurity and privacy as well as simultaneously fostering innovation and the economic competitiveness of American companies in a global marketplace.

[…]

All public safety officials should think of protecting the cybersecurity of the United States as an essential part of their core mission to protect the American people and uphold the Constitution. And they should be doing so even if there will be real and painful costs associated with such a cybersecurity-forward orientation. The stakes are too high and our current cybersecurity situation too grave to adopt a different approach.

Basically, he argues that the security value of strong encryption greatly outweighs the security value of encryption that can be bypassed. He endorses a “defense dominant” strategy for Internet security.

Keep in mind that Baker led the FBI’s legal case against Apple regarding the San Bernardino shooter’s encrypted iPhone. In writing this piece, Baker joins the growing list of former law enforcement and national security senior officials who have come out in favor of strong encryption over backdoors: Michael Hayden, Michael Chertoff, Richard Clarke, Ash Carter, William Lynn, and Mike McConnell.

Edward Snowden also agrees.

EDITED TO ADD: Good commentary from Cory Doctorow.

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Dark Web Site Taken Down without Breaking Encryption

The US Department of Justice unraveled a dark web child-porn website, leading to the arrest of 337 people in at least 18 countries. This was all accomplished not through any backdoors in communications systems, but by analyzing the bitcoin transactions and following the money:

Welcome to Video made money by charging fees in bitcoin, and gave each user a unique bitcoin wallet address when they created an account. Son operated the site as a Tor hidden service, a dark web site with a special address that helps mask the identity of the site’s host and its location. But Son and others made mistakes that allowed law enforcement to track them. For example, according to the indictment, very basic assessments of the Welcome to Video website revealed two unconcealed IP addresses managed by a South Korean internet service provider and assigned to an account that provided service to Son’s home address. When agents searched Son’s residence, they found the server running Welcome to Video.

To “follow the money,” as officials put it in Wednesday’s press conference, law enforcement agents sent fairly small amounts of bitcoin­ — roughly equivalent at the time to $125 to $290­ — to the bitcoin wallets Welcome to Video listed for payments. Since the bitcoin blockchain leaves all transactions visible and verifiable, they could observe the currency in these wallets being transferred to another wallet. Law enforcement learned from a bitcoin exchange that the second wallet was registered to Son with his personal phone number and one of his personal email addresses.

Remember this the next time some law enforcement official tells us that they’re powerless to investigate crime without breaking cryptography for everyone.

More news articles. The indictment is here. Some of it is pretty horrifying to read.

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NordVPN Breached

There was a successful attack against NordVPN:

Based on the command log, another of the leaked secret keys appeared to secure a private certificate authority that NordVPN used to issue digital certificates. Those certificates might be issued for other servers in NordVPN’s network or for a variety of other sensitive purposes. The name of the third certificate suggested it could also have been used for many different sensitive purposes, including securing the server that was compromised in the breach.

The revelations came as evidence surfaced suggesting that two rival VPN services, TorGuard and VikingVPN, also experienced breaches that leaked encryption keys. In a statement, TorGuard said a secret key for a transport layer security certificate for *.torguardvpnaccess.com was stolen. The theft happened in a 2017 server breach. The stolen data related to a squid proxy certificate.

TorGuard officials said on Twitter that the private key was not on the affected server and that attackers “could do nothing with those keys.” Monday’s statement went on to say TorGuard didn’t remove the compromised server until early 2018. TorGuard also said it learned of VPN breaches last May, “and in a related development we filed a legal complaint against NordVPN.”

The breach happened nineteen months ago, but the company is only just disclosing it to the public. We don’t know exactly what was stolen and how it affects VPN security. More details are needed.

VPNs are a shadowy world. We use them to protect our Internet traffic when we’re on a network we don’t trust, but we’re forced to trust the VPN instead. Recommendations are hard. NordVPN’s website says that the company is based in Panama. Do we have any reason to trust it at all?

I’m curious what VPNs others use, and why they should be believed to be trustworthy.

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Calculating the Benefits of the Advanced Encryption Standard

NIST has completed a study — it was published last year, but I just saw it recently — calculating the costs and benefits of the Advanced Encryption Standard.

From the conclusion:

The result of performing that operation on the series of cumulated benefits extrapolated for the 169 survey respondents finds that present value of benefits from today’s perspective is approximately $8.9 billion. On the other hand, the present value of NIST’s costs from today’s perspective is $127 million. Thus, the NPV from today’s perspective is $8,772,000,000; the B/C ratio is therefore 70.2/1; and a measure (explained in detail in Section 6.1) of the IRR for the alternative investment perspective is 31%; all are indicators of a substantial economic impact.

Extending the approach of looking back from 2017 to the larger national economy required the selection of economic sectors best represented by the 169 survey respondents. The economic sectors represented by ten or more survey respondents include the following: agriculture; construction; manufacturing; retail trade; transportation and warehousing; information; real estate rental and leasing; professional, scientific, and technical services; management services; waste management; educational services; and arts and entertainment. Looking at the present value of benefits and costs from 2017’s perspective for these economic sectors finds that the present value of benefits rises to approximately $251 billion while the present value of NIST’s costs from today’s perspective remains the same at $127 million. Therefore, the NPV of the benefits of the AES program to the national economy from today’s perspective is $250,473,200,000; the B/C ratio is roughly 1976/1; and the appropriate, alternative (explained in Section 6.1) IRR and investing proceeds at the social rate of return is 53.6%.

The report contains lots of facts and figures relevant to crypto policy debates, including the chaotic nature of crypto markets in the mid-1990s, the number of approved devices and libraries of various kinds since then, other standards that invoke AES, and so on.

There’s a lot to argue with about the methodology and the assumptions. I don’t know if I buy that the benefits of AES to the economy are in the billions of dollars, mostly because we in the cryptographic community would have come up with alternative algorithms to triple-DES that would have been accepted and used. Still, I like seeing this kind of analysis about security infrastructure. Security is an enabling technology; it doesn’t do anything by itself, but instead allows all sorts of things to be done. And I certainly agree that the benefits of a standardized encryption algorithm that we all trust and use outweigh the cost by orders of magnitude.

And this isn’t the first time NIST has conducted economic impact studies. It released a study of the economic impact of DES in 2001.

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Factoring 2048-bit Numbers Using 20 Million Qubits

This theoretical paper shows how to factor 2048-bit RSA moduli with a 20-million qubit quantum computer in eight hours. It’s interesting work, but I don’t want overstate the risk.

We know from Shor’s Algorithm that both factoring and discrete logs are easy to solve on a large, working quantum computer. Both of those are currently beyond our technological abilities. We barely have quantum computers with 50 to 100 qubits. Extending this requires advances not only in the number of qubits we can work with, but in making the system stable enough to read any answers. You’ll hear this called “error rate” or “coherence” — this paper talks about “noise.”

Advances are hard. At this point, we don’t know if they’re “send a man to the moon” hard or “faster-than-light travel” hard. If I were guessing, I would say they’re the former, but still harder than we can accomplish with our current understanding of physics and technology.

I write about all this generally, and in detail, here. (Short summary: Our work on quantum-resistant algorithms is outpacing our work on quantum computers, so we’ll be fine in the short run. But future theoretical work on quantum computing could easily change what “quantum resistant” means, so it’s possible that public-key cryptography will simply not be possible in the long run. That’s not terrible, though; we have a lot of good scalable secret-key systems that do much the same things.)

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More Cryptanalysis of Solitaire

In 1999, I invented the Solitaire encryption algorithm, designed to manually encrypt data using a deck of cards. It was written into the plot of Neal Stephenson’s novel Cryptonomicon, and I even wrote an afterward to the book describing the cipher.

I don’t talk about it much, mostly because I made a dumb mistake that resulted in the algorithm not being reversible. Still, for the short message lengths you’re likely to use a manual cipher for, it’s still secure and will likely remain secure.

Here’s some new cryptanalysis:

Abstract: The Solitaire cipher was designed by Bruce Schneier as a plot point in the novel Cryptonomicon by Neal Stephenson. The cipher is intended to fit the archetype of a modern stream cipher whilst being implementable by hand using a standard deck of cards with two jokers. We find a model for repetitions in the keystream in the stream cipher Solitaire that accounts for the large majority of the repetition bias. Other phenomena merit further investigation. We have proposed modifications to the cipher that would reduce the repetition bias, but at the cost of increasing the complexity of the cipher (probably beyond the goal of allowing manual implementation). We have argued that the state update function is unlikely to lead to cycles significantly shorter than those of a random bijection.

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The Doghouse: Crown Sterling

A decade ago, the Doghouse was a regular feature in both my email newsletter Crypto-Gram and my blog. In it, I would call out particularly egregious — and amusing — examples of cryptographic “snake oil.”

I dropped it both because it stopped being fun and because almost everyone converged on standard cryptographic libraries, which meant standard non-snake-oil cryptography. But every so often, a new company comes along that is so ridiculous, so nonsensical, so bizarre, that there is nothing to do but call it out.

Crown Sterling is complete and utter snake oil. The company sells “TIME AI,” “the world’s first dynamic ‘non-factor’ based quantum AI encryption software,” “utilizing multi-dimensional encryption technology, including time, music’s infinite variability, artificial intelligence, and most notably mathematical constancies to generate entangled key pairs.” Those sentence fragments tick three of my snake-oil warning signs — from 1999! — right there: pseudo-math gobbledygook (warning sign #1), new mathematics (warning sign #2), and extreme cluelessness (warning sign #4).

More: “In March of 2019, Grant identified the first Infinite Prime Number prediction pattern, where the discovery was published on Cornell University’s www.arXiv.org titled: ‘Accurate and Infinite Prime Number Prediction from Novel Quasi-Prime Analytical Methodology.’ The paper was co-authored by Physicist and Number Theorist Talal Ghannam PhD. The discovery challenges today’s current encryption framework by enabling the accurate prediction of prime numbers.” Note the attempt to leverage Cornell’s reputation, even though the preprint server is not peer-reviewed and allows anyone to upload anything. (That should be another warning sign: undeserved appeals to authority.) PhD student Mark Carney took the time to refute it. Most of it is wrong, and what’s right isn’t new.

I first encountered the company earlier this year. In January, Tom Yemington from the company emailed me, asking to talk. “The founder and CEO, Robert Grant is a successful healthcare CEO and amateur mathematician that has discovered a method for cracking asymmetric encryption methods that are based on the difficulty of finding the prime factors of a large quasi-prime numbers. Thankfully the newly discovered math also provides us with much a stronger approach to encryption based on entangled-pairs of keys.” Sounds like complete snake-oil, right? I responded as I usually do when companies contact me, which is to tell them that I’m too busy.

In April, a colleague at IBM suggested I talk with the company. I poked around at the website, and sent back: “That screams ‘snake oil.’ Bet you a gazillion dollars they have absolutely nothing of value — and that none of their tech people have any cryptography expertise.” But I thought this might be an amusing conversation to have. I wrote back to Yemington. I never heard back — LinkedIn suggests he left in April — and forgot about the company completely until it surfaced at Black Hat this year.

Robert Grant, president of Crown Sterling, gave a sponsored talk: “The 2019 Discovery of Quasi-Prime Numbers: What Does This Mean For Encryption?” I didn’t see it, but it was widely criticized and heckled. Black Hat was so embarrassed that it removed the presentation from the conference website. (Parts of it remain on the Internet. Here’s a short video from the company, if you want to laugh along with everyone else at terms like “infinite wave conjugations” and “quantum AI encryption.” Or you can read the company’s press release about what happened at Black Hat, or Grant’s Twitter feed.)

Grant has no cryptographic credentials. His bio — on the website of something called the “Resonance Science Foundation” — is all over the place: “He holds several patents in the fields of photonics, electromagnetism, genetic combinatorics, DNA and phenotypic expression, and cybernetic implant technologies. Mr. Grant published and confirmed the existence of quasi-prime numbers (a new classification of prime numbers) and their infinite pattern inherent to icositetragonal geometry.”

Grant’s bio on the Crown Sterling website contains this sentence, absolutely beautiful in its nonsensical use of mathematical terms: “He has multiple publications in unified mathematics and physics related to his discoveries of quasi-prime numbers (a new classification for prime numbers), the world’s first predictive algorithm determining infinite prime numbers, and a unification wave-based theory connecting and correlating fundamental mathematical constants such as Pi, Euler, Alpha, Gamma and Phi.” (Quasi-primes are real, and they’re not new. They’re numbers with only large prime factors, like RSA moduli.)

Near as I can tell, Grant’s coauthor is the mathematician of the company: “Talal Ghannam — a physicist who has self-published a book called The Mystery of Numbers: Revealed through their Digital Root as well as a comic book called The Chronicles of Maroof the Knight: The Byzantine.” Nothing about cryptography.

There seems to be another technical person. Ars Technica writes: “Alan Green (who, according to the Resonance Foundation website, is a research team member and adjunct faculty for the Resonance Academy) is a consultant to the Crown Sterling team, according to a company spokesperson. Until earlier this month, Green — a musician who was ‘musical director for Davy Jones of The Monkees’ — was listed on the Crown Sterling website as Director of Cryptography. Green has written books and a musical about hidden codes in the sonnets of William Shakespeare.”

None of these people have demonstrated any cryptographic credentials. No papers, no research, no nothing. (And, no, self-publishing doesn’t count.)

After the Black Hat talk, Grant — and maybe some of those others — sat down with Ars Technica and spun more snake oil. They claimed that the patterns they found in prime numbers allows them to break RSA. They’re not publishing their results “because Crown Sterling’s team felt it would be irresponsible to disclose discoveries that would break encryption.” (Snake-oil warning sign #7: unsubstantiated claims.) They also claim to have “some very, very strong advisors to the company” who are “experts in the field of cryptography, truly experts.” The only one they name is Larry Ponemon, who is a privacy researcher and not a cryptographer at all.

Enough of this. All of us can create ciphers that we cannot break ourselves, which means that amateur cryptographers regularly produce amateur cryptography. These guys are amateurs. Their math is amateurish. Their claims are nonsensical. Run away. Run, far, far, away.

But be careful how loudly you laugh when you do. Not only is the company ridiculous, it’s litigious as well. It has sued ten unnamed “John Doe” defendants for booing the Black Hat talk. (It also sued Black Hat, which may have more merit. The company paid good money to have its talk presented amongst actual peer-reviewed talks. For Black Hat to remove its nonsense may very well be a breach of contract.)

Maybe Crown Sterling can file a meritless lawsuit against me instead for this post. I’m sure it would think it’d result in all sorts of positive press coverage. (Although any press is good press, so maybe it’s right.) But it I can prevent others from getting taken in by this stuff, it would be a good thing.

EDITED TO ADD: Crown Sterling paid $115K for that Black Hat sponsorship.

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