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This is the Discrete logarithm problem in the EC group. These parameters have been constructed from a sha1 hash of a public seed but nobody knows how the seed itself has been chosen. It is an algorithm generating an infinite number of pseudo-random sequences from a single seed, taken in the first step or after an explicit reseed. It is unfortunate that SPA and the presentation from Microsoft use conflicting terminology variable names.

So I will use these variables: : Internal seed value. You can also see in the document two functions: and. If we unroll the inductive feedback loop on the first two generated outputs, we get this:. We can see that is the X coordinate of a point, with 16 bits missing we lost the 2 most significant bytes in the output process. So we have at most 17 bits of bruteforce to do to recover the original point A. If you look carefully at the unrolled algorithm, you will notice that if we know we can calculate and we have all the required information to calculate subsequent and.

All we need to do is to guess a value of A based on a bruteforce approach , multiply it by the secret value d , then multiply the resulting scalar with Q, strip two bytes and publish the output. It is also very interesting that if we learn in a practical attack the first 32 bytes generated by this PRNG, the 30 first bytes give us candidates for A and the remaining two bytes can be used to validate our findings. If the X value had been output on 32 bytes, we would have an one over two chance of success because of the two coexisting points on same coordinate X.

Remember from high school, second degree equations can have two solutions. As you have seen before, for our backdoor to work we need to choose the P and Q points in order to have the secret key to the backdoor. This value can be calculated :. The equation to resolve is where r is the order of the EC.

The value of e is the inverse of d modulo r. We can then use that value to generate Q. You can find the proof of concept code on my github. If you remember, we have the 30 least significant bytes of the X coordinate, that means we need to bruteforce our way into A point candidates. There are then two valid points and where is the opposite of the first value modulo p.

Explanation thanks Rod :. This part is pretty straightforward. We import the estimated x and y values, verify that they are in the curve they should! We then multiply Q with the resulting scalar and we get 30 bytes of the next output. If the two first bytes match, we have successfully guessed the 28 remaining bytes. It is quite obvious in light of the recent revelations from Snowden that this weakness was introduced by purpose by the NSA.

It is very elegant and leaks its complete internal state in only 32 bytes of output, which is very impressive knowing it takes 32 bytes of input as a seed. Subscribe to the Techdirt Daily newsletter. Comment Options: Use markdown. Use plain text. Join the Insider Chat. This feature is only available to registered users. Register or sign in to use it. Sign In Register Preferences.

Wed, Sep 11th pm — Mike Masnick. If you liked this post, you may also be interested in Anonymous Coward , 11 Sep pm. I hope not. Better start fresh with a new international body with ZERO influence from governments.

Anonymous Coward , 12 Sep am. They've been part of the work done on the math, the standards, the software, the hardware, the procedures, everything. Should we conclude that they've only done this once? PRMan , 11 Sep pm. Brent Ashley profile , 11 Sep pm. It's all right there in the name, hidden in plain sight. Of course, the C could also stand for Corrupt. Anonymous , 11 Sep pm.

Me , 11 Sep pm. Even open source isn't completely safe if the NSA is running the show in the sense it might take much longer than it otherwise would for the duplicity to be uncovered. The answer is at a minimum to blackball NSA personnel and alumni. OldMugwump profile , 11 Sep pm.

No, don't blackball them. They'll just go undercover. The real lesson is to trust no one. You must assume everyone is cheating and trying to slip a fast one by. Because some of them are, and you'll never know which ones. Why should we trust you? Oh wait, you already said we shouldn't. But if we are to trust you, then we can't trust you, so how Lawrence D'Oliveiro , 11 Sep pm. This whole Dual EC DRBG debacle never got trusted to the point where it could do much damage, simply because there are too many smart people outside the NSA nowadays, who will find holes no matter how cunningly hidden.

For example, look at the SELinux mandatory access control system built into the Linux kernel. It was primarily written by the NSA. Do we trust it? Been saying for years: Canada is evil. But nobody will listen.

Even their healthcare is sinister. As more is revealed, there are probably other standards that they've had their fingers in. There will be fallout for all the revelations that have come from the Snowden releases. US corporations are going to pay a heavy price for this co-operation voluntary or involuntary before it is all over with. Every release reveals more things that need to be looked into.

The NSA has no real place to hide anymore in the sense of just how deep they've been into gaining access to near everything.

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US Patent US, "Elliptic curve random number generation" to Daniel Brown , Scott Vanstone , which describes the "backdoor" in Dual EC DRBG, teaches how the backdoor can be removed by generating Q the second base point randomly after P the basepoint is known by a mechanism not involving point multiplication, thereby ensuring that that P is not a known multiple of Q which is the "backdoor" , nor that Q is known multiple of P.

And further teaching how known secret relations between P and Q may be used as part of a key escrow system. Can anyone cite such a reference? Their statement to Ars is so clearly misleading, that anyone who knows anything about the subject can easily see it. Slow random number generator to thwart attacks? Gimmi a break, this is not how things are done. Was is supposed to be something like " All searches with quote marks. The subject is most well known due to having a backdoor, it doesn't really matter what technology it's based on.

We need a list of products, operating systems, appliances that use this algorithm. Is MS Windows using it? So the existence of this deal is not disputed, and is therefore not "alleged", and I am removing all "alleged" prefixes for this in the article. Thue talk , 2 January UTC.

The New York Times writes :. Cryptographers have long suspected that the agency planted vulnerabilities in a standard adopted in by the National Institute of Standards and Technology and later by the International Organization for Standardization, which has countries as members.

To me, that sounds like as solid a confirmation as you could, without NSA coming straight out and admitting it good luck waiting for that. All the cryptographers believe it, and NIST itself obviously believe it too. Thue talk , 3 January UTC.

Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:. As of February , "External links modified" talk page sections are no longer generated or monitored by InternetArchiveBot. No special action is required regarding these talk page notices, other than regular verification using the archive tool instructions below.

Editors have permission to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the RfC before doing mass systematic removals. I believe by now that a number of programs use Dual EC, but have changed the bad parameters so it's now secure. Not sure how many programs use the fixed version, or if the fixed version is in any way backward compatible a communication from a fixed program being able to send an encrypted message that can be read by an un-fixed program, or visa versa.

From Wikipedia, the free encyclopedia. Redirected from Talk:Dual elliptic curve deterministic random bit generator. This is not a forum for general discussion of the article's subject. Put new text under old text. Click here to start a new topic. New to Wikipedia? In my entire career in cryptography, I've never seen a vulnerability like this.

It's not the first time the NSA has been accused of installing backdoors. Crypto trapdoors, real and imagined, have been part of NSA lore for decades. In some ways the current controversy echoes the long-ago debate over the first U. Data Encryption Standard in the s. The NSA was widely suspected of weakening DES to make it more crackable by the agency by tinkering with a table of numeric constants called an S-Box and shortening the algorithm's key length.

In , though, the NSA was exonerated when it turned out that the agency had actually changed the S-Box numbers to harden DES against a code-breaking technique that had been known only within NSA at the time. In , another case came up that seemed to confirm suspicions about the NSA. The Baltimore Sun reported that year that the NSA had inserted a backdoor into cryptographic machines made by the respected Swiss company Crypto AG, apparently substantiating longstanding rumors to that effect.

Then in , Microsoft inadvertently kicked off another controversy when it leaked its internal name for a cryptographic signing key built into Windows NT. The key was part of Microsoft's compliance with U. The standard discussed four federally sanctioned random number generators approved for use in encrypting government classified and unclassified-but-sensitive communication. Each of the four algorithms was based on a different cryptographic design family.

One was based on hash functions, one on so-called HMAC hash-based message authentication code , one on block ciphers and the fourth one was based on elliptic curves. Elliptic curve algorithms are based on slightly different mathematics than the more common RSA algorithm, and the NSA believes they're the future of cryptography , asserting that elliptic curve algorithms are smaller, faster and offer better security.

But as Shumow and Ferguson examined the properties of the elliptic curve random number generator in the standard, to determine how to incorporate it into the Windows operating system, a couple of strange things stood out. First, the random number generator was very slow - two to three orders of magnitude slower than another algorithm in the standard.

In non-geek speak, there was a weakness that made the random number generator not so random. Good random number generation is at the core of encryption, and a weak RNG can undo the entire encryption system. Random number generators play a role in creating cryptographic keys, in opening secure communications between users and web sites and in resetting passwords for email accounts.

Without assured randomness, an attacker can predict what the system will generate and undermine the algorithm. Shumow and Ferguson found that the obstacles to predicting what the random number generator would generate were low. The standard, which contained guidelines for implementing the algorithm, included a list of constants — static numbers — that were used in the elliptic curve on which the random number generator was based.

Whoever generated the constants, which served as a kind of public key for the algorithm, could have generated a second set of numbers at the same time — a private key. And, Shumow and Ferguson realized, they could predict this after seeing as few as 32 bytes of output from the generator.

With a very small sample, they could crack the entire encryption system used to secure the output. He could then use it for whatever nefarious purpose he wanted. Or he could publish his result, and render every implementation of the random-number generator completely insecure. No one knew who had produced the constants, but it was assumed that because the NSA had pushed the algorithm into the standard, the agency had generated the numbers. The spy agency might also, then, have generated a secret key.

As a result, developers of web sites and software applications wouldn't use it to help secure their products and systems, he said. The U. Microsoft added support for the standard, including the elliptic curve random-number generator, in a Vista update in February , though it did not make the problematic generator the default algorithm.

Microsoft decided to include the algorithm in its operating system because a major customer was asking for it, because it had been sanctioned by NIST, and because it wasn't going to be enabled as the default algorithm in the system, thus having no impact on other customers. Other major companies, like Cisco and RSA, added it as well.

NIST in fact provides a lengthy list of companies that have included it in their libraries , though the list doesn't say which companies made it the default algorithm in their library or which products have been developed that invoke the algorithm.

A Cisco spokesman told WIRED that the algorithm was implemented in its standard crypto library around mid, a library that is used in more than product lines, but the algorithm is not the default, and the default algorithm cannot be changed by users. The company is currently completing an internal audit of all of its products that leverage the NIST standard. RSA, however, made the algorithm the default in its BSafe toolkit for Java and C developers until this week when it told WIRED that it was changing the default following the renewed controversy over it.

The company sent an advisory to developer customers "strongly" urging them to change the default to one of a number of other random number generator algorithms RSA supports. The company is currently doing an internal review of all of its products to see where the algorithm gets invoked in order to change those.

RSA actually added the algorithm to its libraries in or , before NIST approved it for the standard in and before the government made it a requirement for FIPS certification, says Sam Curry, the company's chief technology officer.

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All we need to do is to guess a value of A based on a bruteforce approach , multiply it by the secret value d , then multiply the resulting scalar with Q, strip two bytes and publish the output. It is also very interesting that if we learn in a practical attack the first 32 bytes generated by this PRNG, the 30 first bytes give us candidates for A and the remaining two bytes can be used to validate our findings.

If the X value had been output on 32 bytes, we would have an one over two chance of success because of the two coexisting points on same coordinate X. Remember from high school, second degree equations can have two solutions. As you have seen before, for our backdoor to work we need to choose the P and Q points in order to have the secret key to the backdoor. This value can be calculated :. The equation to resolve is where r is the order of the EC. The value of e is the inverse of d modulo r.

We can then use that value to generate Q. You can find the proof of concept code on my github. If you remember, we have the 30 least significant bytes of the X coordinate, that means we need to bruteforce our way into A point candidates. There are then two valid points and where is the opposite of the first value modulo p.

Explanation thanks Rod :. This part is pretty straightforward. We import the estimated x and y values, verify that they are in the curve they should! We then multiply Q with the resulting scalar and we get 30 bytes of the next output. If the two first bytes match, we have successfully guessed the 28 remaining bytes. It is quite obvious in light of the recent revelations from Snowden that this weakness was introduced by purpose by the NSA.

It is very elegant and leaks its complete internal state in only 32 bytes of output, which is very impressive knowing it takes 32 bytes of input as a seed. Nevertheless having a whole EC point parameter leaked in the output makes it too easy to distinguish from real random and should never have been made into any specs at all. I do not know the secret value used to compute the Q constant, and thus cannot break the default implementation.

The secret value I speak of is the scalar used to generate Q. How long would it be to bruteforce that value? Finding the value e is equivalent to resolving the discrete logarithm problem in NIST-p For my culture, is there a way to perform the opposite deduction, like, for a given state and maybe additional information , would we be able to predict former generated bit sequences?

Hey, very interesting read. Your email address will not be published. The order r: this is the order of the EC and the total number of points into the group. A Generator point defined by Gx and Gy: This point is considered as the base element of the group. This requires re-building the audited source code with a fully open source compiler and making sure the machine code matches.

Reproducible binaries help demonstrate that a backdoor was not inserted in the program's machine code by a malicious person or compiler. In some cases, even this might not be enough. For example, TrueCrypt, like most cryptographic systems, use the system's random number generator to create secret keys.

If an attacker can control or predict the random numbers produced by a system, they can often break otherwise secure cryptographic algorithms. Any predictability in a system's random number generator can render it vulnerable to attacks. Examples of security systems being bypassed using flaws intentionally created or otherwise in random number generators are very common. Some recent examples:. It's absolutely essential to have an unpredictable source of random numbers in secure systems that rely on them.

If you design a random number generator that allows you to predict the output, and convince someone to use it, you can break their system. This kind of algorithmic backdoor is what we will create in this blog post. The digits of pi are quite random looking but they don't make a very good random number generator because they are predictable. Anyone who knows that someone is using the digits of pi as their source of randomness can use that against them. Convincing someone to use a pi-based random number generator is a difficult challenge.

Many pseudo-random number generators start with a number called a seed. The seed is the starting point for the internal state of the algorithm. The algorithm generates a stream of random numbers using some mathematical operation on the internal state. As long as the seed and the subsequent internal state are kept secret, the pseudo-random numbers output by the algorithm are unpredictable to any observer.

Conversely, anyone who knows the state will be able to predict the output. Every time a program requests random data from the system, Linux returns a cryptographic hash of its internal state using the algorithm SHA This hash function is designed to be one-way, it is easy to compute but very difficult to find the input given an output. It is so difficult, no person has ever published an inversion of a SHA-1 hash without knowing the input beforehand.

This keeps the internal state of the random number generator secret. The random data extracted by the hash function is then mixed back into internal state. Periodically, the hashes of the timestamps of "unpredictable" system events like clicks and key presses are also mixed in. This construction is pretty standard. The internal state is kept secret, data is output via a one-way function, and the internal state is updated by mixing the data back into the state.

At any point, if an attacker can figure out the internal state, they can predict the output. The strategic choices for F and G here are what make this construction safe. You do not lose the randomness in the pool by XOR-ing with something else, entropy always goes up. If F and G were chosen to be two completely independent one-way functions, it would probably still be safe. The key here is in the word independent, but first a sidestep into elliptic curves.

In a previous blog post we gave a gentle introduction to elliptic curve cryptography. We talked about how this class of curves can be used for encryption and digital signature algorithms. We also hinted that elliptic curves could be used for generating random numbers. That is what we we will describe here. The reason elliptic curves are used in cryptography is the strongly one way function they enable. As described previously, there is a geometrically intuitive way to define an arithmetic on the points of an elliptic curve.

Any two points on an elliptic curve can be "dotted" "multiplied" together to get a new point on the curve. Dotting a point with itself any number of times is fast easy to do, but going back to the original point takes a lot of computation. This operation can be used to create a nice and simple one-way function from a point P It's hard to go back from m to n, because that would be enough to solve the elliptic curve discrete logarithm problem, which is thought to be very, very hard to do. The metaphor used in the previous post was that the one way function in elliptic curves is like playing a peculiar game of billiards.

If someone were locked alone in a room they could play a certain number of shots and the ball would end up at a particular location. However, if you entered the room at some point and simply saw the position of the ball it would be very difficult to determine the number of shots the player had taken without playing through the whole game again yourself.

With this billiards analogy, we can think of this random number generator as a new bizarro game of pool. Consider two balls on the infinite elliptic curve billiards table, the yellow ball called P1 and the blue ball called P2. These two balls have specific points on the curve where they start. This is a two person game where one person is called the generator and the other is the observer. The generator has a secret number "n". The generator takes the ball P1 and performs n shots, and lets the observer see its final location.

Then it takes P2 and performs n shots, taking the final location of P2 as a new value for n. Then P1 and P2 are reset to their original location and that's the end of the turn. Each turn the observer sees a new pseudo-random location for P1, and that's the output of the game.

In the Linux random number generator example above, SHA-1 is used as the one-way function. Let's consider what happens when we use our elliptic curve one way function instead. Looking back at the construction for a pseudo-random number generator above, we need to choose two functions to serve as F and G. The elliptic curve one-way function above seems to fit the bill, so let's use the functions defined by two points on the curve, P1 and P2.

Each one-way function is hard to reverse, and if P1 and P2 are chosen randomly, they should be independent. So how do we add a backdoor? The key is to choose P1 and P2 so that to any outside observer they look random and independent, but in reality they have a special relationship that only we know. Suppose we choose P2 to be P1 dotted with itself s times, where s is secret number. Then P1 and P2 are related but it is hard to prove how since finding s requires solving the elliptic curve discrete logarithm problem.

Given an initial state n, let's look at what the output becomes and what the state gets updated to. And since we know s and the output and therefore Q , we can calculate the next internal state of the algorithm. The state is revealed and all subsequent bytes can be predicted. In just one round! Since given P1 and P2, finding s requires solving the discrete logarithm problem, you get to be the only one who knows this mathematical backdoor.

This can be described in the terms of the billiards game from the last section. Remember the output of one turn of the game is the location of P1 after n shots and generator's secret number comes from the location of P2 after n shots. Knowing the value s is like knowing how many shots it takes to go from P1 to P2. This lets the observer cheat at the game.

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Large parts of it are based on basic web technology, which is, itself, a piece of shit. Anyone who things that SSL protects their transactions needs their head examined. Consider SSL: the original protocol included bidirectional authentication. That would have been nice, right? And how did the idea of negotiable encryption options get into the protocol? How did that get into SSL? Back in the day hackers would pop the stack and fire up a root shell and pwn the system.

What if, hypothetically, some agency had a secret exploit and instead of starting a shell they just pulled the server side secret key out of the process memory? A key harvester that used an exploit in this manner would not get noticed by That would be useful, I suspect. Anyway, update your blockchain wallet password.

Having a large number of btc secretly in government hands would also allow them to crater the market on demand — another useful capability. I cannot imagine the NSA would not realize that, either. No shit. As of right now, there are 17,, bitcoins that have already been mined. Of those, somewhere in the region of 4 million have been stolen.

SSL was superseded over 20 years ago. It was proven to be fatally broken in , for those who still supported it at the time actually quite a few systems, due to the reticence of our friends in Redmond. RC was a compromise to allow the use of crypto that the NSA could break easily but which they thought that no one else could break at the time. This was a legitimate fiasco, but happened 20 -ish years later and was never part of any SSL or related standard.

If I were to choose the single worst design decision in web applications as we know them, it would be the lack of persistent transport-layer sessions. If we could have a single transport-layer session that lasted the duration of a user session and that all data related to that user session flowed through, then a lot of problems would either go away entirely or be far easier to manage — including problems related to authentication.

Alas, while there was research into the problem and at least a few draft protocols, none of them seem to have ever caught on. People tend to pick crappy passwords, re-use them, and not store them safely, and applications all too frequently pick crappy session secrets and allow them to be stolen. You must be logged in to post a comment.

Top menu. So the existence of this deal is not disputed, and is therefore not "alleged", and I am removing all "alleged" prefixes for this in the article. Thue talk , 2 January UTC. The New York Times writes :. Cryptographers have long suspected that the agency planted vulnerabilities in a standard adopted in by the National Institute of Standards and Technology and later by the International Organization for Standardization, which has countries as members.

To me, that sounds like as solid a confirmation as you could, without NSA coming straight out and admitting it good luck waiting for that. All the cryptographers believe it, and NIST itself obviously believe it too. Thue talk , 3 January UTC. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information.

I made the following changes:. As of February , "External links modified" talk page sections are no longer generated or monitored by InternetArchiveBot. No special action is required regarding these talk page notices, other than regular verification using the archive tool instructions below. Editors have permission to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the RfC before doing mass systematic removals.

I believe by now that a number of programs use Dual EC, but have changed the bad parameters so it's now secure. Not sure how many programs use the fixed version, or if the fixed version is in any way backward compatible a communication from a fixed program being able to send an encrypted message that can be read by an un-fixed program, or visa versa.

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