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Author Nick Sullivan worked for six years at Apple on many of its most important cryptography efforts before recently joining CloudFlare, where he is a systems engineer.

There has been a lot of news lately about nefarious-sounding backdoors being inserted into cryptographic standards and toolkits.

Backdoors can be inserted by lazy programmers who want to bypass their own security systems for debugging reasons, or they can be created to intentionally weaken a system used by others.

Open source is a great tool for understanding how code works but it is not a cure-all for finding backdoors in software.

The translation step between human programming languages and machine code can also be used to insert a backdoor.

Examples of security systems being bypassed using flaws (intentionally created or otherwise) in random number generators are very common. A broken random number generator in Android allowed attackers to hijack thousands of dollars worth of bitcoins.

The version of OpenSSL on the Debian distribution of the Linux operating system had a random number generator problem that could allow attackers to guess private keys created on these systems.

It’s absolutely essential to have an unpredictable source of random numbers in secure systems that rely on them.

The digits of pi are quite random looking but they don’t make a very good random number generator because they are predictable.

At any point, if an attacker can figure out the internal state, they can predict the output. If F and G were chosen to be two completely independent one-way functions, it would probably still be safe.

The reason elliptic curves are used in cryptography is the strong one-way function they enable. Any two points on an elliptic curve can be “dotted” (“multiplied”) together to get a new point on the curve. 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. With this billiards analogy, we can think of this random number generator as a new bizarro game of pool. Looking back at the construction for a pseudo-random number generator above, we need to choose two functions to serve as F and G. 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. This toy random number generator may seem very simple and the backdoor might even seem obvious.

The values for the points P1 and P2 could have been chosen randomly or they could have been chosen with a deliberate relationship.

Up until recently, Dual EC_DRBG was the default random number generator for several cryptographic products from RSA (the security division of EMC), even though cryptographers have long been skeptical of the algorithm’s design. Even secure cryptographic functions can be weakened if there isn't a good source of randomness.

Security-conscious engineers understand this fact and take pains to make sure that the randomness in their cryptographic systems is truly random.

NASA's Michoud Assembly Facility has been under NASA's umbrella since 1961, but many don't know it's right outside New Orleans. The material on this site may not be reproduced, distributed, transmitted, cached or otherwise used, except with the prior written permission of Conde Nast. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Your answer helped me very quickly but I guess Envious Page needs the reuptation more than you do, and his answer is still as good as yours. Not the answer you're looking for?Browse other questions tagged elliptic-curves or ask your own question. He has a degree in mathematics from the University of Waterloo and a Masters in computer science with a concentration in cryptography from the University of Calgary.

One algorithm, a pseudo-random bit generator, Dual_EC_DRBG, was ratified by the National Institute of Standards and Technology (NIST) in 2007 and is attracting a lot of attention for having a potential backdoor.

This is a technical primer that explains what a backdoor is, how easy it can be to create your own, and the dangerous consequences of using a random number generator that was designed to have a backdoor.

A backdoor is an intentional flaw in a cryptographic algorithm or implementation that allows an individual to bypass the security mechanisms the system was designed to enforce. Government agencies have been known to insert backdoors into commonly used software to enable mass surveillance. It can be difficult and time-consuming to fully analyze all the code in a complicated codebase. For example, TrueCrypt, like most cryptographic systems, uses the system’s random number generator to create secret keys. Anyone who knows that someone is using the digits of pi as their source of randomness can use that against them. Every time a program requests random data from the system, Linux returns a cryptographic hash of its internal state using the algorithm SHA-1. Periodically, the hashes of the timestamps of “unpredictable” system events like clicks and key presses are also mixed in. 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.

Having SHA-1 as F and MD5 (a different hash function) as G would not be too unreasonable of a choice. We talked about how this class of curves can be used for encryption and digital signature algorithms. As described previously, there is a geometrically intuitive way to define an arithmetic on the points of an elliptic curve.

Dotting a point with itself any number of times is fast and easy to do, but going back to the original point takes a lot of computation.

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.

Consider two balls on the infinite elliptic curve billiards table, the yellow ball called P1 and the blue ball called P2. This is a two person game where one person is called the generator and the other is the observer. 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.

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. Then P1 and P2 are related but it is hard to prove how since finding s requires solving the elliptic curve discrete logarithm problem. 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. The amazing fact is that our toy random number generator described above is Dual EC_DRBG, almost exactly. This could have easily been done by choosing P1 and P2 as outputs of a hash function, but they did not. A working proof of concept backdoor was published in late 2013 using OpenSSL, and a patent for using the construction as “key escrow” (another term for backdoor) was filed back in 2006. There are reports of impropriety connecting a $10 million investment by the United States government and RSA’s decision to use this obscure and widely maligned algorithm in their widely distributed products. Steps include extracting entropy from the physical world, monitoring system entropy levels, using a hardware random number generator to mix in extra entropy, and not relying on a single random number generator as the source of all randomness.

The variable $a$ is the vertical translation of the curve, $b$ is the $x$-coordinate of the intersection, and $c$ is the $x$-coordinate of the loop. It only describes curves where the crossing is at $(0,0)$, not their translations to the sides.

Elliptic curves offer major advances on older systems such as increased speed, less memory and smaller key sizes. This post was originally written for the CloudFlare blog and has been lightly edited to appear on Ars.

This is the algorithm that the NSA reportedly paid RSA $10 million in exchange for making it the default way for its BSAFE crypto toolkit to generated random numbers. This is necessarily a long technical discussion, but hopefully by the end it should be clear why Dual_EC_DRBG has such a bad reputation. A backdoor is a way for someone to get something out of the system that they otherwise would not be able to.

Backdoors can be built into software, hardware, or even built into the design of an algorithm. The International Obfuscated C Code Contest shows how code can be made extremely hard to understand.

The cryptographic community has recently banded together to audit the open source disk encryption software TrueCrypt for backdoors. If an attacker can control or predict the random numbers produced by a system, they can often break otherwise secure cryptographic algorithms. 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. The algorithm generates a stream of random numbers using some mathematical operation on the internal state.

This hash function is designed to be one-way, as it is easy to compute but very difficult to find the input given an output.

You do not lose the randomness in the pool by XORing with something else, because entropy always goes up.

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. Each one-way function is hard to reverse, and if P1 and P2 are chosen randomly, they should be independent.

If they truly were chosen randomly, then finding the internal state is as difficult as breaking elliptic curve cryptography. As digital signatures become more and more important in the commercial world the use of elliptic curve-based signatures will become all pervasive. That backdoor allows anyone with knowledge of a secret user agent string to log in and modify settings on any router running the vulnerable software. The Underhanded C Contest takes this even further, showing that benign looking code can hide malicious behavior.

One of the key steps in this audit is verifying that the machine code distributed online for TrueCrypt matches the source code.

Any predictability in a system’s random number generator can render it vulnerable to attacks. 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. It is so difficult, no person has ever published an inversion of a SHA-1 hash without knowing the input beforehand.

The generator takes the ball P1 and performs n shots, and lets the observer see its final location. 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. Unfortunately, there is no way to identify if the two points were chosen together or randomly without either solving the elliptic curve discrete logarithm function, or catching the algorithm’s author with the secret backdoor value. This book summarizes knowledge built up within Hewlett-Packard over a number of years, and explains the mathematics behind practical implementations of elliptic curve systems. The D-Link backdoor took a long time to find because the source code for the router software was not available to security researchers to examine.

This requires re-building the audited source code with a fully open source compiler and making sure the machine code matches.

Then it takes P2 and performs n shots, taking the final location of P2 as a new value for n. If you know where P1 lands after n shots, you can shoot s times from that location to get the location of P2 after n shots. Due to the advanced nature of the mathematics there is a high barrier to entry for individuals and companies to this technology. With open source software, a researcher can look directly at the part of the code that verifies authentication and check for backdoors. Reproducible binaries help demonstrate that a backdoor was not inserted in the program’s machine code by a malicious person or compiler.

This gives you the generator’s secret number and allows you to predict the next turn of the game.

Hence this book will be invaluable not only to mathematicians wanting to see how pure mathematics can be applied but also to engineers and computer scientists wishing (or needing) to actually implement such systems.

Each turn the observer sees a new pseudo-random location for P1, and that’s the output of the game.

There has been a lot of news lately about nefarious-sounding backdoors being inserted into cryptographic standards and toolkits.

Backdoors can be inserted by lazy programmers who want to bypass their own security systems for debugging reasons, or they can be created to intentionally weaken a system used by others.

Open source is a great tool for understanding how code works but it is not a cure-all for finding backdoors in software.

The translation step between human programming languages and machine code can also be used to insert a backdoor.

Examples of security systems being bypassed using flaws (intentionally created or otherwise) in random number generators are very common. A broken random number generator in Android allowed attackers to hijack thousands of dollars worth of bitcoins.

The version of OpenSSL on the Debian distribution of the Linux operating system had a random number generator problem that could allow attackers to guess private keys created on these systems.

It’s absolutely essential to have an unpredictable source of random numbers in secure systems that rely on them.

The digits of pi are quite random looking but they don’t make a very good random number generator because they are predictable.

At any point, if an attacker can figure out the internal state, they can predict the output. If F and G were chosen to be two completely independent one-way functions, it would probably still be safe.

The reason elliptic curves are used in cryptography is the strong one-way function they enable. Any two points on an elliptic curve can be “dotted” (“multiplied”) together to get a new point on the curve. 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. With this billiards analogy, we can think of this random number generator as a new bizarro game of pool. Looking back at the construction for a pseudo-random number generator above, we need to choose two functions to serve as F and G. 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. This toy random number generator may seem very simple and the backdoor might even seem obvious.

The values for the points P1 and P2 could have been chosen randomly or they could have been chosen with a deliberate relationship.

Up until recently, Dual EC_DRBG was the default random number generator for several cryptographic products from RSA (the security division of EMC), even though cryptographers have long been skeptical of the algorithm’s design. Even secure cryptographic functions can be weakened if there isn't a good source of randomness.

Security-conscious engineers understand this fact and take pains to make sure that the randomness in their cryptographic systems is truly random.

NASA's Michoud Assembly Facility has been under NASA's umbrella since 1961, but many don't know it's right outside New Orleans. The material on this site may not be reproduced, distributed, transmitted, cached or otherwise used, except with the prior written permission of Conde Nast. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Your answer helped me very quickly but I guess Envious Page needs the reuptation more than you do, and his answer is still as good as yours. Not the answer you're looking for?Browse other questions tagged elliptic-curves or ask your own question. He has a degree in mathematics from the University of Waterloo and a Masters in computer science with a concentration in cryptography from the University of Calgary.

One algorithm, a pseudo-random bit generator, Dual_EC_DRBG, was ratified by the National Institute of Standards and Technology (NIST) in 2007 and is attracting a lot of attention for having a potential backdoor.

This is a technical primer that explains what a backdoor is, how easy it can be to create your own, and the dangerous consequences of using a random number generator that was designed to have a backdoor.

A backdoor is an intentional flaw in a cryptographic algorithm or implementation that allows an individual to bypass the security mechanisms the system was designed to enforce. Government agencies have been known to insert backdoors into commonly used software to enable mass surveillance. It can be difficult and time-consuming to fully analyze all the code in a complicated codebase. For example, TrueCrypt, like most cryptographic systems, uses the system’s random number generator to create secret keys. Anyone who knows that someone is using the digits of pi as their source of randomness can use that against them. Every time a program requests random data from the system, Linux returns a cryptographic hash of its internal state using the algorithm SHA-1. Periodically, the hashes of the timestamps of “unpredictable” system events like clicks and key presses are also mixed in. 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.

Having SHA-1 as F and MD5 (a different hash function) as G would not be too unreasonable of a choice. We talked about how this class of curves can be used for encryption and digital signature algorithms. As described previously, there is a geometrically intuitive way to define an arithmetic on the points of an elliptic curve.

Dotting a point with itself any number of times is fast and easy to do, but going back to the original point takes a lot of computation.

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.

Consider two balls on the infinite elliptic curve billiards table, the yellow ball called P1 and the blue ball called P2. This is a two person game where one person is called the generator and the other is the observer. 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.

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. Then P1 and P2 are related but it is hard to prove how since finding s requires solving the elliptic curve discrete logarithm problem. 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. The amazing fact is that our toy random number generator described above is Dual EC_DRBG, almost exactly. This could have easily been done by choosing P1 and P2 as outputs of a hash function, but they did not. A working proof of concept backdoor was published in late 2013 using OpenSSL, and a patent for using the construction as “key escrow” (another term for backdoor) was filed back in 2006. There are reports of impropriety connecting a $10 million investment by the United States government and RSA’s decision to use this obscure and widely maligned algorithm in their widely distributed products. Steps include extracting entropy from the physical world, monitoring system entropy levels, using a hardware random number generator to mix in extra entropy, and not relying on a single random number generator as the source of all randomness.

The variable $a$ is the vertical translation of the curve, $b$ is the $x$-coordinate of the intersection, and $c$ is the $x$-coordinate of the loop. It only describes curves where the crossing is at $(0,0)$, not their translations to the sides.

Elliptic curves offer major advances on older systems such as increased speed, less memory and smaller key sizes. This post was originally written for the CloudFlare blog and has been lightly edited to appear on Ars.

This is the algorithm that the NSA reportedly paid RSA $10 million in exchange for making it the default way for its BSAFE crypto toolkit to generated random numbers. This is necessarily a long technical discussion, but hopefully by the end it should be clear why Dual_EC_DRBG has such a bad reputation. A backdoor is a way for someone to get something out of the system that they otherwise would not be able to.

Backdoors can be built into software, hardware, or even built into the design of an algorithm. The International Obfuscated C Code Contest shows how code can be made extremely hard to understand.

The cryptographic community has recently banded together to audit the open source disk encryption software TrueCrypt for backdoors. If an attacker can control or predict the random numbers produced by a system, they can often break otherwise secure cryptographic algorithms. 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. The algorithm generates a stream of random numbers using some mathematical operation on the internal state.

This hash function is designed to be one-way, as it is easy to compute but very difficult to find the input given an output.

You do not lose the randomness in the pool by XORing with something else, because entropy always goes up.

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. Each one-way function is hard to reverse, and if P1 and P2 are chosen randomly, they should be independent.

If they truly were chosen randomly, then finding the internal state is as difficult as breaking elliptic curve cryptography. As digital signatures become more and more important in the commercial world the use of elliptic curve-based signatures will become all pervasive. That backdoor allows anyone with knowledge of a secret user agent string to log in and modify settings on any router running the vulnerable software. The Underhanded C Contest takes this even further, showing that benign looking code can hide malicious behavior.

One of the key steps in this audit is verifying that the machine code distributed online for TrueCrypt matches the source code.

Any predictability in a system’s random number generator can render it vulnerable to attacks. 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. It is so difficult, no person has ever published an inversion of a SHA-1 hash without knowing the input beforehand.

The generator takes the ball P1 and performs n shots, and lets the observer see its final location. 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. Unfortunately, there is no way to identify if the two points were chosen together or randomly without either solving the elliptic curve discrete logarithm function, or catching the algorithm’s author with the secret backdoor value. This book summarizes knowledge built up within Hewlett-Packard over a number of years, and explains the mathematics behind practical implementations of elliptic curve systems. The D-Link backdoor took a long time to find because the source code for the router software was not available to security researchers to examine.

This requires re-building the audited source code with a fully open source compiler and making sure the machine code matches.

Then it takes P2 and performs n shots, taking the final location of P2 as a new value for n. If you know where P1 lands after n shots, you can shoot s times from that location to get the location of P2 after n shots. Due to the advanced nature of the mathematics there is a high barrier to entry for individuals and companies to this technology. With open source software, a researcher can look directly at the part of the code that verifies authentication and check for backdoors. Reproducible binaries help demonstrate that a backdoor was not inserted in the program’s machine code by a malicious person or compiler.

This gives you the generator’s secret number and allows you to predict the next turn of the game.

Hence this book will be invaluable not only to mathematicians wanting to see how pure mathematics can be applied but also to engineers and computer scientists wishing (or needing) to actually implement such systems.

Each turn the observer sees a new pseudo-random location for P1, and that’s the output of the game.

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