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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. Recent studies show that hacker groups are becoming more aggressive in attacking worldwide financial organizations. The tactical and strategic studies of hacker attack campaigns and activity patterns performed by financial institution security organizations with the support of the National Security Agency indicates that the security cryptography technologies widely adopted on the Internet need improvement.

The early signs of insufficiency of the older security methods and technology is being addressed by the newer methodologies. The National Institute of Standards and Technology has standardized on a list of fifteen elliptic curves of varying sizes. An innovative new trends sparkle colorful rainbow of Technology News connecting the ends of the world's ocean of innovation.

While conventional public-key cryptosystems (RSA, Diffie-Hellman and DSA) operate directly on large integers, an Elliptic Curve Cryptography (ECC) operates over points on an elliptic curve. Elliptic curves are mathematical constructions from number theory and algebraic geometry, which in recent years have found numerous applications in cryptography.

An elliptic curve can be defined over any field (for example, real, rational, complex), though elliptic curves used in cryptography are mainly defined over finite fields. The set of points on an elliptic curve forms a group under addition, where addition of two points on an elliptic curve is defined according to a set of simple rules.

EMC builds information infrastructures and virtual infrastructures to help people and businesses around the world unleash the power of their digital information. We are an Equal Employment Opportunity employer that values the strength diversity brings to the workplace. A fast cryptographic method between two entities exchanging data via a non-secure communication channel.

Advances in Cryptology—Eurocrypt, International Conference on the Theory and Application of Cryptographic Techniques, DE, Berlin, Springer, May 12, 1996, pp. Knudsen E W: “Elliptic Scalar Multiplication Using Point Halving” Advances in Cryptology—Asiacrypt'99. FIELDThe invention relates to a cryptographic method employed between two entities exchanging information over a non-secure communication channel, for example a cable or radio network, the method assuring the confidentiality and the integrity of information transfer between the two entities.

It is known that if P=(x,y) is on the elliptic curve E, it is possible to define a “product” or “scalar multiplication” of the point P of E by an integer m.

Doubling a chosen point P on this kind of elliptic curve in a Diffie-Hellmann key exchange algorithm is known in the art.

Because E(F2n) is a finite sub-group of E, there exists k??1 such that E(2k) is contained in E(F2n) if and only if k?k?. We next describe how to perform the check, solve the second degree equation and calculate the square root in the algorithm for halving a point rapidly. The time to calculate ? is negligible compared to the time to calculate a multiplication of an inversion in the body. For the check and for solving the second degree equation, we consider F2n as a n-dimensional vectorial space on F2.

For a given x, the equation ?2+?=x has its solutions in F2n if and only if the vector x is in the image of F.

Accordingly, the check can be performed by adding the components of x to which components of w equal to 1 correspond. Application of the principles explained above to scalar multiplication is described below.Let P?E(F2n) be a point of odd order r, c a random integer and m the integer part of log2 (r).

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. Elliptic curves offer major advances on older systems such as increased speed, less memory and smaller key sizes. Over the past three decades, the public key cryptography was considered complete and sufficient for providing reliable security of communications over the Internet and throughout many other forms of electronic messaging and communication streams.

In particular moving to elliptic curve cryptography as a foundation for future Internet security addresses the potential threat posed by eavesdroppers and hackers. From robots to satellites, from cloud computing to internet security, from global positioning navigation to sea exploration, we cover technology news for restless explorer, cycling and hiking enthusiast, runner, swimmer, moto biker, car and truck driver, hunter and dog trainer, power boat and sail yacht captain, marine engineer, geologist and scientist, fisherman and kayaker, geocaching enthusiast and mountain adventurer. EMC offerings in backup and recovery, enterprise content management, unified storage, big data, enterprise storage, data federation, archiving, security, and deduplication help customers move to and build IT trust in their next generation of information management and enable them to offer IT-as-a-Service as part of their journey to cloud computing. The method, for example, forms a common key between two entities (A,B), each having a secret key (a,b) and using a public key (P) formed by a point of an elliptic curve (E), and includes at least multiplying the odd order point (P) by an integer by additions and halving operations. International Conference on the Theory and Applications of Cryptology and Information Security.

A method according to claim 1, further comprising constructing a common key from two secret keys respectively belonging to the aforementioned two entities and a public key consisting of the point P of odd order r of a chosen non-supersingular elliptic curve E.

A method according to claim 7, wherein scalar multiplication using halvings is obtained by the following operations: (e) if said scalar of the multiplication is denoted S, choose m+1 values So . A method according to claim 1, further comprising calculating a signature between two entities based on a pair of permanent keys belonging to one of the entities, one secret (a) and the other public (Q), by scalar multiplication of the secret key (a) by another public key consisting of the point (P) of odd order r of a chosen non-supersingular elliptic curve (E). A method according to claim 1, wherein said integer is decomposed as a set of values using powers of half said order, and said addition and halving operations are implemented dependent on said set of values. The invention relates more particularly to an improvement to cryptosystems employing calculations on an elliptic curve. This operation is known as “point doubling” and is part of an iterative double-and-add process. It is well known in the art that E can be given an abelian group structure by taking the point at infinity as a neutral element.

As Q1 is determined by the equations (i), (ii) and (iii), we have to study the operations used in solving these equations, which are not internal to the body but have their result on a super-body of F2n. We consider the normal basis and the polynomial basis.The normal basis results are known in the art. As the time to calculate a solution of the second degree equation is negligible, the check can be effected as follows: calculate a candidate ? from x and check if ?2+?=x. This is why the time to calculate a square root in a polynomial basis is equivalent to half the time to calculate a multiplication in the body.

One implementation consists of precalculating the matrix representing G in the basis {1, T, . 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. The public key cryptography has been solid foundation for cryptography key management and digital signatures.

The collection of attack weapons commonly used by hackers include viruses, worms, spyware, malware, spam, phishing and pharming. The new elliptic curve algorithms will process stronger elliptic curves over finite fields with large prime moduli of 256, 384, and 521 bits. Point p1 plus point p2 is equal to point p4 = (x,-y), where (x,y) = p3 is the third point on the intersection of the elliptic curve and the line L through p1 and p2. The improvement mainly reduces the calculation time.BACKGROUNDThe Diffie-Hellmann key exchange cryptographic protocol is used to exchange keys securely between two entities. Any such doubling takes time.The slowest part of the Diffie-Hellman key exchange protocol is multiplying an unknown point on the curve by a random scalar.

For a given curve F2n the minimal two-torsion elliptic curves constitute exactly half of the set of elliptic curves defined on F2n. The only possible instance is that of solving the second degree equation (i): we must also calculate a square root to calculate the first coordinate of Q1, but in characteristic-two finding the square root is an operation internal to the body. For a given basis of F2n and the corresponding scalar product there exists a single non-trivial vector orthogonal to all the vectors of Im(F).

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. In key management, public key cryptography is the main method of distributing the secret keys used in major cryptographic algorithms. One of the main security improvements offered by elliptic curve cryptography is its scalability and the better performance of encryption algorithms generating more protected keys for larger distributed environments. The United States, the United Kingdom, Canada and other members of NATO are incorporating new forms of elliptic curve cryptography for future Internet based and electronic communication systems to protect classified information throughout and between their governments.

The addition operation in an elliptic curve is the counterpart to modular multiplication in common public-key cryptosystems, and multiple addition is the counterpart to modular exponentiation.

This is why, although it is not totally general, the fastest version of the method described applies to a good proportion of the curves in interest in cryptography.

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. For digital signatures, public key cryptography is used to authenticate the origin of data and protect the integrity of that data while in transit. While Internet commerce and banking services grow within cloud computing environments, at the same time allowing access to greater computing resources for massive audiences of mobile users Elliptic Curve Cryptography begin to offer dramatic improvements and cost savings over the older, first generation public key cryptography techniques and methods.

The Cryptographic Modernization Initiative in the US Department of Defense aims at replacing over a million of existing cryptographic units over the next decade while developing the highly secure Global Information Grid communication network. Elliptic curves are covered in more recent texts on cryptography, including an informative text by Koblitz [Kob94]. In a normal basis, the square root is calculated by a left circular shift and squaring is effected by a right circular shift. 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. Internet communication fabric has been secured by the first generation of public key cryptographic algorithms developed twenty to thirty years ago. The functionality of mobile user communicator and mobile personal assistant devices are exploding, widely utilizing the global positioning systems GPS navigational services, and, therefore the security threats could affect vital aspects of the positioning information provided by the GPS satellite constellation to the mobile GPS navigator, car navigation system or Marine Autopilot GPS enabled electronic compass.

Most of the new installations of the networking infrastructure for the cloud computing commercial, financial and government institutions include the new generation of cryptographic equipment and algorithms that use elliptic curve cryptography for key management and digital signatures. However, it is obviously preferable, at least in principle, to be able to choose the curve to be used from a class of curves that is as general as possible. In characteristic-two, the multiplication of a matrix by a vector is reduced to adding columns of the matrix to which a component of the vector equal to 1 corresponds. Each turn the observer sees a new pseudo-random location for P1, and that’s the output of the game. Notably, they form the basis for key management and authentication for Internet Protocol encryption, web traffic secure socket layer protection and securing of electronic mail.

In choosing an elliptic curve as the foundation of a public key system there are a variety of different choices. The fastest version of the method in accordance with the invention is applied to half the elliptic curves. 1 is a graph showing a very particular elliptic curve that can be represented geometrically and is used hereinafter to explain elementary operations employed in the context of the invention;FIG. As cloud computing backbone brings new suites of web services capable to scale towards millions of new mobile users the traditional security scalability becomes a bottleneck.

2 is a diagram showing exchanges of information in accordance with the invention between two entities;FIGS. Before the theory of the method is described, the basic concepts are reviewed.For simplicity, consider the elliptic curve (E) that can be represented geometrically and is defined for the set R of real numbers by the equation y2+y=x3?x2 shown in FIG. 7 is a block diagram of another system for exchanging information between two entities A and B which can employ a cryptographic method according to the invention.DETAILED DESCRIPTIONWe will show how to calculate [?] P?G from P?G.

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. Recent studies show that hacker groups are becoming more aggressive in attacking worldwide financial organizations. The tactical and strategic studies of hacker attack campaigns and activity patterns performed by financial institution security organizations with the support of the National Security Agency indicates that the security cryptography technologies widely adopted on the Internet need improvement.

The early signs of insufficiency of the older security methods and technology is being addressed by the newer methodologies. The National Institute of Standards and Technology has standardized on a list of fifteen elliptic curves of varying sizes. An innovative new trends sparkle colorful rainbow of Technology News connecting the ends of the world's ocean of innovation.

While conventional public-key cryptosystems (RSA, Diffie-Hellman and DSA) operate directly on large integers, an Elliptic Curve Cryptography (ECC) operates over points on an elliptic curve. Elliptic curves are mathematical constructions from number theory and algebraic geometry, which in recent years have found numerous applications in cryptography.

An elliptic curve can be defined over any field (for example, real, rational, complex), though elliptic curves used in cryptography are mainly defined over finite fields. The set of points on an elliptic curve forms a group under addition, where addition of two points on an elliptic curve is defined according to a set of simple rules.

EMC builds information infrastructures and virtual infrastructures to help people and businesses around the world unleash the power of their digital information. We are an Equal Employment Opportunity employer that values the strength diversity brings to the workplace. A fast cryptographic method between two entities exchanging data via a non-secure communication channel.

Advances in Cryptology—Eurocrypt, International Conference on the Theory and Application of Cryptographic Techniques, DE, Berlin, Springer, May 12, 1996, pp. Knudsen E W: “Elliptic Scalar Multiplication Using Point Halving” Advances in Cryptology—Asiacrypt'99. FIELDThe invention relates to a cryptographic method employed between two entities exchanging information over a non-secure communication channel, for example a cable or radio network, the method assuring the confidentiality and the integrity of information transfer between the two entities.

It is known that if P=(x,y) is on the elliptic curve E, it is possible to define a “product” or “scalar multiplication” of the point P of E by an integer m.

Doubling a chosen point P on this kind of elliptic curve in a Diffie-Hellmann key exchange algorithm is known in the art.

Because E(F2n) is a finite sub-group of E, there exists k??1 such that E(2k) is contained in E(F2n) if and only if k?k?. We next describe how to perform the check, solve the second degree equation and calculate the square root in the algorithm for halving a point rapidly. The time to calculate ? is negligible compared to the time to calculate a multiplication of an inversion in the body. For the check and for solving the second degree equation, we consider F2n as a n-dimensional vectorial space on F2.

For a given x, the equation ?2+?=x has its solutions in F2n if and only if the vector x is in the image of F.

Accordingly, the check can be performed by adding the components of x to which components of w equal to 1 correspond. Application of the principles explained above to scalar multiplication is described below.Let P?E(F2n) be a point of odd order r, c a random integer and m the integer part of log2 (r).

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. Elliptic curves offer major advances on older systems such as increased speed, less memory and smaller key sizes. Over the past three decades, the public key cryptography was considered complete and sufficient for providing reliable security of communications over the Internet and throughout many other forms of electronic messaging and communication streams.

In particular moving to elliptic curve cryptography as a foundation for future Internet security addresses the potential threat posed by eavesdroppers and hackers. From robots to satellites, from cloud computing to internet security, from global positioning navigation to sea exploration, we cover technology news for restless explorer, cycling and hiking enthusiast, runner, swimmer, moto biker, car and truck driver, hunter and dog trainer, power boat and sail yacht captain, marine engineer, geologist and scientist, fisherman and kayaker, geocaching enthusiast and mountain adventurer. EMC offerings in backup and recovery, enterprise content management, unified storage, big data, enterprise storage, data federation, archiving, security, and deduplication help customers move to and build IT trust in their next generation of information management and enable them to offer IT-as-a-Service as part of their journey to cloud computing. The method, for example, forms a common key between two entities (A,B), each having a secret key (a,b) and using a public key (P) formed by a point of an elliptic curve (E), and includes at least multiplying the odd order point (P) by an integer by additions and halving operations. International Conference on the Theory and Applications of Cryptology and Information Security.

A method according to claim 1, further comprising constructing a common key from two secret keys respectively belonging to the aforementioned two entities and a public key consisting of the point P of odd order r of a chosen non-supersingular elliptic curve E.

A method according to claim 7, wherein scalar multiplication using halvings is obtained by the following operations: (e) if said scalar of the multiplication is denoted S, choose m+1 values So . A method according to claim 1, further comprising calculating a signature between two entities based on a pair of permanent keys belonging to one of the entities, one secret (a) and the other public (Q), by scalar multiplication of the secret key (a) by another public key consisting of the point (P) of odd order r of a chosen non-supersingular elliptic curve (E). A method according to claim 1, wherein said integer is decomposed as a set of values using powers of half said order, and said addition and halving operations are implemented dependent on said set of values. The invention relates more particularly to an improvement to cryptosystems employing calculations on an elliptic curve. This operation is known as “point doubling” and is part of an iterative double-and-add process. It is well known in the art that E can be given an abelian group structure by taking the point at infinity as a neutral element.

As Q1 is determined by the equations (i), (ii) and (iii), we have to study the operations used in solving these equations, which are not internal to the body but have their result on a super-body of F2n. We consider the normal basis and the polynomial basis.The normal basis results are known in the art. As the time to calculate a solution of the second degree equation is negligible, the check can be effected as follows: calculate a candidate ? from x and check if ?2+?=x. This is why the time to calculate a square root in a polynomial basis is equivalent to half the time to calculate a multiplication in the body.

One implementation consists of precalculating the matrix representing G in the basis {1, T, . 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. The public key cryptography has been solid foundation for cryptography key management and digital signatures.

The collection of attack weapons commonly used by hackers include viruses, worms, spyware, malware, spam, phishing and pharming. The new elliptic curve algorithms will process stronger elliptic curves over finite fields with large prime moduli of 256, 384, and 521 bits. Point p1 plus point p2 is equal to point p4 = (x,-y), where (x,y) = p3 is the third point on the intersection of the elliptic curve and the line L through p1 and p2. The improvement mainly reduces the calculation time.BACKGROUNDThe Diffie-Hellmann key exchange cryptographic protocol is used to exchange keys securely between two entities. Any such doubling takes time.The slowest part of the Diffie-Hellman key exchange protocol is multiplying an unknown point on the curve by a random scalar.

For a given curve F2n the minimal two-torsion elliptic curves constitute exactly half of the set of elliptic curves defined on F2n. The only possible instance is that of solving the second degree equation (i): we must also calculate a square root to calculate the first coordinate of Q1, but in characteristic-two finding the square root is an operation internal to the body. For a given basis of F2n and the corresponding scalar product there exists a single non-trivial vector orthogonal to all the vectors of Im(F).

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. In key management, public key cryptography is the main method of distributing the secret keys used in major cryptographic algorithms. One of the main security improvements offered by elliptic curve cryptography is its scalability and the better performance of encryption algorithms generating more protected keys for larger distributed environments. The United States, the United Kingdom, Canada and other members of NATO are incorporating new forms of elliptic curve cryptography for future Internet based and electronic communication systems to protect classified information throughout and between their governments.

The addition operation in an elliptic curve is the counterpart to modular multiplication in common public-key cryptosystems, and multiple addition is the counterpart to modular exponentiation.

This is why, although it is not totally general, the fastest version of the method described applies to a good proportion of the curves in interest in cryptography.

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. For digital signatures, public key cryptography is used to authenticate the origin of data and protect the integrity of that data while in transit. While Internet commerce and banking services grow within cloud computing environments, at the same time allowing access to greater computing resources for massive audiences of mobile users Elliptic Curve Cryptography begin to offer dramatic improvements and cost savings over the older, first generation public key cryptography techniques and methods.

The Cryptographic Modernization Initiative in the US Department of Defense aims at replacing over a million of existing cryptographic units over the next decade while developing the highly secure Global Information Grid communication network. Elliptic curves are covered in more recent texts on cryptography, including an informative text by Koblitz [Kob94]. In a normal basis, the square root is calculated by a left circular shift and squaring is effected by a right circular shift. 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. Internet communication fabric has been secured by the first generation of public key cryptographic algorithms developed twenty to thirty years ago. The functionality of mobile user communicator and mobile personal assistant devices are exploding, widely utilizing the global positioning systems GPS navigational services, and, therefore the security threats could affect vital aspects of the positioning information provided by the GPS satellite constellation to the mobile GPS navigator, car navigation system or Marine Autopilot GPS enabled electronic compass.

Most of the new installations of the networking infrastructure for the cloud computing commercial, financial and government institutions include the new generation of cryptographic equipment and algorithms that use elliptic curve cryptography for key management and digital signatures. However, it is obviously preferable, at least in principle, to be able to choose the curve to be used from a class of curves that is as general as possible. In characteristic-two, the multiplication of a matrix by a vector is reduced to adding columns of the matrix to which a component of the vector equal to 1 corresponds. Each turn the observer sees a new pseudo-random location for P1, and that’s the output of the game. Notably, they form the basis for key management and authentication for Internet Protocol encryption, web traffic secure socket layer protection and securing of electronic mail.

In choosing an elliptic curve as the foundation of a public key system there are a variety of different choices. The fastest version of the method in accordance with the invention is applied to half the elliptic curves. 1 is a graph showing a very particular elliptic curve that can be represented geometrically and is used hereinafter to explain elementary operations employed in the context of the invention;FIG. As cloud computing backbone brings new suites of web services capable to scale towards millions of new mobile users the traditional security scalability becomes a bottleneck.

2 is a diagram showing exchanges of information in accordance with the invention between two entities;FIGS. Before the theory of the method is described, the basic concepts are reviewed.For simplicity, consider the elliptic curve (E) that can be represented geometrically and is defined for the set R of real numbers by the equation y2+y=x3?x2 shown in FIG. 7 is a block diagram of another system for exchanging information between two entities A and B which can employ a cryptographic method according to the invention.DETAILED DESCRIPTIONWe will show how to calculate [?] P?G from P?G.

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