This post is the third in the series ECC: a gentle introduction. In the previous posts, we have seen what an elliptic curve is and we have defined a group law in order to do some math with the points of elliptic curves. Finally, we have seen that scalar multiplication in finite fields is an “easy” problem, while the discrete logarithm problem seems to be “hard”. Now we’elliptic Curve Cryptography (ECC) see how all of this applies to cryptography.
Domain parameters Our elliptic curve algorithms will work in a cyclic subgroup of an elliptic curve over a finite field. Random curves When I said that the discrete logarithm problem was “hard”, I wasn’t entirely right. There are some classes of elliptic curves that are particularly weak and allow the use of special purpose algorithms to solve the discrete logarithm problem efficiently. Now, suppose that I give you the domain parameters of a curve.
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There’s the possibility that I’ve discovered a new class of weak curves that nobody knows, and probably I have built a “fast” algorithm for computing discrete logarithms on the curve I gave you. How can I convince you of the contrary, i. I’m not aware of any vulnerability? Hashes, as we know, are “easy” to compute, but “hard” to reverse. A simple sketch of how a random curve is generated from a seed: the hash of a random number is used to calculate different parameters of the curve. This trick should give some sort of assurance that the curve has not been specially crafted to expose vulnerabilities known to the author.
The reason why I say “relatively” will be explained in the next post. A standardized algorithm for generating and checking random curves is described in ANSI X9. 62 and is based on SHA-1. I’ve created a tiny Python script that verifies all the random curves currently shipped with OpenSSL. I strongly recommend you to check it out! Elliptic Curve Cryptography It took us a long time, but finally here we are! Encryption with ECDH ECDH is a variant of the Diffie-Hellman algorithm for elliptic curves.
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It is actually a key-agreement protocol, more than an encryption algorithm. How to actually encrypt data using such keys is up to us. This is one of the principles behind TLS, just to give you an example. First, Alice and Bob generate their own private and public keys.
The latter form is used in the original Diffie-Hellman algorithm, based on modular arithmetic. The Diffie-Hellman key exchange: Alice and Bob can “easily” calculate the shared secret, the Man in the Middle has to solve a “hard” problem. The Diffie-Hellman problem for elliptic curves is assumed to be a “hard” problem. It is believed to be as “hard” as the discrete logarithm problem, although no mathematical proofs are available.
What we can tell for sure is that it can’t be “harder”, because solving the logarithm problem is a way of solving the Diffie-Hellman problem. Now that Alice and Bob have obtained the shared secret, they can exchange data with symmetric encryption. Unlike all the examples we have seen till now, this script makes use of a standardized curve, rather than a simple curve on a small field. This same curve is also used by Bitcoin for digital signatures. These numbers were taken from OpenSSL source code. Of course, you are free to modify the script to use other curves and domain parameters, just be sure to use prime fields and curves Weierstrass normal form, otherwise the script won’t work. The script is really simple and includes some of the algorithms we have described so far: point addition, double and add, ECDH.
I recommend you to read and run it. Ephemeral ECDH Some of you may have heard of ECDHE instead of ECDH. The “E” in ECHDE stands for “Ephemeral” and refers to the fact that the keys exchanged are temporary, rather than static. Nobody but Alice should be able to produce valid signatures. Everyone should be able to check signatures. Again, Alice and Bob are using the same domain parameters. The algorithm we are going to see is ECDSA, a variant of the Digital Signature Algorithm applied to elliptic curves.
ECDSA works on the hash of the message, rather than on the message itself. The choice of the hash function is up to us, but it should be obvious that a cryptographically-secure hash function should be chosen. If a subgroup has a non-prime order, ECDSA can’t be used. Correctness of the algorithm The logic behind this algorithm may not seem obvious at a first sight, however if we put together all the equations we have written so far, things will be clearer. 2 of the signature generation algorithm! This is why the algorithm works. Playing with ECDSA Of course, I’ve created a Python script for signature generation and verification.
Lastly, it tries to verify the signature against the correct message, but using another random public key and verification fails again. This is the kind of mistake made by Sony a few years ago. Apparently, Sony’s random number generator was inspired by either XKCD or Dilbert. Have a great weekend I really hope you enjoyed what I’ve written here. As usual, don’t hesitate to leave a comment or send me a poke if you need help with something.
Next week I’ll publish the fourth and last article of this series. It’ll be about techniques for solving discrete logarithms, some important problems of Elliptic Curve cryptography, and how ECC compares with RSA. Those of you who know what public-key cryptography is may have already heard of ECC, ECDH or ECDSA. The first is an acronym for Elliptic Curve Cryptography, the others are names for algorithms based on it. Today, we can find elliptic curves cryptosystems in TLS, PGP and SSH, which are just three of the main technologies on which the modern web and IT world are based.
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Not to mention Bitcoin and other cryptocurrencies. Before ECC become popular, almost all public-key algorithms were based on RSA, DSA, and DH, alternative cryptosystems based on modular arithmetic. RSA and friends are still very important today, and often are used alongside ECC. With a series of blog posts I’m going to give you a gentle introduction to the world of elliptic curve cryptography. ECC is and why it is considered secure, without losing time on long mathematical proofs or boring implementation details. In order to understand what’s written here, you’ll need to know some basic stuff of set theory, geometry and modular arithmetic, and have familiarity with symmetric and asymmetric cryptography.
Lastly, you need to have a clear idea of what an “easy” problem is, what a “hard” problem is, and their roles in cryptography. Elliptic Curves First of all: what is an elliptic curve? The equation above is what is called Weierstrass normal form for elliptic curves. Groups are nice because, if we can demonstrate that those four properties hold, we get some other properties for free. Either directly or indirectly, these and other facts about groups will be very important for us later. The group law for elliptic curves We can define a group over elliptic curves. The sum of three aligned point is 0.
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Note that with the last rule, we only require three aligned points, and three points are aligned without respect to order. But how do we actually compute the sum of two arbitrary points? This geometric method works but needs some refinement. In this case, the line going through the two points is vertical, and does not intersect any third point. In this case, there are infinitely many lines passing through the point. Here things start getting a bit more complicated.
We are in a case very similar to the previous one. If our line intersects just two points, then it means that it’s tangent to the curve. The geometric method is now complete and covers all cases. With a pencil and a ruler we are able to perform addition involving every point of any elliptic curve.
Algebraic addition If we want a computer to perform point addition, we need to turn the geometric method into an algebraic method. Transforming the rules described above into a set of equations may seem straightforward, but actually it can be really tedious because it requires solving cubic equations. For this reason, here I will report only the results. First, let’s get get rid of the most annoying corner cases.
Although the procedure to derive them can be really tedious, our equations are pretty compact. This is thanks to Weierstrass normal form: without it, these equations could have been really long and complicated! I’ve written a visual tool for scalar multiplication too, if you want to play with that. Its principle of operation can be better explained with an example. But what about the other way round? This problem is known as the logarithm problem.
I don’t know of any “easy” algorithm for the logarithm problem, however playing with multiplication it’s easy to see some patterns. But there’s a variant of the logarithm problem: the discrete logarithm problem. As we will see in the next post, if we reduce the domain of our elliptic curves, scalar multiplication remains “easy”, while the discrete logarithm becomes a “hard” problem. This duality is the key brick of elliptic curve cryptography. See you next week That’s all for today, I hope you enjoyed this post!
Next week we will discover finite fields and the discrete logarithm problem, along with examples and tools to play with. If this stuff sounds interesting to you, then stay tuned! Enter the characters you see below Sorry, we just need to make sure you’re not a robot. Since ECC requires fewer bits than RSA to achieve the same cipher strength, it is frequently used in embedded applications. The operations necessary for the ECC cannot be efficiently implemented on an embedded CPU, however, typically requiring hundreds of milliseconds of the CPU time for signature verification. The design is fully synchronous and available in multiple configurations varying in bus widths, set of elliptic curves supported and throughput. The use of elliptic curves in cryptography was suggested independently by Neal Koblitz and Victor S.
The elements of this field are the bit strings of length m, and the field arithmetic is implemented in terms of operations on the bits. The set of points on such a curve — all solutions of the above equation together with a point at infinity — form an Abelian group, with the point at infinity as identity element and a generator element G. P given an integer k and point P. This operation is by far the most computationally intensive, and its hardware implementation improves the performance of the ECC algorithm and power consumed by the device. P indeed belongs to the curve. This is important, because the value of kP for a point P that does not belong to the curve, might reveal the secret random number k. For comparison, the bit length of comparable RSA public key solutions is also provided.
For each field degree m, NIST defined a pseudo-random curve along with a Koblitz curve. The core implements the point multiplication operation and the point verification operations. The operands for the multiplication: k, Px, Py are placed into the shared memory before the start of operation. Once the operation is complete, the results are placed back into the shared memory. Design of ECC1 allows sharing the arbitrated CPU memory to store the arguments and results of operations.
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If sharing memory is not desirable, an optional dedicated memory can be used. The core is subject to the US export regulations. US government sites and licensing details. 2008, All Right Reserved, IP Cores, Inc.
Elliptic curves are applicable for key agreement, digital signatures, pseudo-random generators and other tasks. Indirectly, they can be used for encryption by combining the key agreement with a symmetric encryption scheme. Public-key cryptography is based on the intractability of certain mathematical problems. The primary benefit promised by elliptic curve cryptography is a smaller key size, reducing storage and transmission requirements, i.
While the RSA patent expired in 2000, there may be patents in force covering certain aspects of ECC technology. The use of elliptic curves in cryptography was suggested independently by Neal Koblitz and Victor S. Elliptic curve cryptography algorithms entered wide use in 2004 to 2005. The coordinates here are to be chosen from a fixed finite field of characteristic not equal to 2 or 3, or the curve equation will be somewhat more complicated.
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This set together with the group operation of elliptic curves is an Abelian group, with the point at infinity as identity element. Suite B which exclusively uses ECC for digital signature generation and key exchange. The suite is intended to protect both classified and unclassified national security systems and information. Recently, a large number of cryptographic primitives based on bilinear mappings on various elliptic curve groups, such as the Weil and Tate pairings, have been introduced.
To use ECC, all parties must agree on all the elements defining the elliptic curve, that is, the domain parameters of the scheme. The field is defined by p in the prime case and the pair of m and f in the binary case. The elliptic curve is defined by the constants a and b used in its defining equation. Unless there is an assurance that domain parameters were generated by a party trusted with respect to their use, the domain parameters must be validated before use. The generation of domain parameters is not usually done by each participant because this involves computing the number of points on a curve which is time-consuming and troublesome to implement. As a result, several standard bodies published domain parameters of elliptic curves for several common field sizes.
SECG test vectors are also available. NIST has approved many SECG curves, so there is a significant overlap between the specifications published by NIST and SECG. EC domain parameters may be either specified by value or by name. Select the number of points and generate a curve with this number of points using complex multiplication technique. 112-bit key for the prime field case and a 109-bit key for the binary field case. 5 months using this cluster when running continuously. A current project is aiming at breaking the ECC2K-130 challenge by Certicom, by using a wide range of different hardware: CPUs, GPUs, FPGA.
Fortunately, points on a curve can be represented in different coordinate systems which do not require an inversion operation to add two points. Note that there may be different naming conventions, for example, IEEE P1363-2000 standard uses “projective coordinates” to refer to what is commonly called Jacobian coordinates. An additional speed-up is possible if mixed coordinates are used. Compared to Barrett reduction, there can be an order of magnitude speed-up.
Mersenne p are recommended by NIST. 3, which improves addition in Jacobian coordinates. According to Bernstein and Lange, many of the efficiency-related decisions in NIST FIPS 186-2 are sub-optimal. Other curves are more secure and run just as fast. Elliptic curves are applicable for encryption, digital signatures, pseudo-random generators and other tasks. In 1999, NIST recommended 15 elliptic curves.
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192, 224, 256, 384, and 521 bits. For each of the prime fields, one elliptic curve is recommended. 163, 233, 283, 409, and 571. For each of the binary fields, one elliptic curve and one Koblitz curve was selected.
The NIST recommendation thus contains a total of 5 prime curves and 10 binary curves. The curves were ostensibly chosen for optimal security and implementation efficiency. NIST national standard due to the influence of NSA, which had included a deliberate weakness in the algorithm and the recommended elliptic curve. In his 2014 book, Mastering Bitcoin Andreas Antonopoulos asserts that bitcoin uses “elliptic curve multiplication as the basis for its cryptography”. Cryptographic experts have expressed concerns that the National Security Agency has inserted a kleptographic backdoor into at least one elliptic curve-based pseudo random generator.
Shor’s algorithm can be used to break elliptic curve cryptography by computing discrete logarithms on a hypothetical quantum computer. 2330 qubits and 126 billion Toffoli gates. In August 2015, NSA announced that it planned to transition “in the not distant future” to a new cipher suite that is resistant to quantum attacks. Unfortunately, the growth of elliptic curve use has bumped up against the fact of continued progress in the research on quantum computing, necessitating a re-evaluation of our cryptographic strategy. Commercial National Security Algorithm Suite and Quantum Computing FAQ U. Fact Sheet NSA Suite B Cryptography”. Archived from the original on 2009-02-07.