Elliptic curve cryptography

Elliptic curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. Elliptic curves are also used in several integer factorization algorithms that have applications in cryptography, such as Lenstra elliptic curve factorization.

The use of elliptic curves in cryptography was suggested independently by Neal Koblitz and Victor S. Miller in 1985. Elliptic curve cryptography algorithms entered wide use in 2004 to 2005.

Introduction

Public-key cryptography is based on the intractability of certain mathematical problems. Early public-key systems are secure assuming that it is difficult to factor a large integer composed of two or more large prime factors. For elliptic-curve-based protocols, it is assumed that finding the discrete logarithm of a random elliptic curve element with respect to a publicly known base point is infeasible –this is the "elliptic curve discrete logarithm problem" or ECDLP. The entire security of ECC depends on the ability to compute a point multiplication and the inability to compute the multiplicand given the original and product points. The size of the elliptic curve determines the difficulty of the problem.

The primary benefit promised by ECC is a smaller key size, reducing storage and transmission requirements, i.e. that an elliptic curve group could provide the same level of security afforded by an RSA-based system with a large modulus and correspondingly larger key – e.g., a 256-bit ECC public key should provide comparable security to a 3072-bit RSA public key (see key sizes below).

For current cryptographic purposes, an elliptic curve is a plane curve which consists of the points satisfying the equation

$y^{2}=x^{3}+ax+b,\,$

along with a distinguished point at infinity, denoted ∞. (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.)

This set together with the group operation of the elliptic group theory form an Abelian group, with the point at infinity as identity element. The structure of the group is inherited from the divisor group of the underlying algebraic variety. As for other popular public key cryptosystems, no mathematical proof of security has been published for ECC Template:As of.

The U.S. National Institute of Standards and Technology (NIST) has endorsed ECC by including schemes based on ECC in its Suite B set of recommended algorithms and allows their use for protecting information classified up to top secret with 384-bit keys.Template:Year needed

While the RSA patent expired in 2000, there are patents in force covering certain aspects of ECC technology, though some (including RSA Laboratories and Daniel J. Bernstein) argue that the Federal elliptic curve digital signature standard (ECDSA; NIST FIPS 186-3) and certain practical ECC-based key exchange schemes (including ECDH) can be implemented without infringing them.

Cryptographic schemes

Several discrete logarithm-based protocols have been adapted to elliptic curves, replacing the group $(\mathbb {Z} _{p})^{\times }$ with an elliptic curve:

• the elliptic curve Diffie–Hellman (ECDH) key agreement scheme is based on the Diffie–Hellman scheme,
• the Elliptic Curve Integrated Encryption Scheme (ECIES), also known as Elliptic Curve Augmented Encryption Scheme or simply the Elliptic Curve Encryption Scheme,
• the Elliptic Curve Digital Signature Algorithm (ECDSA) is based on the Digital Signature Algorithm,
• the ECMQV key agreement scheme is based on the MQV key agreement scheme.
• the ECQV implicit certificate scheme.

At the RSA Conference 2005, the National Security Agency (NSA) announced 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. Schemes based on these primitives provide efficient identity-based encryption as well as pairing-based signatures, signcryption, key agreement, and proxy re-encryption.

Implementation considerations

Although the details of each particular elliptic curve scheme are described in the article referenced above, some common implementation considerations are discussed here.

Domain parameters

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. Finally, the cyclic subgroup is defined by its generator (aka. base point) G. For cryptographic application the order of G, that is the smallest non-negative number n such that $nG=\infty$, is normally prime. Since n is the size of a subgroup of $E(\mathbb {F} _{p})$ it follows from Lagrange's theorem that the number $h={\frac {|E(\mathbb {F} _{p})|}{n}}$ is an integer. In cryptographic applications this number h, called the cofactor, must be small ($h\leq 4$) and, preferably, $h=1$. Let us summarize: in the prime case the domain parameters are $(p,a,b,G,n,h)$ and in the binary case they are $(m,f,a,b,G,n,h)$.

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 since this involves counting 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. Such domain parameters are commonly known as "standard curves" or "named curves"; a named curve can be referenced either by name or by the unique object identifier defined in the standard documents:

• NIST, Recommended Elliptic Curves for Government Use
• SECG, SEC 2: Recommended Elliptic Curve Domain Parameters
• ECC Brainpool, ECC Brainpool Standard Curves and Curve Generation

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.

If one (despite the above) wants to construct one's own domain parameters, one should select the underlying field and then use one of the following strategies to find a curve with appropriate (i.e., near prime) number of points using one of the following methods:

• select a random curve and use a general point-counting algorithm, for example, Schoof's algorithm or Schoof–Elkies–Atkin algorithm,
• select a random curve from a family which allows easy calculation of the number of points (e.g., Koblitz curves), or
• select the number of points and generate a curve with this number of points using complex multiplication technique.

Several classes of curves are weak and should be avoided:

• curves over $\mathbb {F} _{2^{m}}$ with non-prime m are vulnerable to Weil descent attacks.
• curves such that n divides $p^{B}-1$ (where p is the characteristic of the field – q for a prime field, or $2$ for a binary field) for sufficiently small B are vulnerable to Menezes-Okamoto-Vanstone (MOV) attack which applies usual Discrete Logarithm Problem (DLP) in a small degree extension field of $\mathbb {F} _{p}$ to solve ECDLP. The bound B should be chosen so that discrete logarithms in the field $\mathbb {F} _{p^{B}}$ are at least as difficult to compute as discrete logs on the elliptic curve $E(\mathbb {F} _{q})$.
• curves such that $|E(\mathbb {F} _{q})|=q$ are vulnerable to the attack that maps the points on the curve to the additive group of $\mathbb {F} _{q}$

Key sizes

DTZ's public sale group in Singapore auctions all forms of residential, workplace and retail properties, outlets, homes, lodges, boarding homes, industrial buildings and development websites. Auctions are at present held as soon as a month.

We will not only get you a property at a rock-backside price but also in an space that you've got longed for. You simply must chill out back after giving us the accountability. We will assure you 100% satisfaction. Since we now have been working in the Singapore actual property market for a very long time, we know the place you may get the best property at the right price. You will also be extremely benefited by choosing us, as we may even let you know about the precise time to invest in the Singapore actual property market.

The Hexacube is offering new ec launch singapore business property for sale Singapore investors want to contemplate. Residents of the realm will likely appreciate that they'll customize the business area that they wish to purchase as properly. This venture represents one of the crucial expansive buildings offered in Singapore up to now. Many investors will possible want to try how they will customise the property that they do determine to buy by means of here. This location has offered folks the prospect that they should understand extra about how this course of can work as well.

Singapore has been beckoning to traders ever since the value of properties in Singapore started sky rocketing just a few years again. Many businesses have their places of work in Singapore and prefer to own their own workplace area within the country once they decide to have a everlasting office. Rentals in Singapore in the corporate sector can make sense for some time until a business has discovered a agency footing. Finding Commercial Property Singapore takes a variety of time and effort but might be very rewarding in the long term.

is changing into a rising pattern among Singaporeans as the standard of living is increasing over time and more Singaporeans have abundance of capital to invest on properties. Investing in the personal properties in Singapore I would like to applaud you for arising with such a book which covers the secrets and techniques and tips of among the profitable Singapore property buyers. I believe many novice investors will profit quite a bit from studying and making use of some of the tips shared by the gurus." – Woo Chee Hoe Special bonus for consumers of Secrets of Singapore Property Gurus Actually, I can't consider one other resource on the market that teaches you all the points above about Singapore property at such a low value. Can you? Condominium For Sale (D09) – Yong An Park For Lease

In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value.

Since all the fastest known algorithms that allow one to solve the ECDLP (baby-step giant-step, Pollard's rho, etc.), need $O({\sqrt {n}})$ steps, it follows that the size of the underlying field should be roughly twice the security parameter. For example, for 128-bit security one needs a curve over $\mathbb {F} _{q}$, where $q\approx 2^{256}$. This can be contrasted with finite-field cryptography (e.g., DSA) which requires 3072-bit public keys and 256-bit private keys, and integer factorization cryptography (e.g., RSA) which requires 3072-bit public and private keys.

The hardest ECC scheme (publicly) broken to date had a 112-bit key for the prime field case and a 109-bit key for the binary field case. For the prime field case this was broken in July 2009 using a cluster of over 200 PlayStation 3 game consoles and could have been finished in 3.5 months using this cluster when running continuously (see ). For the binary field case, it was broken in April 2004 using 2600 computers for 17 months.

A current project is aiming at breaking the ECC2K-130 challenge by Certicom, by using a wide range of different hardware: CPUs, GPUs, FPGA.

Projective coordinates

A close examination of the addition rules shows that in order to add two points one needs not only several additions and multiplications in $\mathbb {F} _{q}$ but also an inversion operation. The inversion (for given $x\in \mathbb {F} _{q}$ find $y\in \mathbb {F} _{q}$ such that $xy=1$) is one to two orders of magnitude slower than multiplication. Fortunately, points on a curve can be represented in different coordinate systems which do not require an inversion operation to add two points. Several such systems were proposed: in the projective system each point is represented by three coordinates $(X,Y,Z)$ using the following relation: $x={\frac {X}{Z}}$, $y={\frac {Y}{Z}}$; in the Jacobian system a point is also represented with three coordinates $(X,Y,Z)$, but a different relation is used: $x={\frac {X}{Z^{2}}}$, $y={\frac {Y}{Z^{3}}}$; in the López–Dahab system the relation is $x={\frac {X}{Z}}$, $y={\frac {Y}{Z^{2}}}$; in the modified Jacobian system the same relations are used but four coordinates are stored and used for calculations $(X,Y,Z,aZ^{4})$; and in the Chudnovsky Jacobian system five coordinates are used $(X,Y,Z,Z^{2},Z^{3})$. 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.

Fast reduction (NIST curves)

Reduction modulo p (which is needed for addition and multiplication) can be executed much faster if the prime p is a pseudo-Mersenne prime that is $p\approx 2^{d}$, for example, $p=2^{521}-1$ or $p=2^{256}-2^{32}-2^{9}-2^{8}-2^{7}-2^{6}-2^{4}-1.$ Compared to Barrett reduction there can be an order of magnitude speedup. The speedup here is a practical rather than theoretical one, and derives from the fact that the moduli of numbers against numbers near powers of two can be performed efficiently by computers operating on binary numbers with bitwise operations.

The curves over $\mathbb {F} _{p}$ with pseudo-Mersenne p are recommended by NIST. Yet another advantage of the NIST curves is the fact that they use a = −3 which improves addition in Jacobian coordinates.

Many of the efficiency-related decisions in NIST FIPS 186-2 are sub-optimal. Other curves are more secure and run just as fast.

NIST-recommended elliptic curves

NIST recommendedTemplate:Year needed fifteen elliptic curves. Specifically, FIPS 186-3 has ten recommended finite fields:

• Five prime fields $\mathbb {F} _{p}$ for certain primes p of sizes 192, 224, 256, 384, and 521 bits. For each of the prime fields, one elliptic curve is recommended.
• Five binary fields $\mathbb {F} _{2^{m}}$ for m equal 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 five prime curves and ten binary curves. The curves were ostensibly chosen for optimal security and implementation efficiency.

In 2013, the New York Times revealed that Dual Elliptic Curve Deterministic Random Bit Generation (or Dual_EC_DRBG) had been included as a NIST national standard due to the influence of NSA, which had included a deliberate weakness in the algorithm and the recommended elliptic curve. RSA Security in September 2013 issued an advisory recommending that its customers discontinue using any software based on Dual_EC_DRBG. In the wake of the exposure of Dual_EC_DRBG as "an NSA undercover operation", cryptography experts have also expressed concern over the security of the NIST recommended elliptic curves, suggesting a return to encryption based on the discrete logarithms.

Side-channel attacks

Unlike DLP systems (where it is possible to use the same procedure for squaring and multiplication) the EC addition is significantly different for doubling ($P=Q$) and general addition ($P\neq Q$) depending on the coordinate system used. Consequently, it is important to counteract side channel attacks (e.g., timing or simple/differential power analysis attacks) using, for example, fixed pattern window (aka. comb) methodsTemplate:Clarify (note that this does not increase the computation time). Another concern for ECC-systems is the danger of fault attacks, especially when running on smart cards.

Cryptographic experts have also expressed concerns that the National Security Agency has inserted a backdoor into at least one elliptic curve-based pseudo random generator. One analysis of the possible backdoor concluded that an adversary in possession of the algorithm's secret key could obtain encryption keys given only 32 bytes of ciphertext.

Quantum computing attacks

Elliptic curve cryptography is vulnerable to a modified Shor's algorithm for solving the discrete logarithm problem on elliptic curves.

Patents

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. At least one ECC scheme (ECMQV) and some implementation techniques are covered by patents.

Alternative representations of elliptic curves

• Hessian curves
• Edwards curves
• Twisted curves
• Twisted Hessian curves
• Twisted Edwards curve
• Doubling-oriented Doche–Icart–Kohel curve
• Tripling-oriented Doche–Icart–Kohel curve
• Jacobian curve
• Montgomery curve

See also

• DNSCurve
• ECC patents
• ECDH
• ECDSA
• ECMQV
• Curve25519
• Public-key cryptography
• Quantum cryptography
• Pairing-based cryptography
• Homomorphic Signatures for Network Coding
• Elliptic Curve Point Multiplication

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

• Standards for Efficient Cryptography Group (SECG), SEC 1: Elliptic Curve Cryptography, Version 1.0, September 20, 2000.
• D. Hankerson, A. Menezes, and S.A. Vanstone, Guide to Elliptic Curve Cryptography, Springer-Verlag, 2004.
• I. Blake, G. Seroussi, and N. Smart, Elliptic Curves in Cryptography, London Mathematical Society 265, Cambridge University Press, 1999.
• I. Blake, G. Seroussi, and N. Smart, editors, Advances in Elliptic Curve Cryptography, London Mathematical Society 317, Cambridge University Press, 2005.
• L. Washington, Elliptic Curves: Number Theory and Cryptography, Chapman & Hall / CRC, 2003.
• The Case for Elliptic Curve Cryptography, National Security Agency
• Online Elliptic Curve Cryptography Tutorial, Certicom Corp.
• K. Malhotra, S. Gardner, and R. Patz, Implementation of Elliptic-Curve Cryptography on Mobile Healthcare Devices, Networking, Sensing and Control, 2007 IEEE International Conference on, London, 15–17 April 2007 Page(s):239–244
• Saikat Basu, A New Parallel Window-Based Implementation of the Elliptic Curve Point Multiplication in Multi-Core Architectures, International Journal of Network Security, Vol. 13, No. 3, 2011, Page(s):234-241
• Christof Paar, Jan Pelzl, "Elliptic Curve Cryptosystems", Chapter 9 of "Understanding Cryptography, A Textbook for Students and Practitioners". (companion web site contains online cryptography course that covers elliptic curve cryptography), Springer, 2009.

External links

Most brokers paid and post their listings at these online property categorised portals but fail to realise that there is simply extra to it. They fail to leverage on one of the vital well-liked on-line advertising and marketing tool of their marketing campaign and that is through the social media.

In case you are among the many few who've passed the grueling Actual Estate Salesperson (RES) course, congratulations. So what next? Which agency should you be a part of? Earlier than taking the plunge, you will need to choose the proper mentor who can educate you the ropes in actual estate. An excellent mentor will allow you to navigate the complex world of real property by instructing you the way to get listings, advertising and marketing methods, real estate contracts and methods to closing your deal. He will even caution you on errors to keep away from that would land you in trouble. This has been reflected within the Industrial Production reading index for prescribed drugs. In June, the reading fell to one hundred thirty from 287. Effectively if you do not, you then're simply leaving your actual property enterprise to chance. Commons for rent

Thanks to hirepropertyagent.com.sg, i've discovered myself a good agent. He did a great job promoting my property and it was bought at an excellent value." JLL appointed unique agent for the sale of 2, 4 and 6 Dunlop Street by Expression of Curiosity. Uncommon Industrial Growth inside Pandan Meals Zone space up for sale conserving you updated with the property market 3. Work @ Residence IT Solutions As property costs cool in Hong Kong and Singapore, which have lengthy been magnets for Chinese language funding, extra money is flowing to actual property markets comparable to New York, London and Sydney. Chinese language have overtaken Russians house for sale in singapore the primary time as the biggest buyers of flats in Manhattan, in response to actual estate brokers. Condominium For Lease – Tribeca by the Waterfront (D09)

Property developer and residential landlord for flats and homes for lease and sale. Most property firms share the same database of property listings in Singapore. Due to this fact it is best to solely use only ONE agent at a time. In case you approach many agents at the similar time, very likely that they'll present you the same property. A lot confusion and embarrassment will arise should you engage many brokers. One of the best, and most of the time only, strategy to discover a good property agent in Singapore is phrase-of-mouth. Ask your friends and colleagues for reference. It is very simple to provide you with a couple of candidates since a lot of the expatriates dwelling in Singapore for a long time can have several good agent contacts to guide you. Toa Payoh, Singapore Singapore 319378 Estate

This is precisely what happened to me and my husband at the moment, to not point out a very unscrupulous developer operating in a really unprofessional manner. I need to share this story with everyone here, and please pass the message round particularly among expats communities, beware while you want to purchase property developed by VicLand Pte Ltd and if developer's agent is ECG property. There was only one unit left on the market by developer, 03-09, a 3 bed room flat. On the time my husband was out of town, and initially I liked what I saw so I instructed the developer's agent and my agent we should come back with my husband in two weeks to view it once more and make a decision after ward. Complaint / Suggestions about lousy property agent Darren Ng from Dennis Wee

This bought me thinking and I started to surprise – how much does a property agent really earn? We often hear or read about sure brokers making million dollar commissions, however is that the exception or the norm? That piqued my curiosity. Like any job, those who put in time and effort will do well and rise to the top. The ethics of exhausting work apply to the true property market as nicely. For individuals who are pondering of making a career change to develop into a property agent, you should be ready to invest the trouble to do properly. Otherwise you may just add to the statistic of brokers who eventually drop out of the realtor game. Properties that do not fall within the definition of residential properties stated above are non-residential properties Web site - www.riaschool.com.sg

Ought to you are on the lookout for new properties for investment or for own stay, we offer property recommendation and search services tailored to your needs. We have represented many together with worldwide and local buyers in efficiently finishing their property purchases. We work with main builders to bring you the latest and one of the best prime properties in Singapore. We are a one-cease service that may full your property cycle from purchase to sale. Property agents for Singapore Land Authority protecting among the government colonial properties for rent. Property leases for expatriates and foreigners. Also helps expats to purchase and promote their properties as well as property investment opportunities in Singapore and China. The Restaurant Affiliation of Singapore

• Certicom ECC Tutorial
• a relatively easy to understand primer on elliptic curve cryptography
• Elliptic curves and their implementation by Adam Langley (OpenSSL dev).
• Interactive introduction to elliptic curves and elliptic curve cryptography with SAGE
• How the NSA Was Able to Snoop Our Email explained by Mathematician Edward Frenkel