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99-777: Weaknesses in the cryptographic security of the algorithm were known and publicly criticised well before the algorithm became part of a formal standard endorsed by the ANSI , ISO , and formerly by the National Institute of Standards and Technology (NIST). One of the weaknesses publicly identified was the potential of the algorithm to harbour a cryptographic backdoor advantageous to those who know about it—the United States government's National Security Agency (NSA)—and no one else . In 2013, The New York Times reported that documents in their possession but never released to

198-419: A and b in K . The curve is required to be non-singular , which means that the curve has no cusps or self-intersections . (This is equivalent to the condition 4 a + 27 b ≠ 0 , that is, being square-free in x .) It is always understood that the curve is really sitting in the projective plane , with the point O being the unique point at infinity . Many sources define an elliptic curve to be simply

297-434: A field means X = 0 {\displaystyle X=0} . Y {\displaystyle Y} on the other hand can take any value thus all triplets ( 0 , Y , 0 ) {\displaystyle (0,Y,0)} satisfy the equation. In projective geometry this set is simply the point O = [ 0 : 1 : 0 ] {\displaystyle O=[0:1:0]} , which

396-473: A 160-bit elliptic curve group, the number of potential points R in the list is about 2, and searching the list would be about as hard as solving the discrete logarithm problem. The cost of this method is that the ECRNG is made half as efficient, because the output length is effectively halved. According to John Kelsey, the option in the standard to choose a verifiably random Q was added as an option in response to

495-400: A CSPRNG, even if Q had not been chosen to contain a backdoor. The standard says that implementations "should" use the small max_outlen provided, but gives the option of outputting a multiple of 8 fewer bits. Appendix C of the standard gives a loose argument that outputting fewer bits will make the output less uniformly distributed. Brown's 2006 security proof relies on outlen being much smaller

594-514: A CSPRNG. Members of the ANSI standard group to which Dual_EC_DRBG was first submitted were aware of the exact mechanism of the potential backdoor and how to disable it, but did not elect to disable or publicize the backdoor. The general cryptographic community was initially not aware of the potential backdoor, until Dan Shumow and Niels Ferguson 's publication, or of Certicom 's Daniel R. L. Brown and Scott Vanstone's 2005 patent application describing

693-626: A backdoor could occur, since they filed a patent application in January 2005 on exactly how to insert or prevent the backdoor in DUAL_EC_DRBG. The working of the "trap door" mentioned in the patent is identical to the one later confirmed in Dual_EC_DRBG. Writing about the patent in 2014, commentator Matthew Green describes the patent as a " passive aggressive " way of spiting NSA by publicizing the backdoor, while still criticizing everybody on

792-785: A cryptographic standard", but according to the New York Times story, the NSA had been spending $ 250 million per year to insert backdoors in software and hardware as part of the Bullrun program . A Presidential advisory committee subsequently set up to examine NSA's conduct recommended among other things that the US government "fully support and not undermine efforts to create encryption standards". On April 21, 2014, NIST withdrew Dual_EC_DRBG from its draft guidance on random number generators recommending "current users of Dual_EC_DRBG transition to one of

891-404: A cubic at three points when accounting for multiplicity. For a point P , − P is defined as the unique third point on the line passing through O and P . Then, for any P and Q , P + Q is defined as − R where R is the unique third point on the line containing P and Q . For an example of the group law over a non-Weierstrass curve, see Hessian curves . A curve E defined over

990-412: A curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic curves ; see § Elliptic curves over a general field below.) An elliptic curve is an abelian variety – that is, it has a group law defined algebraically, with respect to which it is an abelian group – and O serves as

1089-742: A library. At least RSA Security (BSAFE library), OpenSSL , Microsoft, and Cisco have libraries which included Dual_EC_DRBG, but only BSAFE used it by default. According to the Reuters article which revealed the secret $ 10 million deal between RSA Security and NSA, RSA Security's BSAFE was the most important distributor of the algorithm. There was a flaw in OpenSSL's implementation of Dual_EC_DRBG that made it non-working outside test mode, from which OpenSSL's Steve Marquess concludes that nobody used OpenSSL's Dual_EC_DRBG implementation. A list of products which have had their CSPRNG-implementation FIPS 140-2 validated

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1188-430: A national standards organization. According to Adam Stanton, the first permanent secretary and head of staff in 1919, AESC started as an ambitious program and little else. Staff for the first year consisted of one executive, Clifford B. LePage, who was on loan from a founding member, ASME. An annual budget of $ 7,500 was provided by the founding bodies. In 1931, the organization (renamed ASA in 1928) became affiliated with

1287-619: A secret deal with the NSA to insert a "known flawed random number generator" into its BSAFE toolkit Following the New York Times story asserting that Dual_EC_DRBG contained a backdoor, Brown (who had applied for the backdoor patent and published the security reduction) wrote an email to an IETF mailing list defending the Dual_EC_DRBG standard process: 1. Dual_EC_DRBG, as specified in NIST SP 800-90A and ANSI X9.82-3, allows an alternative choice of constants P and Q . As far as I know,

1386-447: A user option in the community. Only after widespread concern about the backdoor was there an effort to find software which used Dual_EC_DRBG, of which BSAFE was by far the most prominent found. After the 2013 revelations, RSA security Chief of Technology Sam Curry provided Ars Technica with a rationale for originally choosing the flawed Dual EC DRBG standard as default over the alternative random number generators. The technical accuracy of

1485-437: A valid alternative (assuming implementors disable the obvious backdoor). Note that Daniel R.L. Brown works for Certicom, the main owner of elliptic curve cryptography patents, so there may be a conflict of interest in promoting an EC CSPRNG. The alleged NSA backdoor would allow the attacker to determine the internal state of the random number generator from looking at the output from a single round (32 bytes); all future output of

1584-422: A way that leaves little room for adjustment. However, Dual_EC_DRBG did not specify how the default P and Q constants were chosen, possibly because they were constructed by NSA to be backdoored. Because the standard committee were aware of the potential for a backdoor, a way for an implementer to choose their own secure P and Q was included. But the exact formulation in the standard was written such that use of

1683-642: Is copyright infringement for them to be provided to the public by others free of charge. These assertions have been the subject of criticism and litigation. ANSI was most likely formed in 1918, when five engineering societies and three government agencies founded the American Engineering Standards Committee ( AESC ). In 1928, the AESC became the American Standards Association ( ASA ). In 1966,

1782-418: Is "Up to a thousand times slower" than the alternatives. The potential for a backdoor in Dual_EC_DRBG was not widely publicised outside of internal standard group meetings. It was only after Dan Shumow and Niels Ferguson 's 2007 presentation that the potential for a backdoor became widely known. Shumow and Ferguson had been tasked with implementing Dual_EC_DRBG for Microsoft, and at least Furguson had discussed

1881-525: Is a plane curve defined by an equation of the form after a linear change of variables ( a and b are real numbers). This type of equation is called a Weierstrass equation, and said to be in Weierstrass form, or Weierstrass normal form. The definition of elliptic curve also requires that the curve be non-singular . Geometrically, this means that the graph has no cusps , self-intersections, or isolated points . Algebraically, this holds if and only if

1980-472: Is a smooth , projective , algebraic curve of genus one, on which there is a specified point O . An elliptic curve is defined over a field K and describes points in K , the Cartesian product of K with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions ( x , y ) for: for some coefficients

2079-475: Is a fixed representant of P in E ( Q )/2 E ( Q ), the height of P 1 is about ⁠ 1 / 4 ⁠ of the one of P (more generally, replacing 2 by any m > 1, and ⁠ 1 / 4 ⁠ by ⁠ 1 / m ⁠ ). Redoing the same with P 1 , that is to say P 1 = 2 P 2 + Q 2 , then P 2 = 2 P 3 + Q 3 , etc. finally expresses P as an integral linear combination of points Q i and of points whose height

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2178-408: Is a group, because properties of polynomial equations show that if P is in E ( K ) , then − P is also in E ( K ) , and if two of P , Q , R are in E ( K ) , then so is the third. Additionally, if K is a subfield of L , then E ( K ) is a subgroup of E ( L ) . The above groups can be described algebraically as well as geometrically. Given the curve y = x + bx + c over

2277-510: Is a hard problem if P and Q are set ahead of time, but it's easier if P and Q are chosen. e is a secret key presumably known only by NSA, and the alleged backdoor is a kleptographic asymmetric hidden backdoor. Matthew Green's blog post The Many Flaws of Dual_EC_DRBG has a simplified explanation of how the alleged NSA backdoor works by employing the discrete-log kleptogram introduced in Crypto 1997. NSA first introduced Dual_EC_DRBG in

2376-506: Is also a group isomorphism . Elliptic curves are especially important in number theory , and constitute a major area of current research; for example, they were used in Andrew Wiles's proof of Fermat's Last Theorem . They also find applications in elliptic curve cryptography (ECC) and integer factorization . An elliptic curve is not an ellipse in the sense of a projective conic, which has genus zero: see elliptic integral for

2475-625: Is available at the NIST. The validated CSPRNGs are listed in the Description/Notes field. Note that even if Dual_EC_DRBG is listed as validated, it may not have been enabled by default. Many implementations come from a renamed copy of a library implementation. The BlackBerry software is an example of non-default use. It includes support for Dual_EC_DRBG, but not as default. BlackBerry Ltd has however not issued an advisory to any of its customers who may have used it, because they do not consider

2574-553: Is based on computational hardness assumptions from number theory. A mathematical security reduction proof can then prove that as long as the number theoretical problems are hard, the random number generator itself is secure. However, the makers of Dual_EC_DRBG did not publish a security reduction for Dual_EC_DRBG, and it was shown soon after the NIST draft was published that Dual_EC_DRBG was indeed not secure, because it output too many bits per round. The output of too many bits (along with carefully chosen elliptic curve points P and Q )

2673-417: Is bounded by a fixed constant chosen in advance: by the weak Mordell–Weil theorem and the second property of the height function P is thus expressed as an integral linear combination of a finite number of fixed points. The theorem however doesn't provide a method to determine any representatives of E ( Q )/ mE ( Q ). The rank of E ( Q ), that is the number of copies of Z in E ( Q ) or, equivalently,

2772-596: Is given where the constant b is The points on the curve are E ( F p ) {\displaystyle E({\displaystyle F_{p}})} . Two of these points are given as the fixed points P and Q Their coordinates are A function to extract the x-coordinate is used. It "converts" from elliptic curve points to elements of the field. Output integers are truncated before being output The functions g P {\displaystyle g_{P}} and g Q {\displaystyle g_{Q}} . These functions raise

2871-442: Is given by the tangent to the curve at ( x P , y P ). A more general expression for s {\displaystyle s} that works in both case 1 and case 2 is where equality to ⁠ y P − y Q / x P − x Q ⁠ relies on P and Q obeying y = x + bx + c . For the curve y = x + ax + bx + c (the general form of an elliptic curve with characteristic 3),

2970-521: Is not defined on the line at infinity , but we can multiply by Z 3 {\displaystyle Z^{3}} to get one that is : This resulting equation is defined on the whole projective plane, and the curve it defines projects onto the elliptic curve of interest. To find its intersection with the line at infinity, we can just posit Z = 0 {\displaystyle Z=0} . This implies X 3 = 0 {\displaystyle X^{3}=0} , which in

3069-492: Is not proven which of them have higher rank than the others or which is the true "current champion". As for the groups constituting the torsion subgroup of E ( Q ), the following is known: the torsion subgroup of E ( Q ) is one of the 15 following groups ( a theorem due to Barry Mazur ): Z / N Z for N = 1, 2, ..., 10, or 12, or Z /2 Z × Z /2 N Z with N = 1, 2, 3, 4. Examples for every case are known. Moreover, elliptic curves whose Mordell–Weil groups over Q have

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3168-435: Is simply the point opposite itself, i.e. itself. [REDACTED] Let K be a field over which the curve is defined (that is, the coefficients of the defining equation or equations of the curve are in K ) and denote the curve by E . Then the K - rational points of E are the points on E whose coordinates all lie in K , including the point at infinity. The set of K -rational points is denoted by E ( K ) . E ( K )

3267-463: Is the difficulty of balancing "the interests of both the nation's industrial and commercial sectors and the nation as a whole." Although ANSI itself does not develop standards, the Institute oversees the development and use of standards by accrediting the procedures of standards developing organizations. ANSI accreditation signifies that the procedures used by standards developing organizations meet

3366-529: Is thus the unique intersection of the curve with the line at infinity. Since the curve is smooth, hence continuous , it can be shown that this point at infinity is the identity element of a group structure whose operation is geometrically described as follows: Since the curve is symmetric about the x -axis, given any point P , we can take − P to be the point opposite it. We then have − O = O {\displaystyle -O=O} , as O {\displaystyle O} lies on

3465-464: Is used as an escrow key and stored by for a security domain. The administrator logs the output of the generator to reconstruct the random number with the escrow key. 2) Small output truncation [0041] Another alternative method for preventing a key escrow attack on the output of an ECRNG, shown in Figures 3 and 4 is to add a truncation function to ECRNG to truncate the ECRNG output to approximately half

3564-436: Is useful in a more advanced study of elliptic curves.) The real graph of a non-singular curve has two components if its discriminant is positive, and one component if it is negative. For example, in the graphs shown in figure to the right, the discriminant in the first case is 64, and in the second case is −368. When working in the projective plane , the equation in homogeneous coordinates becomes : This equation

3663-453: Is what makes the NSA backdoor possible, because it enables the attacker to revert the truncation by brute force guessing. The output of too many bits was not corrected in the final published standard, leaving Dual_EC_DRBG both insecure and backdoored. In many other standards, constants that are meant to be arbitrary are chosen by the nothing up my sleeve number principle, where they are derived from pi or similar mathematical constants in

3762-417: The decisional Diffie–Hellman assumption (which is generally accepted to be hard), and two newer less-known problems which are not generally accepted to be hard: the truncated point problem , and the x-logarithm problem . Dual_EC_DRBG was quite slow compared to many alternative CSPRNGs (which don't have security reductions), but Daniel R.L. Brown argues that the security reduction makes the slow Dual_EC_DRBG

3861-612: The ANSI X9.82 DRBG in the early 2000s, including the same parameters which created the alleged backdoor, and Dual_EC_DRBG was published in a draft ANSI standard. Dual_EC_DRBG also exists in the ISO 18031 standard. According to John Kelsey (who together with Elaine Barker was listed as author of NIST SP 800-90A), the possibility of the backdoor by carefully chosen P and Q was brought up at an ANSI X9F1 Tool Standards and Guidelines Group meeting. When Kelsey asked Don Johnson of Cygnacom about

3960-472: The XZ -plane, so that − O {\displaystyle -O} is also the symmetrical of O {\displaystyle O} about the origin, and thus represents the same projective point. If P and Q are two points on the curve, then we can uniquely describe a third point P + Q in the following way. First, draw the line that intersects P and Q . This will generally intersect

4059-399: The discriminant , Δ {\displaystyle \Delta } , is not equal to zero. The discriminant is zero when a = − 3 k 2 , b = 2 k 3 {\displaystyle a=-3k^{2},b=2k^{3}} . (Although the factor −16 is irrelevant to whether or not the curve is non-singular, this definition of the discriminant

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4158-416: The quotient group E ( Q )/ mE ( Q ) is finite (this is the weak Mordell–Weil theorem). Second, introducing a height function h on the rational points E ( Q ) defined by h ( P 0 ) = 0 and h ( P ) = log max(| p |, | q |) if P (unequal to the point at infinity P 0 ) has as abscissa the rational number x = p / q (with coprime p and q ). This height function h has

4257-477: The 3 sets of constants available) and have fixed output length. The algorithm operates exclusively over a prime finite field F p {\displaystyle F_{p}} ( Z / p Z {\displaystyle \mathbb {Z} /p\mathbb {Z} } ) where p is prime. The state, the seed and the random numbers are all elements of this field. Field size is An elliptic curve over F p {\displaystyle F_{p}}

4356-449: The API. Bruce Schneier has pointed out that even if not enabled by default, having a backdoored CSPRNG implemented as an option can make it easier for NSA to spy on targets which have a software-controlled command-line switch to select the encryption algorithm, or a " registry " system, like most Microsoft products, such as Windows Vista : A Trojan is really, really big. You can’t say that

4455-671: The ASA was reorganized and became United States of America Standards Institute ( USASI ). The present name was adopted in 1969. Prior to 1918, these five founding engineering societies: had been members of the United Engineering Society (UES). At the behest of the AIEE, they invited the U.S. government Departments of War, Navy (combined in 1947 to become the Department of Defense or DOD) and Commerce to join in founding

4554-477: The Dual_EC_DRBG's design, with the design of Dual_EC_DRBG having the unusual property that it was theoretically impossible for anyone but Dual_EC_DRBG's designers (NSA) to confirm the backdoor's existence. Bruce Schneier concluded shortly after standardization that the "rather obvious" backdoor (along with other deficiencies) would mean that nobody would use Dual_EC_DRBG. The backdoor would allow NSA to decrypt for example SSL/TLS encryption which used Dual_EC_DRBG as

4653-605: The ISO and the IEC, and administers many key committees and subgroups. In many instances, U.S. standards are taken forward to ISO and IEC, through ANSI or the USNC, where they are adopted in whole or in part as international standards. Adoption of ISO and IEC standards as American standards increased from 0.2% in 1986 to 15.5% in May 2012. The Institute administers nine standards panels: Each of

4752-476: The New York Times reported that Dual_EC_DRBG contained a backdoor by the NSA, RSA Security said they had not been aware of any backdoor when they made the deal with NSA, and told their customers to switch CSPRNG. In the 2014 RSA Conference keynote, RSA Security Executive Chairman Art Coviello explained that RSA had seen declining revenue from encryption, and had decided to stop being "drivers" of independent encryption research, but to instead to "put their trust behind"

4851-693: The U.S. National Committee of the International Electrotechnical Commission ( IEC ), which had been formed in 1904 to develop electrical and electronics standards. ANSI's members are government agencies, organizations, academic and international bodies, and individuals. In total, the Institute represents the interests of more than 270,000 companies and organizations and 30 million professionals worldwide. ANSI's market-driven, decentralized approach has been criticized in comparison with more planned and organized international approaches to standardization. An underlying issue

4950-420: The United States. The organization also coordinates U.S. standards with international standards so that American products can be used worldwide. ANSI accredits standards that are developed by representatives of other standards organizations , government agencies , consumer groups , companies, and others. These standards ensure that the characteristics and performance of products are consistent, that people use

5049-514: The adoption of international standards as national standards where appropriate. The institute is the official U.S. representative to the two major international standards organizations, the International Organization for Standardization (ISO), as a founding member, and the International Electrotechnical Commission (IEC), via the U.S. National Committee (USNC). ANSI participates in almost the entire technical program of both

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5148-483: The alleged backdoored P and Q was required for FIPS 140-2 validation, so the OpenSSL project chose to implement the backdoored P and Q , even though they were aware of the potential backdoor and would have preferred generating their own secure P and Q . New York Times would later write that NSA had worked during the standardization process to eventually become the sole editor of the standard. A security proof

5247-545: The alternatives do not admit a known feasible backdoor. In my view, it is incorrect to imply that Dual_EC_DRBG always has a backdoor, though I admit a wording to qualify the affected cases may be awkward. 2. Many things are obvious in hindsight. I'm not sure if this was obvious. [...] 8. All considered, I don't see how the ANSI and NIST standards for Dual_EC_DRBG can be viewed as a subverted standard, per se. But maybe that's just because I'm biased or naive. Implementations which used Dual_EC_DRBG would usually have gotten it via

5346-417: The backdoor mechanism. In September 2013, The New York Times reported that internal NSA memos leaked by Edward Snowden indicated that the NSA had worked during the standardization process to eventually become the sole editor of the Dual_EC_DRBG standard, and concluded that the Dual_EC_DRBG standard did indeed contain a backdoor for the NSA. In response, NIST stated that "NIST would not deliberately weaken

5445-421: The committee for not actually disabling the backdoor they obviously were aware of. Brown and Vanstone's patent list two necessary conditions for the backdoor to exist: 1) Chosen Q An elliptic curve random number generator avoids escrow keys by choosing a point Q on the elliptic curve as verifiably random. Intentional use of escrow keys can provide for back up functionality. The relationship between P and Q

5544-416: The cubic at a third point, R . We then take P + Q to be − R , the point opposite R . This definition for addition works except in a few special cases related to the point at infinity and intersection multiplicity. The first is when one of the points is O . Here, we define P + O = P = O + P , making O the identity of the group. If P = Q we only have one point, thus we cannot define

5643-627: The default P and Q . OpenSSL chose to implement Dual_EC_DRBG despite its dubious reputation for completeness, noting that OpenSSL tried to be complete and implements many other insecure algorithms. OpenSSL did not use Dual_EC_DRBG as the default CSPRNG, and it was discovered in 2013 that a bug made the OpenSSL implementation of Dual_EC_DRBG non-functioning, meaning that no one could have been using it. Bruce Schneier reported in December 2007 that Microsoft added Dual_EC_DRBG support to Windows Vista, though not enabled by default, and Schneier warned against

5742-463: The default in BSAFE since 2004. OpenSSL implemented all of NIST SP 800-90A including Dual_EC_DRBG at the request of a client. The OpenSSL developers were aware of the potential backdoor because of Shumow and Ferguson's presentation, and wanted to use the method included in the standard to choose a guaranteed non-backdoored P and Q , but were told that to get FIPS 140-2 validation they would have to use

5841-406: The default max_outlen value in the standard. The ANSI X9F1 Tool Standards and Guidelines Group which discussed the backdoor also included three employees from the prominent security company RSA Security. In 2004, RSA Security made an implementation of Dual_EC_DRBG which contained the NSA backdoor the default CSPRNG in their RSA BSAFE as a result of a secret $ 10 million deal with NSA. In 2013, after

5940-414: The equations have identical y values at these values. which is equivalent to Since x P , x Q , and x R are solutions, this equation has its roots at exactly the same x values as and because both equations are cubics they must be the same polynomial up to a scalar. Then equating the coefficients of x in both equations and solving for the unknown x R . y R follows from

6039-425: The field K (whose characteristic we assume to be neither 2 nor 3), and points P = ( x P , y P ) and Q = ( x Q , y Q ) on the curve, assume first that x P ≠ x Q (case 1 ). Let y = sx + d be the equation of the line that intersects P and Q , which has the following slope: The line equation and the curve equation intersect at the points x P , x Q , and x R , so

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6138-409: The field of rational numbers is also defined over the field of real numbers. Therefore, the law of addition (of points with real coordinates) by the tangent and secant method can be applied to E . The explicit formulae show that the sum of two points P and Q with rational coordinates has again rational coordinates, since the line joining P and Q has rational coefficients. This way, one shows that

6237-432: The fixed elliptic curve point P . g Q ( x ) {\displaystyle g_{Q}(x)} is similar except that it uses the point Q . The points P and Q stay constant for a particular implementation of the algorithm. The algorithm allows for different constants, variable output length and other customization. For simplicity, the one described here will use the constants from curve P-256 (one of

6336-456: The fixed points to a power. "Raising to a power" in this context, means using the special operation defined for points on elliptic curves . The generator is seeded with an element from F p {\displaystyle F_{p}} The k -th state and random number The random numbers The stated purpose of including the Dual_EC_DRBG in NIST SP 800-90A is that its security

6435-424: The formulas are similar, with s = ⁠ x P + x P x Q + x Q + ax P + ax Q + b / y P + y Q ⁠ and x R = s − a − x P − x Q . For a general cubic curve not in Weierstrass normal form, we can still define a group structure by designating one of its nine inflection points as the identity O . In the projective plane, each line will intersect

6534-420: The identity element. If y = P ( x ) , where P is any polynomial of degree three in x with no repeated roots, the solution set is a nonsingular plane curve of genus one, an elliptic curve. If P has degree four and is square-free this equation again describes a plane curve of genus one; however, it has no natural choice of identity element. More generally, any algebraic curve of genus one, for example

6633-411: The identity on each trajectory curve. Topologically , a complex elliptic curve is a torus , while a complex ellipse is a sphere . Although the formal definition of an elliptic curve requires some background in algebraic geometry , it is possible to describe some features of elliptic curves over the real numbers using only introductory algebra and geometry . In this context, an elliptic curve

6732-442: The institute's requirements for openness, balance, consensus, and due process. ANSI also designates specific standards as American National Standards, or ANS, when the Institute determines that the standards were developed in an environment that is equitable, accessible and responsive to the requirements of various stakeholders. Voluntary consensus standards quicken the market acceptance of products while making clear how to improve

6831-447: The intersection of two quadric surfaces embedded in three-dimensional projective space, is called an elliptic curve, provided that it is equipped with a marked point to act as the identity. Using the theory of elliptic functions , it can be shown that elliptic curves defined over the complex numbers correspond to embeddings of the torus into the complex projective plane . The torus is also an abelian group , and this correspondence

6930-556: The known potential backdoor. Windows 10 and later will silently replace calls to Dual_EC_DRBG with calls to CTR_DRBG based on AES. On September 9, 2013, following the Snowden leak, and the New York Times report on the backdoor in Dual_EC_DRBG, the National Institute of Standards and Technology (NIST) ITL announced that in light of community security concerns, it was reissuing SP 800-90A as draft standard, and re-opening SP800-90B/C for public comment. NIST now "strongly recommends" against

7029-399: The length of a compressed elliptic curve point. Preferably, this operation is done in addition to the preferred method of Figure 1 and 2, however, it will be appreciated that it may be performed as a primary measure for preventing a key escrow attack. The benefit of truncation is that the list of R values associated with a single ECRNG output r is typically infeasible to search. For example, for

7128-400: The line between them. In this case, we use the tangent line to the curve at this point as our line. In most cases, the tangent will intersect a second point R and we can take its opposite. If P and Q are opposites of each other, we define P + Q = O . Lastly, If P is an inflection point (a point where the concavity of the curve changes), we take R to be P itself and P + P

7227-501: The line equation and this is an element of K , because s is. If x P = x Q , then there are two options: if y P = − y Q (case 3 ), including the case where y P = y Q = 0 (case 4 ), then the sum is defined as 0; thus, the inverse of each point on the curve is found by reflecting it across the x -axis. If y P = y Q ≠ 0 , then Q = P and R = ( x R , y R ) = −( P + P ) = −2 P = −2 Q (case 2 using P as R ). The slope

7326-532: The method of tangents and secants detailed above , starting with a finite number of rational points. More precisely the Mordell–Weil theorem states that the group E ( Q ) is a finitely generated (abelian) group. By the fundamental theorem of finitely generated abelian groups it is therefore a finite direct sum of copies of Z and finite cyclic groups. The proof of the theorem involves two parts. The first part shows that for any integer m  > 1,

7425-448: The most important distributor of the insecure algorithm. RSA responded that they "categorically deny" that they had ever knowingly colluded with the NSA to adopt an algorithm that was known to be flawed, but also stated "we have never kept [our] relationship [with the NSA] a secret". Sometime before its first known publication in 2004, a possible kleptographic backdoor was discovered with

7524-580: The number of independent points of infinite order, is called the rank of E . The Birch and Swinnerton-Dyer conjecture is concerned with determining the rank. One conjectures that it can be arbitrarily large, even if only examples with relatively small rank are known. The elliptic curve with the currently largest exactly-known rank is It has rank 20, found by Noam Elkies and Zev Klagsbrun in 2020. Curves of rank higher than 20 have been known since 1994, with lower bounds on their ranks ranging from 21 to 29, but their exact ranks are not known and in particular it

7623-603: The origin of Q , Johnson answered in a 27 October 2004 email to Kelsey that NSA had prohibited the public discussion of generation of an alternative Q to the NSA-supplied one. At least two members of the Members of the ANSI X9F1 Tool Standards and Guidelines Group which wrote ANSI X9.82, Daniel R. L. Brown and Scott Vanstone from Certicom , were aware of the exact circumstances and mechanism in which

7722-654: The origin of the term. However, there is a natural representation of real elliptic curves with shape invariant j ≥ 1 as ellipses in the hyperbolic plane H 2 {\displaystyle \mathbb {H} ^{2}} . Specifically, the intersections of the Minkowski hyperboloid with quadric surfaces characterized by a certain constant-angle property produce the Steiner ellipses in H 2 {\displaystyle \mathbb {H} ^{2}} (generated by orientation-preserving collineations). Further,

7821-412: The orthogonal trajectories of these ellipses comprise the elliptic curves with j ≤ 1 , and any ellipse in H 2 {\displaystyle \mathbb {H} ^{2}} described as a locus relative to two foci is uniquely the elliptic curve sum of two Steiner ellipses, obtained by adding the pairs of intersections on each orthogonal trajectory. Here, the vertex of the hyperboloid serves as

7920-539: The panels works to identify, coordinate, and harmonize voluntary standards relevant to these areas. In 2009, ANSI and the National Institute of Standards and Technology (NIST) formed the Nuclear Energy Standards Coordination Collaborative (NESCC). NESCC is a joint initiative to identify and respond to the current need for standards in the nuclear industry. Elliptic curve In mathematics , an elliptic curve

8019-411: The possible backdoor in a 2005 X9 meeting. Bruce Schneier wrote in a 2007 Wired article that the Dual_EC_DRBG's flaws were so obvious that nobody would use Dual_EC_DRBG: "It makes no sense as a trap door: It's public, and rather obvious. It makes no sense from an engineering perspective: It's too slow for anyone to willingly use it." Schneier was apparently unaware that RSA Security had used Dual_EC_DRBG as

8118-538: The probable backdoor a vulnerability. Jeffrey Carr quotes a letter from Blackberry: The Dual EC DRBG algorithm is only available to third party developers via the Cryptographic APIs on the [Blackberry] platform. In the case of the Cryptographic API, it is available if a 3rd party developer wished to use the functionality and explicitly designed and developed a system that requested the use of

8217-427: The property that h ( mP ) grows roughly like the square of m . Moreover, only finitely many rational points with height smaller than any constant exist on E . The proof of the theorem is thus a variant of the method of infinite descent and relies on the repeated application of Euclidean divisions on E : let P ∈ E ( Q ) be a rational point on the curve, writing P as the sum 2 P 1 + Q 1 where Q 1

8316-538: The public "appear to confirm" that the backdoor was real, and had been deliberately inserted by the NSA as part of its Bullrun decryption program. In December 2013, a Reuters news article alleged that in 2004, before NIST standardized Dual_EC_DRBG, NSA paid RSA Security $ 10 million in a secret deal to use Dual_EC_DRBG as the default in the RSA BSAFE cryptography library, which resulted in RSA Security becoming

8415-490: The random number generator can then easily be calculated, until the CSPRNG is reseeded with an external source of randomness. This makes for example SSL/TLS vulnerable, since the setup of a TLS connection includes the sending of a randomly generated cryptographic nonce in the clear. NSA's alleged backdoor would depend on their knowing of the single e such that e Q = P {\displaystyle eQ=P} . This

8514-456: The safety of those products for the protection of consumers. There are approximately 9,500 American National Standards that carry the ANSI designation. The American National Standards process involves: In addition to facilitating the formation of standards in the United States, ANSI promotes the use of U.S. standards internationally, advocates U.S. policy and technical positions in international and regional standards organizations, and encourages

8613-938: The same definitions and terms, and that products are tested the same way. ANSI also accredits organizations that carry out product or personnel certification in accordance with requirements defined in international standards. The organization's headquarters are in Washington, D.C. ANSI's operations office is located in New York City. The ANSI annual operating budget is funded by the sale of publications, membership dues and fees, accreditation services, fee-based programs, and international standards programs. Many ANSI regulations are incorporated by reference into United States federal statutes (i.e. by OSHA regulations referring to individual ANSI specifications). ANSI does not make these standards publicly available, and charges money for access to these documents; it further claims that it

8712-616: The security reduction of Dual_EC_DRBG mentions the need for more output truncation and a randomly chosen Q , but mostly in passing, and does not mention his conclusions from his patent that these two defects in Dual_EC_DRBG together can be used as a backdoor. Brown writes in the conclusion: "Therefore, the ECRNG should be a serious consideration, and its high efficiency makes it suitable even for constrained environments." Note that others have criticised Dual_EC_DRBG as being extremely slow, with Bruce Schneier concluding "It's too slow for anyone to willingly use it", and Matthew Green saying Dual_EC_DRBG

8811-467: The set of rational points of E forms a subgroup of the group of real points of E . This section is concerned with points P = ( x , y ) of E such that x is an integer. For example, the equation y = x + 17 has eight integral solutions with y  > 0: As another example, Ljunggren's equation , a curve whose Weierstrass form is y = x − 2 x , has only four solutions with y  ≥ 0 : Rational points can be constructed by

8910-456: The standard did not use greater truncation, which Brown's patent said could be used as the "primary measure for preventing a key escrow attack". The small truncation was unusual compared to previous EC PRGs, which according to Matthew Green had only output 1/2 to 2/3 of the bits in the output function. The low truncation was in 2006 shown by Gjøsteen to make the RNG predictable and therefore unusable as

9009-461: The standardization process to eventually become the sole editor of the standard. The early usage of Dual_EC_DRBG by RSA Security (for which NSA was later reported to have secretly paid $ 10 million) was cited by the NSA as an argument for Dual_EC_DRBG's acceptance into the NIST SP 800-90A standard. RSA Security subsequently cited Dual_EC_DRBG's acceptance into the NIST standard as a reason they used Dual_EC_DRBG. Daniel R. L. Brown's March 2006 paper on

9108-450: The standards and guidance from standards organizations such as NIST. A draft of NIST SP 800-90A including the Dual_EC_DRBG was published in December 2005. The final NIST SP 800-90A including Dual_EC_DRBG was published in June 2006. Documents leaked by Snowden have been interpreted as suggesting that the NSA backdoored Dual_EC_DRBG, with those making the allegation citing the NSA's work during

9207-403: The statement was widely criticized by cryptographers, including Matthew Green and Matt Blaze . On December 20, 2013, it was reported by Reuters that RSA had accepted a secret payment of $ 10 million from the NSA to set the Dual_EC_DRBG random number generator as the default in two of its encryption products. On December 22, 2013, RSA posted a statement to its corporate blog "categorically" denying

9306-457: The suspected backdoor, though in such a way that FIPS 140-2 validation could only be attained by using the possibly backdoored Q . Steve Marquess (who helped implement NIST SP 800-90A for OpenSSL) speculated that this requirement to use the potentially backdoored points could be evidence of NIST complicity. It is not clear why the standard did not specify the default Q in the standard as a verifiably generated nothing up my sleeve number , or why

9405-410: The three remaining approved algorithms as quickly as possible." The algorithm uses a single integer s as state. Whenever a new random number is requested, this integer is updated. The k -th state is given by The returned random integer r is a function of the state. The k -th random number is The function g P ( x ) {\displaystyle g_{P}(x)} depends on

9504-478: The use of Dual_EC_DRBG, as specified in the January 2012 version of SP 800-90A. The discovery of a backdoor in a NIST standard has been a major embarrassment for the NIST . RSA Security had kept Dual_EC_DRBG as the default CSPRNG in BSAFE even after the wider cryptographic community became aware of the potential backdoor in 2007, but there does not seem to have been a general awareness of BSAFE's usage of Dual_EC_DRBG as

9603-405: Was a mistake. It’s a massive piece of code collecting keystrokes. But changing a bit-one to a bit-two [in the registry to change the default random number generator on the machine] is probably going to be undetected. It is a low conspiracy, highly deniable way of getting a backdoor. So there’s a benefit to getting it into the library and into the product. In December 2013, a proof of concept backdoor

9702-430: Was later published for Dual_EC_DRBG by Daniel R.L. Brown and Kristian Gjøsteen, showing that the generated elliptic curve points would be indistinguishable from uniformly random elliptic curve points, and that if fewer bits were output in the final output truncation, and if the two elliptic curve points P and Q were independent, then Dual_EC_DRBG is secure. The proof relied on the assumption that three problems were hard:

9801-435: Was published that uses the leaked internal state to predict subsequent random numbers, an attack viable until the next reseed. American National Standards Institute The American National Standards Institute ( ANSI / ˈ æ n s i / AN -see ) is a private nonprofit organization that oversees the development of voluntary consensus standards for products, services, processes, systems, and personnel in

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