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Description

Independent Submission M. The domain parameters are consistent with the relevant international standards, and can be used in X. Status of This Memo This document is not an Internet Standards Track specification; it is published for informational purposes. The RFC Editor has chosen to publish this document at its discretion and makes no statement about its value for implementation or deployment. All rights reserved.

Please review these documents carefully, as they describe your rights and restrictions with respect to this document. Scope and Relation to Other Specifications Requirements Language Requirements on the Elliptic Curve Domain Parameters Security Requirements Technical Requirements Domain Parameter Specification Domain Parameters for Bit Curves Object Identifiers and ASN.

Object Identifiers Security Considerations Intellectual Property Rights Normative Prime. curves Informative References Pseudo-Random Generation of Parameters Generation of Prime s Generation of Pseudo-Random Curves Introduction Although several standards for elliptic curves and domain parameters exist e. In particular, the seeds from which the curve parameters were derived have been chosen ad hoc, leaving out an essential part of the security proof. This does not only contradict the approach of pseudo-random parameters, but also increases the risk of implementations violating one of the numerous patents for fast modular arithmetic with special prime.

curves. At least for applications with the highest security demands or under circumstances that complicate a change of parameters in response to new attacks, the inclusion of a corresponding security requirement for domain parameters the class group condition, see Section 2 is justified. In particular, there is no bit curve defined, but only one with a bit length, which may be disadvantageous for some implementations. Furthermore, many of the parameters specified by the existing standards are identical see [ SEC2 ] for a comparison.

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Thus, there is still a need for additional elliptic curve domain parameters that overcome the above limitations. Scope and Relation to Other Specifications This RFC specifies elliptic curve domain parameters over prime fields GF p with p having a length of,, and bits. These parameters were generated in a pseudo-random, yet completely systematic and reproducible, way and have been verified to resist current cryptanalytic approaches.

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Furthermore, this document identifies the security and implementation requirements for the parameters, and describes the methods used for the pseudo-random generation of the parameters. This document does neither address the cryptographic algorithms to be used with the specified parameters nor their application in other standards. Typically, for cryptographic applications, an element G of prime order q is chosen in E GF p.

This choice induces a natural ordering on GF p. Security Requirements The following security requirements are either motivated by known cryptographic analysis or aim to enhance trust in the recommended curves. As this specification aims at a particularly high level of security, a restrictive position is taken here. Nevertheless, it may be sensible to slightly deviate from these requirements for certain applications e. Immunity to attacks using the Weil or Tate Pairing.

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By Fermat's Little Theorem, l divides q The trace is not equal to one. Note that these curves are also excluded by requirement 5 of Section 2. Large class. Although there are no efficient attacks exploiting a small classrecent work [ JMV ] and [ HR ] also may be seen as argument for the class condition.

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Prime group order. Therefore, all groups proposed in this RFC have cofactor 1.

Note that curves with prime order have no point of order 2 and therefore no point with y-coordinate 0. Verifiably pseudo-random. The elliptic curve domain parameters shall be generated in a pseudo-random manner using seeds that are generated in a systematic and comprehensive way. The methods by which the parameters have been obtained are explained in Appendix A. Proof of security. For all curves, a proof should be given that all security requirements are met.

These proofs are provided in [ EBP ].

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However, the circumstances under which these attacks are applicable can be avoided in most applications. Therefore, no corresponding security requirement is stated here. However, it is highly recommended that developers verify the security of their implementations against this kind of attack. Technical Requirements Commercial demands and experience with existing implementations lead to the following technical requirements for the elliptic curve domain parameters.

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For each of the bit lengths,, andone curve shall be proposed. This requirement follows from the need for curves providing different levels of security that are appropriate for the underlying symmetric algorithms. The existing standards specify a bit curve instead of a bit curve. The prime p shall be congruent 3 mod 4. This requirement is not always met by the parameters defined in existing standards.

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Due to this isomorphism, E GF p and E' GF p have the same of points, share the same algebraic structure, and hence offer the same level of security. The prime p must not be of any special form; this requirement is met by a verifiably pseudo-random generation of the parameters see requirement 5 in Section 2.

Although parameters specified by existing standards do not meet this requirement, the need for such curves over pseudo- randomly chosen fields has already been foreseen by the Standards for Efficient Cryptography Group SECGsee [ SEC2 ]. In some cases, even the bit-length of E GF p can exceed the bit-length of p.

B shall be a non-square mod p. Otherwise, the compressed representations of the curve-points 0,0 and 0,Xwith X being the square root of B with a least ificant bit of 0, would be identical. As there are implementations of elliptic curves that encode the point at infinity as 0,0we try to avoid ambiguities. Note that this condition is stable under quadratic twists as described in condition 3 above. Condition 6 makes the attack described in [ G ] impossible.

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It can therefore prime. curves be seen as a security requirement. This constraint has not been specified by existing standards. Domain Parameter Specification In this section, the elliptic curve domain parameters proposed are specified in the following way. For all curves, an ID is given by which it can be referenced. For the twisted curve, we also give the coefficient Z that defines the isomorphism F see requirement 3 in Section 2.

The methods for the generation of the parameters are given in Appendix A. Proofs for the fulfillment of the security requirements specified in Section 2. If the domain parameters are explicitly specified using the type specifiedCurve in the field algorithm.

Although the parameters specified in Section 3 have all been generated by the pseudo-random methods described in Appendix Athese algorithms deviate from those mandated in ANSI X9. In order to avoid rejection of the parameters, the ASN. Security Considerations The level of security provided by symmetric ciphers and hash functions used in conjunction with the elliptic curve domain parameters specified in this RFC should roughly match or exceed the level provided by the domain parameters.

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The following table indicates the minimum key sizes for symmetric ciphers and hash functions providing at least roughly comparable security. Intellectual Property Rights The authors have no knowledge about any intellectual property rights that cover the usage of the domain parameters defined herein. However, readers should be aware that implementations based on these domain parameters may require use of inventions covered by patent rights.

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