Internet-Draft | Address Protection ND for LLN | March 2020 |
Thubert, et al. | Expires 10 September 2020 | [Page] |
This document updates the 6LoWPAN Neighbor Discovery (ND) protocol defined in RFC 6775 and RFC 8505. The new extension is called Address Protected Neighbor Discovery (AP-ND) and it protects the owner of an address against address theft and impersonation attacks in a low-power and lossy network (LLN). Nodes supporting this extension compute a cryptographic identifier (Crypto-ID) and use it with one or more of their Registered Addresses. The Crypto-ID identifies the owner of the Registered Address and can be used to provide proof of ownership of the Registered Addresses. Once an address is registered with the Crypto-ID and a proof-of-ownership is provided, only the owner of that address can modify the registration information, thereby enforcing Source Address Validation.¶
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Neighbor Discovery Optimizations for 6LoWPAN networks [RFC6775] (6LoWPAN ND) adapts the original IPv6 Neighbor Discovery (IPv6 ND) protocols defined in [RFC4861] and [RFC4862] for constrained low-power and lossy network (LLN). In particular, 6LoWPAN ND introduces a unicast host Address Registration mechanism that reduces the use of multicast compared to the Duplicate Address Detection (DAD) mechanism defined in IPv6 ND. 6LoWPAN ND defines a new Address Registration Option (ARO) that is carried in the unicast Neighbor Solicitation (NS) and Neighbor Advertisement (NA) messages exchanged between a 6LoWPAN Node (6LN) and a 6LoWPAN Router (6LR). It also defines the Duplicate Address Request (DAR) and Duplicate Address Confirmation (DAC) messages between the 6LR and the 6LoWPAN Border Router (6LBR). In LLN networks, the 6LBR is the central repository of all the registered addresses in its domain.¶
The registration mechanism in "Neighbor Discovery Optimization for Low-power and Lossy Networks" [RFC6775] (aka 6LoWPAN ND) prevents the use of an address if that address is already registered in the subnet (first come first serve). In order to validate address ownership, the registration mechanism enables the 6LR and 6LBR to validate the association between the registered address of a node, and its Registration Ownership Verifier (ROVR). The ROVR is defined in "Registration Extensions for 6LoWPAN Neighbor Discovery" [RFC8505] and it can be derived from the MAC address of the device (using the 64-bit Extended Unique Identifier EUI-64 address format specified by IEEE). However, the EUI-64 can be spoofed, and therefore, any node connected to the subnet and aware of a registered-address-to-ROVR mapping could effectively fake the ROVR. This would allow the an attacker to steal the address and redirect traffic for that address. [RFC8505] defines an Extended Address Registration Option (EARO) option that allows to transport alternate forms of ROVRs, and is a pre-requisite for this specification.¶
In this specification, a 6LN generates a cryptographic ID (Crypto-ID) and places it in the ROVR field during the registration of one (or more) of its addresses with the 6LR(s). Proof of ownership of the Crypto-ID is passed with the first registration exchange to a new 6LR, and enforced at the 6LR. The 6LR validates ownership of the cryptographic ID before it creates any new registration state, or changes existing information.¶
The protected address registration protocol proposed in this document provides the same conceptual benefit as Source Address Validation (SAVI) [RFC7039] that only the owner of an IPv6 address may source packets with that address. As opposed to [RFC7039], which relies on snooping protocols, the protection is based on a state that is installed and maintained in the network by the owner of the address. With this specification, a 6LN may use a 6LR for forwarding an IPv6 packets if and only if it has registered the address used as source of the packet with that 6LR.¶
With the 6lo adaptation layer in [RFC4944] and [RFC6282], a 6LN can obtain a better compression for an IPv6 address with an Interface ID (IID) that is derived from a Layer-2 address. As a side note, this is incompatible with Secure Neighbor Discovery (SeND) [RFC3971] and Cryptographically Generated Addresses (CGAs) [RFC3972], since they derive the IID from cryptographic keys, whereas this specification separates the IID and the key material.¶
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all capitals, as shown here.¶
This document uses the following abbreviations:¶
The reader may get additional context for this specification from the following references:¶
Section 5.3 of [RFC8505] introduces the ROVR that is used to detect and reject duplicate registrations in the DAD process. The ROVR is a generic object that is designed for both backward compatibility and the capability to introduce new computation methods in the future. Using a Crypto-ID per this specification is the RECOMMENDED method. Section 7.3 discusses collisions when heterogeneous methods to compute the ROVR field coexist inside a same network.¶
This specification introduces a new token called a cryptographic identifier (Crypto-ID) that is transported in the ROVR field and used to prove indirectly the ownership of an address that is being registered by means of [RFC8505]. The Crypto-ID is derived from a cryptographic public key and additional parameters.¶
The overall mechanism requires the support of Elliptic Curve Cryptography (ECC) and of a hash function as detailed in Section 6.2. To enable the verification of the proof, the registering node needs to supply certain parameters including a nonce and a signature that will demonstrate that the node possesses the private-key corresponding to the public-key used to build the Crypto-ID.¶
The elliptic curves and the hash functions listed in Table 2 in Section 8.3 can be used with this specification; more may be added in the future to the IANA registry. The signature scheme that specifies which combination is used (including the curve and the representation conventions) is signaled by a Crypto-Type in a new IPv6 ND Crypto-ID Parameters Option (CIPO, see Section 4.3) that contains the parameters that are necessary for the proof, a Nonce option ([RFC3971]) and a NDP Signature option (Section 4.4). The NA(EARO) is modified to enable a challenge and transport a Nonce option.¶
The Crypto-ID is transported in the ROVR field of the EARO option and the EDAR message, and is associated with the Registered Address at the 6LR and the 6LBR. The ownership of a Crypto-ID can be demonstrated by cryptographic mechanisms, and by association, the ownership of the Registered Address can be ascertained.¶
A node in possession of the necessary cryptographic primitives SHOULD use Crypto-ID by default as ROVR in its registrations. Whether a ROVR is a Crypto-ID is indicated by a new "C" flag in the NS(EARO) message.¶
The Crypto-ID is derived from the public key and a modifier as follows:¶
At the time of this writing, a minimal size for the Crypto-ID of 128 bits is RECOMMENDED unless backward compatibility is needed [RFC8505]. This value is bound to augment in the future.¶
This specification updates the EARO option to enable the use of the ROVR field to transport the Crypto-ID. The resulting format is as follows:¶
This specification uses Status values "Validation Requested" and "Validation Failed", which are defined in [RFC8505].¶
this specification does not define any new Status value.¶
This specification defines the Crypto-ID Parameters Option (CIPO). The CIPO carries the parameters used to form a Crypto-ID.¶
In order to provide cryptographic agility [BCP 201], this specification supports different elliptic curves, indicated by a Crypto-Type field:¶
The implementation of multiple hash functions in a constrained devices may consume excessive amounts of program memory. This specification enables the use of SHA-256 [RFC6234] for all the supported ECC curves.¶
Some code factorization is also possible for the ECC computation itself. [CURVE-REPRESENTATIONS] provides information on how to represent Montgomery curves and (twisted) Edwards curves as curves in short-Weierstrass form and illustrates how this can be used to implement elliptic curve computations using existing implementations that already provide, e.g., ECDSA and ECDH using NIST [FIPS186-4] prime curves. For more details on representation conventions, we refer to Appendix B.¶
This specification defines the NDP Signature Option (NDPSO). The NDPSO carries the signature that proves the ownership of the Crypto-ID. The format of the NDPSO is illustrated in Figure 3.¶
As opposed to the RSA Signature Option (RSAO) defined in section 5.2. of SEND [RFC3971], the NDPSO does not have a key hash field. Instead, the leftmost 128 bits of the ROVR field in the EARO are used as hash to retrieve the CIPO that contains the key material used for signature verification, left-padded if needed.¶
Another difference is that the NDPSO signs a fixed set of fields as opposed to all options that appear prior to it in the ND message that bears the signature. This allows to elide a CIPO that the 6LR already received, at the expense of the capability to add arbitrary options that would signed with a RSAO.¶
An ND message that carries an NDPSO MUST have one and only one EARO. The EARO MUST contain a Crypto-ID in the ROVR field, and the Crypto-ID MUST be associated with the keypair used for the Digital Signature in the NDPSO.¶
The CIPO may be present in the same message as the NDPSO. If it is not present, it can be found in an abstract table that was created by a previous message and indexed by the hash.¶
The scope of the protocol specified here is a 6LoWPAN LLN, typically a stub network connected to a larger IP network via a Border Router called a 6LBR per [RFC6775]. A 6LBR has sufficient capability to satisfy the needs of duplicate address detection.¶
The 6LBR maintains registration state for all devices in its attached LLN. Together with the first-hop router (the 6LR), the 6LBR assures uniqueness and grants ownership of an IPv6 address before it can be used in the LLN. This is in contrast to a traditional network that relies on IPv6 address auto-configuration [RFC4862], where there is no guarantee of ownership from the network, and each IPv6 Neighbor Discovery packet must be individually secured [RFC3971].¶
In a mesh network, the 6LR is directly connected to the host device. This specification mandates that the peer-wise layer-2 security is deployed so that all the packets from a particular host are securely identifiable by the 6LR. The 6LR may be multiple hops away from the 6LBR. Packets are routed between the 6LR and the 6LBR via other 6LRs. This specification mandates that a chain of trust is established so that a packet that was validated by the first 6LR can be safely routed by other on-path 6LRs to the 6LBR.¶
The 6LR/6LBR ensures first-come/first-serve by storing the ROVR associated to the address being registered upon the first registration and rejecting a registration with a different ROVR value. A 6LN can claim any address as long as it is the first to make that claim. After a successful registration, the 6LN becomes the owner of the registered address and the address is bound to the ROVR value in the 6LR/6LBR registry.¶
This specification enables the 6LR to challenge the 6LN to verify its ownership of the binding by placing a Crypto-ID in the ROVR. The challenge can happen at any time at the discretion of the 6LR. The 6LR MUST challenge the 6LN when it creates a binding and when a new registration attempts to change a parameter of the binding that identifies the 6LN, for instance its Source Link-Layer Address. The verification protects against a rogue that would steal an address and attract its traffic, or use it as source address.¶
The challenge can also triggered by the 6LBR, e.g., to enforce a global policy. In that case, the 6LBR returns a status of "Validation Requested" in the DAR/DAC exchange, which is echoed by the 6LR in the NA (EARO) back to the registering node. A valid registration in the 6LR or the 6LBR MUST NOT be altered until the challenge is complete.¶
A node may use more than one IPv6 address at the same time. The separation of the address and the cryptographic material avoids the need for the constrained device to compute multiple keys for multiple addresses. The 6LN MAY use the same Crypto-ID to prove the ownership of multiple IPv6 addresses. The 6LN MAY also derive multiple Crypto-IDs from a same key.¶
A 6LN registers to a 6LR that is one hop away from it with the "C" flag set in the EARO, indicating that the ROVR field contains a Crypto-ID. The Target Address in the NS message indicates the IPv6 address that the 6LN is trying to register [RFC8505]. The on-link (local) protocol interactions are shown in Figure 5. If the 6LR does not have a state with the 6LN that is consistent with the NS(EARO), then it replies with a challenge NA (EARO, status=Validation Requested) that contains a Nonce Option (shown as NonceLR in Figure 5).¶
The Nonce option contains a nonce value that, to the extent possible for the implementation, was never employed in association with the key pair used to generate the Crypto-ID. This specification inherits from [RFC3971] that simply indicates that the nonce is a random value. Ideally, an implementation uses an unpredictable cryptographically random value [BCP 106]. But that may be impractical in some LLN scenarios where the devices do not have a guaranteed sense of time and for which computing complex hashes is detrimental to the battery lifetime. Alternatively, the device may use an always-incrementing value saved in the same stable storage as the key, so they are lost together, and starting at a best effort random value, either as the nonce value or as a component to its computation.¶
The 6LN replies to the challenge with an NS(EARO) that includes a new Nonce option (shown as NonceLN in Figure 5), the CIPO (Section 4.3), and the NDPSO containing the signature. Both Nonces are included in the signed material. This provides a "contributory behavior", so that either party that knows it generates a good quality nonce knows that the protocol will be secure.¶
The 6LR MUST store the information associated to a Crypto-ID on the first NS exchange where it appears in a fashion that the CIPO parameters can be retrieved from the Crypto-ID alone.¶
The steps for the registration to the 6LR are as follows:¶
The signature generated by the 6LN to provide proof-of-ownership of the private-key is carried in the NDP Signature Option (NDPSO). It is generated by the 6LN in a fashion that depends on the Crypto-Type (see Table 2 in Section 8.3) chosen by the 6LN as follows:¶
Concatenate the following in the order listed:¶
The 6LR on receiving the NDPSO and CIPO options first checks that the EARO Length in the CIPO matches the length of the EARO. If so it regenerates the Crypto-ID based on the CIPO to make sure that the leftmost bits up to the size of the ROVR match.¶
If and only if the check is successful, it tries to verify the signature in the NDPSO option using the following:¶
Concatenate the following in the order listed:¶
A new 6LN that joins the network auto-configures an address and performs an initial registration to a neighboring 6LR with an NS message that carries an Address Registration Option (EARO) [RFC8505].¶
In a multihop 6LoWPAN, the registration with Crypto-ID is propagated to 6LBR as shown in Figure 6, which illustrates the registration flow all the way to a 6LowPAN Backbone Router (6BBR) [BACKBONE-ROUTER].¶
The 6LR and the 6LBR communicate using ICMPv6 Extended Duplicate Address Request (EDAR) and Extended Duplicate Address Confirmation (EDAC) messages [RFC8505] as shown in Figure 6. This specification extends EDAR/EDAC messages to carry cryptographically generated ROVR.¶
The assumption is that the 6LR and the 6LBR maintain a security association to authenticate and protect the integrity of the EDAR and EDAC messages, so there is no need to propagate the proof of ownership to the 6LBR. The 6LBR implicitly trusts that the 6LR performs the verification when the 6LBR requires it, and if there is no further exchange from the 6LR to remove the state, that the verification succeeded.¶
Observations regarding the following threats to the local network in [RFC3971] also apply to this specification.¶
The threats and mediations discussed in 6LoWPAN ND [RFC6775][RFC8505] also apply here, in particular denial-of-service attacks against the registry at the 6LR or 6LBR.¶
Secure ND [RFC3971] forces the IPv6 address to be cryptographic since it integrates the CGA as the IID in the IPv6 address. In contrast, this specification saves about 1Kbyte in every NS/NA message. Also, this specification separates the cryptographic identifier from the registered IPv6 address so that a node can have more than one IPv6 address protected by the same cryptographic identifier.¶
With this specification the 6LN can freely form its IPv6 address(es) in any fashion, thereby enabling either 6LoWPAN compression for IPv6 addresses that are derived from Layer-2 addresses, or temporary addresses, e.g., formed pseudo-randomly and released in relatively short cycles for privacy reasons [RFC8064][RFC8065], that cannot be compressed.¶
This specification provides added protection for addresses that are obtained following due procedure [RFC8505] but does not constrain the way the addresses are formed or the number of addresses that are used in parallel by a same entity. A rogue may still perform denial-of-service attack against the registry at the 6LR or 6LBR, or attempt to deplete the pool of available addresses at Layer-2 or Layer-3.¶
A collision of Registration Ownership Verifiers (ROVR) (i.e., the Crypto-ID in this specification) is possible, but it is a rare event. Assuming in the calculations/discussion below that the hash used for calculating the Crypto-ID is a well-behaved cryptographic hash and thus that random collisions are the only ones possible, the formula (birthday paradox) for calculating the probability of a collision is 1 - e^{-k^2/(2n)} where n is the maximum population size (2^64 here, 1.84E19) and k is the actual population (number of nodes, assuming one Crypto-ID per node).¶
If the Crypto-ID is 64-bits (the least possible size allowed), the chance of a collision is 0.01% for network of 66 million nodes. Moreover, the collision is only relevant when this happens within one stub network (6LBR). In the case of such a collision, a third party node would be able to claim the registered address of an another legitimate node, provided that it wishes to use the same address. To prevent address disclosure and avoid the chances of collision on both the ROVR and the address, it is RECOMMENDED that nodes do not derive the address being registered from the ROVR.¶
The signature schemes referenced in this specification comply with NIST [FIPS186-4] or Crypto Forum Research Group (CFRG) standards [RFC8032] and offer strong algorithmic security at roughly 128-bit security level. These signature schemes use elliptic curves that were either specifically designed with exception-free and constant-time arithmetic in mind [RFC7748] or where one has extensive implementation experience of resistance to timing attacks [FIPS186-4]. However, careless implementations of the signing operations could nevertheless leak information on private keys. For example, there are micro-architectural side channel attacks that implementors should be aware of [breaking-ed25519]. Implementors should be particularly aware that a secure implementation of Ed25519 requires a protected implementation of the hash function SHA-512, whereas this is not required with implementations of SHA-256 used with ECDSA.¶
The keypair used in this specification can be self-generated and the public key does not need to be exchanged, e.g., through certificates, with a third party before it is used. New keypairs can be formed for new registration as the node desires. On the other hand, it is safer to allocate a keypair that is used only for the address protection and only for one instantiation of the signature scheme (which includes choice of elliptic curve domain parameters, used hash function, and applicable representation conventions). The same private key MUST NOT be reused with more than one instantiation of the signature scheme in this specification. The same private key MUST NOT be used for anything other than computing NDPSO signatures per this specification.¶
This specification distributes the challenge and its validation at the edge of the network, between the 6LN and its 6LR. This protects against DOS attacks targeted at that central 6LBR. This also saves back and forth exchanges across a potentially large and constrained network. The downside is that the 6LBR needs to trust the 6LR for performing the checking adequately, and the communication between the 6LR and the 6LBR must be protected to avoid tempering with the result of the test. If a 6LR is compromised, and provided that it knows the ROVR field used by the real owner of the address, the 6LR may pretend that the owner has moved, is now attached to it and has successfully passed the Crpto-ID validation. The 6LR may then attract and inject traffic at will on behalf of that address or let a rogue take ownership of the address.¶
The ROVR field in the EARO introduced in [RFC8505] extends the EUI-64 field of the ARO defined in [RFC6775]. One of the drawbacks of using an EUI-64 as ROVR is that an attacker that is aware of the registrations can correlate traffic for a same 6LN across multiple addresses. Section 3 of [RFC8505] indicates that the ROVR and the address being registered are decoupled. A 6LN may use a same ROVR for multiple registrations or a different ROVR per registration, and the IID must not derive from the ROVR. In theory different 6LNs could use a same ROVR as long as they do not attempt to register the same address.¶
The Modifier used in the computation of the Crypto-ID enables a 6LN to build different Crypto-IDs for different addresses with a same keypair. Using that facility improves the privacy of the 6LN as the expense of storage in the 6LR, which will need to store multiple CIPOs that contain the same public key. Note that if the attacker is the 6LR, then the Modifier alone does not provide a protection, and the 6LN would need to use different keys and MAC addresses in an attempt to obfuscate its multiple ownership.¶
This document defines a new 128-bit value under the CGA Message Type [RFC3972] name space: 0x8701 55c8 0cca dd32 6ab7 e415 f148 84d0.¶
This document registers two new ND option types under the subregistry "IPv6 Neighbor Discovery Option Formats":¶
Option Name | Suggested Value | Reference |
---|---|---|
NDP Signature Option (NDPSO) | 38 | This document |
Crypto-ID Parameters Option (CIPO) | 39 | This document |
IANA is requested to create a new subregistry "Crypto-Type Subregistry" in the "Internet Control Message Protocol version 6 (ICMPv6) Parameters". The registry is indexed by an integer in the interval 0..255 and contains an Elliptic Curve, a Hash Function, a Signature Algorithm, and Representation Conventions, as shown in Table 2, which together specify a signature scheme. The following Crypto-Type values are defined in this document:¶
Crypto-Type value | 0 (ECDSA256) | 1 (Ed25519) | 2 (ECDSA25519) |
---|---|---|---|
Elliptic curve | NIST P-256 [FIPS186-4] | Curve25519 [RFC7748] | Curve25519 [RFC7748] |
Hash function | SHA-256 [RFC6234] | SHA-512 [RFC6234] | SHA-256 [RFC6234] |
Signature algorithm | ECDSA [FIPS186-4] | Ed25519 [RFC8032] | ECDSA [FIPS186-4] |
Representation conventions | Weierstrass, uncompressed, MSB/msb first | Edwards, compressed, LSB/lsb first | Weierstrass, compressed, MSB/msb first |
Defining specification | This document | This document | This document |
New Crypto-Type values providing similar or better security may be defined in the future.¶
Assignment of new values for new Crypto-Type MUST be done through IANA with either "Specification Required" or "IESG Approval" as defined in BCP 26 [RFC8126].¶
The "Defining specification" column indicates the document that defines the length and computation of the digital signature, which could be this for values defined through "IESG Approval".¶
Code points are requested for curve Wei25519 and its use with ECDSA, using the representation conventions of this document.¶
This section registers the following value in the IANA "COSE Elliptic Curves" registry [IANA.COSE.Curves].¶
(Note that The "kty" value for Wei25519 may be "OKP" or "EC2".)¶
This section registers the following value in the IANA "COSE Algorithms" registry [IANA.COSE.Algorithms].¶
This section registers the following value in the IANA "JSON Web Key Elliptic Curve" registry [IANA.JOSE.Curves].¶
This section registers the following value in the IANA "JSON Web Signature and Encryption Algorithms" registry [IANA.JOSE.Algorithms].¶
Many thanks to Charlie Perkins for his in-depth review and constructive suggestions. The authors are also especially grateful to Robert Moskowitz and Benjamin Kaduk for their comments and discussions that led to many improvements. The authors wish to also thank Roman Danyliw, Alissa Cooper, Mirja Kuhlewind, Eric Vyncke, Vijay Gurbani, Al Morton and Adam Montville for their constructive reviews during the IESG process.¶
In this section we state requirements of a secure neighbor discovery protocol for low-power and lossy networks.¶
The signature scheme ECDSA256 corresponding to Crypto-Type 0 is ECDSA, as specified in [FIPS186-4], instantiated with the NIST prime curve P-256, as specified in Appendix B of [FIPS186-4], and the hash function SHA-256, as specified in [RFC6234], where points of this NIST curve are represented as points of a short-Weierstrass curve (see [FIPS186-4]) and are encoded as octet strings in most-significant-bit first (msb) and most-significant-byte first (MSB) order. The signature itself consists of two integers (r and s), which are each encoded as fixed-size octet strings in most-significant-bit first and most-significant-byte first order. For details on ECDSA, see [FIPS186-4]; for details on the integer encoding, see Appendix B.2.¶
The signature scheme Ed25519 corresponding to Crypto-Type 1 is EdDSA, as specified in [RFC8032], instantiated with the Montgomery curve Curve25519, as specified in [RFC7748], and the hash function SHA-512, as specified in [RFC6234], where points of this Montgomery curve are represented as points of the corresponding twisted Edwards curve (see Appendix B.3) and are encoded as octet strings in least-significant-bit first (lsb) and least-significant-byte first (LSB) order. The signature itself consists of a bit string that encodes a point of this twisted Edwards curve, in compressed format, and an integer encoded in least-significant-bit first and least-significant-byte first order. For details on EdDSA and on the encoding conversions, see the specification of pure Ed25519 in [RFC8032].¶
The signature scheme ECDSA25519 corresponding to Crypto-Type 2 is ECDSA, as specified in [FIPS186-4], instantiated with the Montgomery curve Curve25519, as specified in [RFC7748], and the hash function SHA-256, as specified in [RFC6234], where points of this Montgomery curve are represented as points of a corresponding curve in short-Weierstrass form (see Appendix B.3) and are encoded as octet strings in most-significant-bit first and most-significant-byte first order. The signature itself consists of a bit string that encodes two integers, each encoded as fixed-size octet strings in most-significant-bit first and most-significant-byte first order. For details on ECDSA, see [FIPS186-4]; for details on the integer encoding, see Appendix B.2¶
With ECDSA, each signature is an ordered pair (r, s) of integers [FIPS186-4]. Each integer is encoded as a fixed-size 256-bit bit string, where each integer is represented according to the Field Element to Octet String and Octet String to Bit String conversion rules in [SEC1] and where the ordered pair of integers is represented as the rightconcatenation of the resulting representation values. The inverse operation follows the corresponding Bit String to Octet String and Octet String to Field Element conversion rules of [SEC1].¶
The elliptic curve Curve25519, as specified in [RFC7748], is a so-called Montgomery curve. Each point of this curve can also be represented as a point of a twisted Edwards curve or as a point of an elliptic curve in short-Weierstrass form, via a coordinate transformation (a so-called isomorphic mapping). The parameters of the Montgomery curve and the corresponding isomorphic curves in twisted Edwards curve and short-Weierstrass form are as indicated below. Here, the domain parameters of the Montgomery curve Curve25519 and of the twisted Edwards curve Edwards25519 are as specified in [RFC7748]; the domain parameters of the elliptic curve Wei25519 in short-Weierstrass curve comply with Section 6.1.1 of [FIPS186-4]. For details of the coordinate transformations referenced above, see [RFC7748] and [CURVE-REPRESENTATIONS].¶
General parameters (for all curve models):¶
Montgomery curve-specific parameters (for Curve25519):¶
Twisted Edwards curve-specific parameters (for Edwards25519):¶
Weierstrass curve-specific parameters (for Wei25519):¶