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AREA WG Working Group Internet-Draft This document describes extensions to existing signature schemes for key blinding. This functionality guarantees that a blinded public key and all signatures produced using the blinded key pair are unlinkable to the unblinded key pair. Moreover, signatures produced using blinded key pairs are indistinguishable from signatures produced using unblinded key pairs. About This Document The latest revision of this draft can be found at . Status information for this document may be found at . Discussion of this document takes place on the CFRG Working Group mailing list (), which is archived at . Source for this draft and an issue tracker can be found at .

Introduction EdDSA is a type of Schnorr signature algorithm based on Edwards curves. The specification describes several variants of EdDSA with parameter sets for the edwards25519 and edwards448 curves as described in . According to the specification, private keys are randomly generated seeds, which are then used to derive scalar elements and their corresponding public group element for signing and verifying messages, respectively. Given an EdDSA private and public key pair (sk, pk), any message signed by sk is linkable to pk. One simply checks whether the message signature is valid under pk. In some settings, in is useful to produce signatures with a given key pair (sk, pk) such that the resulting signature is not linkable to pk without knowledge of a particular witness r. That is, given pk corresponding to sk, witness r, and a message signature, one can determine if the signature was indeed produced using sk. In effect, the witness "blinds" the key pair associated with a message signature. This functionality is also possible with other signature schemes, including and some post-quantum signature schemes . This document describes a modification to the EdDSA key generation and signing procedures in to support this blinding operation, referred to as key blinding. It also specifies an extension to that enables the same functionality.

DISCLAIMER This document is a work in progress and is still undergoing security analysis. As such, it MUST NOT be used for real world applications. See for additional information.

Conventions and Definitions 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 when, and only when, they appear in all capitals, as shown here. The following terms are used throughout this document to describe the blinding modification.

• G: The standard base point.
• sk: A signature scheme private key. For EdDSA, this is a a randomly generated private seed of length 32 bytes or 57 bytes according to or , respectively. For , sk is a random scalar in the prime-order elliptic curve group.
• pk(sk): The public key corresponding to the private key sk.
• concat(x0, ..., xN): Concatenation of byte strings. concat(0x01, 0x0203, 0x040506) = 0x010203040506.
• ScalarMult(pk, k): Multiply the public key pk by scalar k, producing a new public key as a result.
• ModInverse(x, L): Compute the multiplicative inverse of x modulo L.

In pseudocode descriptions below, integer multiplication of two scalar values is denoted by the * operator. For example, the product of two scalars x and y is denoted as x * y.

Key Blinding At a high level, a signature scheme with key blinding allows signers to blind their signing key such that any signature produced under the blinded signing key is unlinkable from the unblinded signing key. Similar to the signing key, the blind is also a private key that remains secret. For example, the blind is a 32-byte or 57-byte random seed for Ed25519 or Ed448 variants, respectively, whereas the blind for ECDSA over P-256 is a random scalar in the P-256 group. Key blinding introduces three new functionalities for the signature scheme:

• BlindPublicKey(pkS, skB): Blind the public key pkS using the private key skB.
• UnblindPublicKey(pkM, skB): Unblind the public key pkM using the private key skB.
• BlindKeySign(skS, skB, msg): Sign a message msg using the private key skS with the private blind skB.

Correctness requires the following equivalence to hold: Security requires that signatures produced using BlindKeySign are unlinkable from signatures produced using the standard signature generation function with the same private key.

Ed25519ph, Ed25519ctx, and Ed25519 This section describes implementations of BlindPublicKey, UnblindPublicKey, and BlindKeySign as modifications of routines in .

BlindPublicKey and UnblindPublicKey BlindPublicKey transforms a private blind skB into a scalar for the edwards25519 group and then multiplies the target key by this scalar. UnblindPublicKey performs essentially the same steps except that it multiplies the target public key by the multiplicative inverse of the scalar, where the inverse is computed using the order of the group L, described in . More specifically, BlindPublicKey(pk, skB) works as follows.

1. Hash the 32-byte private key skB using SHA-512, storing the digest in a 64-octet large buffer, denoted h. Only the lower 32 bytes are used for generating the public key.
2. Interpret the buffer as a little-endian integer, forming a secret scalar s. Note that this explicitly skips the buffer pruning step in . Perform a scalar multiplication ScalarMult(pk, s), and output the encoding of the resulting point as the public key.

UnblindPublicKey(pkM, skB) works as follows.

1. Compute the secret scalar s from skB as in BlindPublicKey.
2. Compute the sInv = ModInverse(s, L), where L is as defined in .
3. Perform a scalar multiplication ScalarMult(pk, sInv), and output the encoding of the resulting point as the public key.

BlindKeySign BlindKeySign transforms a private key skB into a scalar for the edwards25519 group and a message prefix to blind both the signing scalar and the prefix of the message used in the signature generation routine. More specifically, BlindKeySign(skS, skB, msg) works as follows:

1. Hash the private key skS, 32 octets, using SHA-512. Let h denote the resulting digest. Construct the secret scalar s1 from the first half of the digest, and the corresponding public key A1, as described in . Let prefix1 denote the second half of the hash digest, h[32],...,h[63].
2. Hash the 32-byte private key skB using SHA-512, storing the digest in a 64-octet large buffer, denoted b. Interpret the lower 32 bytes buffer as a little-endian integer, forming a secret scalar s2. Let prefix2 denote the second half of the hash digest, b[32],...,b[63].
3. Compute the signing scalar s = s1 * s2 (mod L) and the signing public key A = ScalarMult(G, s).
4. Compute the signing prefix as concat(prefix1, prefix2).
5. Run the rest of the Sign procedure in from step (2) onwards using the modified scalar s, public key A, and string prefix.

Ed448ph and Ed448 This section describes implementations of BlindPublicKey, UnblindPublicKey, and BlindKeySign as modifications of routines in .

BlindPublicKey and UnblindPublicKey BlindPublicKey and UnblindPublicKey for Ed448ph and Ed448 are implemented just as these routines are for Ed25519ph, Ed25519ctx, and Ed25519, except that SHAKE256 is used instead of SHA-512 for hashing the secret blind to a 114-byte buffer and the order of the edwards448 group L is as defined in .

BlindKeySign BlindKeySign for Ed448ph and Ed448 is implemented just as this routine for Ed25519ph, Ed25519ctx, and Ed25519, except in how the scalars (s1, s2), public keys (A1, A2), and message strings (prefix1, prefix2) are computed. More specifically, BlindKeySign(skS, skB, msg) works as follows:

1. Hash the private key skS, 57 octets, using SHAKE256(skS, 117). Let h denote the resulting digest. Construct the secret scalar s1 from the first half of the digest, and the corresponding public key A1, as described in . Let prefix1 denote the second half of the hash digest, h[57],...,h[113].
2. Perform the same routine to transform the secret blind skB into a secret scalar s2, public key A2, and prefix2.
3. Compute the signing scalar s = s1 * s2 (mod L) and the signing public key A = ScalarMult(A1, s2).
4. Compute the signing prefix as concat(prefix1, prefix2).
5. Run the rest of the Sign procedure in from step (2) onwards using the modified scalar s, public key A, and string prefix.

ECDSA [[DISCLAIMNER: Multiplicative blinding for ECDSA is known to be NOT be SUF-CMA-secure in the presence of an adversary that controls the blinding value. describes this in the context of related-key attacks. This variant may likely be removed in followup versions of this document based on further analysis.]] This section describes implementations of BlindPublicKey, UnblindPublicKey, and BlindKeySign as functions implemented on top of an existing implementation. In the descriptions below, let p be the order of the corresponding elliptic curve group used for ECDSA. For example, for P-256, p = 115792089210356248762697446949407573529996955224135760342422259061068512044369.

BlindPublicKey and UnblindPublicKey BlindPublicKey multiplies the public key pkS by an augmented private key skB yielding a new public key pkR. UnblindPublicKey inverts this process by multiplying the input public key by the multiplicative inverse of the augmented skB. Augmentation here maps the private key skB to another scalar using hash_to_field as defined in , with DST set to "ECDSA Key Blind", L set to the value corresponding to the target curve, e.g., 48 for P-256 and 72 for P-384, expand_message_xmd with a hash function matching that used for the corresponding digital signature algorithm, and prime modulus equal to the order p of the corresponding curve. Letting HashToScalar denote this augmentation process, BlindPublicKey and UnblindPublicKey are then implemented as follows: ~~~ BlindPublicKey(pk, skB) = ScalarMult(pk, HashToScalar(skB)) UnblindPublicKey(pk, skB) = ScalarMult(pk, ModInverse(HashToScalar(skB), p)) ~~~

BlindKeySign BlindKeySign transforms the signing key skS by the private key skB into a new signing key, skR, and then invokes the existing ECDSA signing procedure. More specifically, skR = skS * HashToScalar(skB) (mod p).

Security Considerations The signature scheme extensions in this document aim to achieve unforgeability and unlinkability. Informally, unforgeability means that one cannot produce a valid (message, signature) pair for any blinding key without access to the private signing key. Similarly, unlinkability means that one cannot distinguish between two signatures produced from two separate key signing keys, and two signatures produced from the same signing key but with different blinds. Security analysis of the extensions in this document with respect to these two properties is currently underway. Preliminary analysis has been done for a variant of these extensions used for identity key blinding routine used in Tor's Hidden Service feature . For EdDSA, further analysis is needed to ensure this is compliant with the signature algorithm described in . The constructions in this document assume that both the signing and blinding keys are private, and, as such, not controlled by an attacker. demonstrate that ECDSA with attacker-controlled multiplicative blinding for producing related keys can be abused to produce forgeries. In particular, if an attacker can control the private blinding key used in BlindKeySign, they can construct a forgery over a different message that validates under a different public key. Further analysis is needed to determine whether or not it is safe to keep this functionality in the specification given this problem.

IANA Considerations This document has no IANA actions.

Test Vectors This section contains test vectors for a subset of the signature schemes covered in this document.

Ed25519 Test Vectors This section contains test vectors for Ed25519 as described in . Each test vector lists the private key and blind seeds, denoted skS and skB and encoded as hexadecimal strings, along with their corresponding public keys pkS and pkB encoded has hexadecimal strings according to . Each test vector also includes the blinded public key pkR computed from skS and skB, denoted pkR and encoded has a hexadecimal string. Finally, each vector includes the message and signature values, each encoded as hexadecimal strings.

ECDSA(P-256, SHA-256) Test Vectors This section contains test vectors for ECDSA with P-256 and SHA-256, as described in . Each test vector lists the signing and blinding keys, denoted skS and skB, each serialized as a big-endian integers and encoded as hexadecimal strings. Each test vector also lists the unblinded and blinded public keys, denoted pkS and pkB and encoded as compressed elliptic curve points according to . Finally, each vector lists message and signature values, where the message is encoded as a hexadecimal string, and the signature value is serialized as the concatenation of scalars (r, s) and encoded as a hexadecimal string.

References Normative References American National Standards Institute This document describes elliptic curve signature scheme Edwards-curve Digital Signature Algorithm (EdDSA). The algorithm is instantiated with recommended parameters for the edwards25519 and edwards448 curves. An example implementation and test vectors are provided. In many standards track documents several words are used to signify the requirements in the specification. These words are often capitalized. This document defines these words as they should be interpreted in IETF documents. This document specifies an Internet Best Current Practices for the Internet Community, and requests discussion and suggestions for improvements. RFC 2119 specifies common key words that may be used in protocol specifications. This document aims to reduce the ambiguity by clarifying that only UPPERCASE usage of the key words have the defined special meanings. Informative References This memo specifies two elliptic curves over prime fields that offer a high level of practical security in cryptographic applications, including Transport Layer Security (TLS). These curves are intended to operate at the ~128-bit and ~224-bit security level, respectively, and are generated deterministically based on a list of required properties. Cloudflare, Inc. Cornell Tech Cloudflare, Inc. Stanford University Cloudflare, Inc. This document specifies a number of algorithms for encoding or hashing an arbitrary string to a point on an elliptic curve. This document is a product of the Crypto Forum Research Group (CFRG) in the IRTF.

Acknowledgments The authors would like to thank Dennis Jackson for helpful discussions that informed the development of this draft.