Deterministic ECDSA and EdDSA Signatures with Additional Randomness

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Table of Contents

1. Introduction

In Elliptic-Curve Cryptography (ECC) signature algorithms, the per-message secret number has traditionally been generated from a random number generator (RNG). The security of such algorithms depends on the cryptographic quality of the random number generation and biases in the randomness may have catastrophic effects such as compromising private keys (see e.g., [Bernstein19]). Repeated per-message secret numbers have caused several severe security accidents in practice. As stated in [RFC6979], the need for a cryptographically secure source of randomness is also a hindrance to deployment of randomized ECDSA [FIPS-186-4] in architectures where secure random number generation is challenging, in particular, embedded IoT systems and smartcards. [ABFJLM17] does however state that smartcards typically has a high-quality RNG on board, which makes it significantly easier and faster to use the RNG instead of doing a hash computation.¶

In deterministic ECC signatures schemes such as Deterministic Elliptic Curve Digital Signature Algorithm (ECDSA) [RFC6979] and Edwards-curve Digital Signature Algorithm (EdDSA) [RFC8032], the per-message secret number is instead generated in a fully-deterministic way as a function of the message and the private key. Except for key generation, the security of such deterministic signatures does not depend on a source of high-quality randomness. As they are presumed to be safer, deterministic signatures have gained popularity and are referenced and recommended by a large number of recent RFCs [RFC8037] [RFC8080] [RFC8152] [RFC8225] [RFC8387] [RFC8410] [RFC8411] [RFC8419] [RFC8420] [RFC8422] [RFC8446] [RFC8463] [RFC8550] [RFC8591] [RFC8624] [RFC8208] [RFC8608].¶

Side-channel attacks are potential attack vectors for implementations of cryptographic algorithms. Side-Channel attacks can in general be classified along three orthogonal axes: passive vs. active, physical vs. logical, and local vs. remote [SideChannel]. It has been demonstrated how side-channel attacks such as power analysis [BCPST14] and timing attacks [Minerva19] [TPM-Fail19] allow for practical recovery of the private key in some existing implementations of randomized ECDSA. [BSI] summarizes minimum requirements for evaluating side-channel attacks of elliptic curve implementations and writes that deterministic ECDSA and EdDSA requires extra care. The deterministic ECDSA specification [RFC6979] notes that the deterministic generation of per-message secret numbers may be useful to an attacker in some forms of side-channel attacks and as stated in [Minerva19], deterministic signatures like [RFC6979] and [RFC8032] might help an attacker to reduce the noise in the side-channel when the same message it signed multiple times. Recent research [SH16] [BP16] [RP17] [ABFJLM17] [SBBDS17] [PSSLR17] [SB18] [WPB19] [AOTZ19] [FG19] have theoretically and experimentally analyzed the resistance of deterministic ECC signature algorithms against side-channel and fault injection attacks. The conclusions are that deterministic signature algorithms have theoretical weaknesses against certain instances of these types of attacks and that the attacks are practically feasibly in some environments. These types of attacks may be of particular concern for hardware implementations such as embedded IoT devices and smartcards where the adversary can be assumed to have access to the device to induce faults and measure its side-channels such as timing information with low signal-to-noise ratio, power consumption, electromagnetic leaks, or sound. A good summary of fault attacks in given by [Cao20]. See also the discussions and references in [Comments-186-5].¶

Fault attacks may also be possible without physical access to the device. RowHammer [RowHammer14] showed how an attacker to induce DRAM bit-flips in memory areas the attacker should not have access to and Plundervolt [Plundervolt19] showed how an attacker with root access can use frequency and voltage scaling interfaces to induce faults that bypasses even secure execution technologies. RowHammer can e.g. be used in operating systems with several processes or cloud scenarios with virtualized servers. Protocols like TLS, SSH, and IKEv2 that adds a random number to the message to be signed mitigate some types of attacks [PSSLR17].¶

Government agencies are clearly concerned about these attacks. In [Notice-186-5] and [Draft-186-5], NIST warns about side-channel and fault injection attacks, but states that deterministic ECDSA may be desirable for devices that lack good randomness. BSI has published [BSI] and researchers from BSI have co-authored two research papers [ABFJLM17] [PSSLR17] on attacks on deterministic signatures. For many industries it is important to be compliant with both RFCs and government publications, alignment between IETF, NIST, and BSI recommendations would be preferable.¶

One countermeasure to entropy failures, side-channel attacks, and fault injection attacks recommended by [Langley13] [RP17] [ABFJLM17] [SBBDS17] [PSSLR17] [SB18] [AOTZ19] [FG19] and implemented in [OpenSSL13a] [OpenSSL13b] [XEdDSA] [libSodium] [libHydrogen] is to generate the per-message secret number from a random string, a secret key, and the message. This combines the security benefits of fully-randomized per-message secret numbers with the security benefits of fully-deterministic secret numbers. Such a construction protects against key compromise due to weak random number generation, but still effectively prevents many side-channel and fault injection attacks that exploit determinism. Such a constuction require minor changes to the implementation and does not increase the number of elliptic curve point multiplications and is therefore suitable for constained IoT. Adding randomness to EdDSA is not compliant with [RFC8032]. [Kampanakis16] describes an alternative [FIPS-186-4] compliant approach where message specific pseudo-random information is used as an additional input to the random number generation to create per-message secret number. [Bernstein14] states that generation of the per-message secret number from a subset of a random string, a secret key, the message, and a message counter is common in DSA/ECDSA implementations.¶

This document updates [RFC6979] and [RFC8032] to recommend constructions with additional randomness for deployments where side-channel and fault injection attacks are a concern. Produced signatures remain fully compatible with unmodified ECDSA and EdDSA verifiers and existing key pairs can continue to be used. As the precise use of the noise is specified, test vectors can still be produced and implementations can be tested against them.¶

2. Updates to RFC 8032 (EdDSA)

For Ed25519ph, Ed25519ctx, and Ed25519: In deployments where side-channel and fault injection attacks are a concern, the following step is RECOMMENDED instead of step (2) in Section 5.1.6 of [RFC8032]:¶

   2.  Compute SHA-512(dom2(F, C) || Z || prefix || 000... || PH(M)),
       where M is the message to be signed, Z is 32 octets of random
       data, the number of zeroes 000... is chosen so that the length
       of (dom2(F, C) || Z || prefix || 000...)  is 1024 bytes.
       Interpret the 64-octet digest as a little-endian integer r.

For Ed448ph and Ed448: In deployments where side-channel and fault injection attacks are a concern, the following step is RECOMMENDED instead of step (2) in Section 5.3.6 of [RFC8032]:¶

   2.  Compute SHAKE256(dom4(F, C) || Z || prefix || 000... || PH(M),
       114), where M is the message to be signed, and Z is 57 octets
       of random data, the number of zeroes 000... is chosen so that
       the length of (dom4(F, C) || Z || prefix || 000...) is 1088
       bytes. F is 1 for Ed448ph, 0 for Ed448, and C is the context
       to use. Interpret the 114-octet digest as a little-endian
       integer r.

3. Updates to RFC 6979 (Deterministic ECDSA)

For Deterministic ECDSA: In existing ECDSA deployments where side-channel and fault injection attacks are a concern, the following steps are RECOMMENDED instead of steps (d) and (f) in Section 3.2 of [RFC6979]:¶

  d.  Set:

      K = HMAC_K(V || 0x00 || Z || int2octets(x) || 000... ||
      bits2octets(h1)) where '||' denotes concatenation.  In other
      words, we compute HMAC with key K, over the concatenation of
      the following, in order: the current value of V, a sequence of
      eight bits of value 0, random data Z (of the same length as
      int2octets(x)), the encoding of the (EC)DSA private key x, a
      sequence of zero bits 000... chosen so that the length of
      (V || 0x00 || Z || int2octets(x) || 000...) is equal to the
      block size of the hash function, and the hashed message
      (possibly truncated and extended as specified by the
      bits2octets transform).  The HMAC result is the new value of K.
      Note that the private key x is in the [1, q-1] range, hence a
      proper input for int2octets, yielding rlen bits of output,
      i.e., an integral number of octets (rlen is a multiple of 8).

  f.  Set:

      K = HMAC_K(V || 0x01 || Z || int2octets(x) || 000... ||
                 bits2octets(h1))

In new deployments, where side-channel and fault injection attacks are a concern, EdDSA with additional randomness as specified in Section 2 is RECOMMENDED.¶

4. Security Considerations

The constructions in this document follows the high-level approach in [XEdDSA] to calculate the per-message secret number from the hash of the private key and the message, but add additional randomness into the calculation for greater resilience. This does not re-introduce the strong security requirement of randomness needed by randomized ECDSA [FIPS-186-4]. The randomness of Z does not need to be perfect, but SHALL be generated by a cryptographically secure pseudo random number generator (PRNG) and SHALL be secret. Even if the same random number Z is used to sign two different messages, the security will be the same as deterministic ECDSA and EdDSA and an attacker will not be able to compromise the private key with algebraic means as in fully-randomized ECDSA [FIPS-186-4]. With the construction specified in this document, two signatures over two equal messages are different which prevents information leakage in use cases where signatures but not messages are public. The construction in this document place the additional randomness before the message to align with randomized hashing methods.¶

[SBBDS17] states that [XEdDSA] would not prevent their attack due to insufficient mixing of the hashed private key with the additional randomness. [SBBDS17] suggest a construction where the randomness is padded with zeroes so that the first 1024-bit SHA-512 block is composed only of the hashed private key and the random value, but not the message. The construction in this document follows this recommendation and pads with zeroes so that the first block is composed only of the hashed private key and the random value, but not the message.¶

Another countermeasure to fault attacks is to force the signer to verify the signature in the last step of the signature generation or to calculate the signature twice and compare the results. These countermeasure would catch a single fault but would not protect against attackers that are able to precisely inject faults several times [RP17] [PSSLR17] [SB18]. Adding an additional sign or verification operation would also significantly affect performance, especially verification which is a heavier operation than signing in ECDSA and EdDSA.¶

[ABFJLM17] suggests using both additional randomness and a counter, which makes the signature generation stateful. While most used signatures have traditionally been stateless, stateful signatures like XMSS [RFC8391] and LMS [RFC8554] have now been standardized and deployed. [RFC8937] specifies a PRNG construction with a random seed, a secret key, a context string, and a nonce, which makes the random number generation stateful. The generation of the per-message secret number in this document is not stateful, but it can be used with a stateful PRNG. The exact construction in [RFC8937] is however not recommended in deployments where side-channel and fault injection attacks are a concern as it relies on deterministic signatures.¶

With the construction in this document, the repetition of the same per-message secret number for two different messages is highly unlikely even with an imperfect random number generator, but not impossible. As an extreme countermeasure, previously used secret numbers can be tracked to ensure their uniqueness for a given key, and a different random number can be used if a collision is detected. This document does not mandate nor stop an implementation from taking such a precaution.¶

Implementations need to follow best practices on how to protect against all side-channel attacks, not just attacks that exploits determinism, see for example [BSI].¶

5. For discussion (to be removed in the future)

• Amount of randomness - The current construction uses random data of the same length as 'prefix' or 'int2octets(x)' which means 32 bytes of randomness for Ed25519. XEdDSA uses 64 bytes of randomness which might be overkill. As discussed in [SBBDS17], the amount of randomness needed depends on the targeted security level. 32 bytes of randomness should be enough for Ed448 and 16 bytes of randomness should be enough for Ed25519. Even less than that is likely sufficient to prevent practical attacks.¶
• Deterministic ECDSA with SHAKE - NIST is planning to approve SHAKE128(M,128) and SHAKE256(M,256) for use in ECDSA [Draft-186-5]. Deterministic ECDSA as specified in [RFC6979] would then use HMAC-SHAKE instead of a more optimal KMAC, which would be the prefered keyed hash function for use with SHAKE. It should be discussed if IETF (or NIST) should specify that the resulting HMAC-SHAKE128(K, M) and HMAC-SHAKE256(K, M) in deterministic ECDSA should be replaced with KMAC128(K,M,128) and KMAC256(K,M,128).¶

Acknowledgments

The authors want to thank Tony Arcieri, Uri Blumenthal, and Quynh Dang for their valuable comments and feedback.¶

Authors' Addresses