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Deterministic ECDSA and EdDSA Signatures with Additional Randomness
draft-mattsson-cfrg-det-sigs-with-noise-04

Document Type Active Internet-Draft (cfrg RG)
Authors John Preuß Mattsson , Erik Thormarker , Sini Ruohomaa
Last updated 2022-04-08 (Latest revision 2022-02-15)
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draft-mattsson-cfrg-det-sigs-with-noise-04
Network Working Group                                  J. Preuß Mattsson
Internet-Draft                                             E. Thormarker
Updates: 6979, 8032 (if approved)                            S. Ruohomaa
Intended status: Informational                                  Ericsson
Expires: 19 August 2022                                 15 February 2022

  Deterministic ECDSA and EdDSA Signatures with Additional Randomness
               draft-mattsson-cfrg-det-sigs-with-noise-04

Abstract

   Deterministic elliptic-curve signatures such as deterministic ECDSA
   and EdDSA have gained popularity over randomized ECDSA as their
   security do not depend on a source of high-quality randomness.
   Recent research has however found that implementations of these
   signature algorithms may be vulnerable to certain side-channel and
   fault injection attacks due to their determinism.  One countermeasure
   to such attacks is to re-add randomness to the otherwise
   deterministic calculation of the per-message secret number.  This
   document updates RFC 6979 and RFC 8032 to recommend constructions
   with additional randomness for deployments where side-channel attacks
   and fault injection attacks are a concern.  The updates are invisible
   to the validator of the signature and compatible with existing ECDSA
   and EdDSA validators.

Status of This Memo

   This Internet-Draft is submitted in full conformance with the
   provisions of BCP 78 and BCP 79.

   Internet-Drafts are working documents of the Internet Engineering
   Task Force (IETF).  Note that other groups may also distribute
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   Drafts is at https://datatracker.ietf.org/drafts/current/.

   Internet-Drafts are draft documents valid for a maximum of six months
   and may be updated, replaced, or obsoleted by other documents at any
   time.  It is inappropriate to use Internet-Drafts as reference
   material or to cite them other than as "work in progress."

   This Internet-Draft will expire on 19 August 2022.

Copyright Notice

   Copyright (c) 2022 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

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   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents (https://trustee.ietf.org/
   license-info) in effect on the date of publication of this document.
   Please review these documents carefully, as they describe your rights
   and restrictions with respect to this document.  Code Components
   extracted from this document must include Revised BSD License text as
   described in Section 4.e of the Trust Legal Provisions and are
   provided without warranty as described in the Revised BSD License.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
   2.  Conventions Used in This Document . . . . . . . . . . . . . .   5
   3.  Updates to RFC 8032 (EdDSA) . . . . . . . . . . . . . . . . .   5
   4.  Updates to RFC 6979 (Deterministic ECDSA) . . . . . . . . . .   5
   5.  Security Considerations . . . . . . . . . . . . . . . . . . .   6
   6.  For discussion (to be removed in the future)  . . . . . . . .   8
   7.  References  . . . . . . . . . . . . . . . . . . . . . . . . .   8
     7.1.  Normative References  . . . . . . . . . . . . . . . . . .   8
     7.2.  Informative References  . . . . . . . . . . . . . . . . .   9
   Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . .  15
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  15

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
   have 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.  This makes
   verification of implementations easier.  As they are presumed to be

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   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,
   power consumption, electromagnetic leaks, or sound with low signal-
   to-noise ratio.  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.  Plundervolt [Plundervolt19] showed how an attacker with root
   access can use frequency and voltage scaling interfaces to induce
   faults that bypass 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].

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   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.

   Note that deriving per-message secret number deterministically, is
   also insecure in a multi-party signature setting
   [I-D.irtf-cfrg-frost].

   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 construction require minor
   changes to the implementation and does not increase the number of
   elliptic curve point multiplications and is therefore suitable for
   constrained 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.  The updates are
   invisible to the validator of the signature.  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.

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2.  Conventions Used in This Document

   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.

3.  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 a multiple of 128
          octets. 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 a
          multiple of 136 octets. 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.

4.  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]:

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     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))

   When ECDSA is used with SHAKE [SHA3] the HMAC construction above MAY
   be used but it is RECOMMENDED to use the more efficient KMAC
   construction [KMAC].  SHAKE is a variable-length hash function
   defined as SHAKE(M, d) where the output is a d-bits-long digest of
   message M.  When ECDSA is used with SHAKE128(M, d), it is RECOMMENDED
   to replace HMAC(K, M) with KMAC128(K, M, d, "").  When ECDSA is used
   with SHAKE256(M, d), it is RECOMMENDED to replace HMAC(K, M) with
   KMAC256(K, M, d, "").  [RFC8692] and [Draft-186-5] define the use of
   SHAKE128 with an output length of 256 bits and SHAKE256 with an
   output length or 512 bits.

   In new deployments, where side-channel and fault injection attacks
   are a concern, EdDSA with additional randomness as specified in
   Section 3 is RECOMMENDED.

5.  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

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   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.

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   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 exploit
   determinism, see for example [BSI].

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

   *  removal of "noise" from filename.  Will be done if/when the draft
      is uploaded as adopted (draft-irtf-....)

   *  Strong consensus to change the name "Deterministic ECDSA and EdDSA
      Signatures with Additional Randomness".  The signatures are
      obliously not deteministic anymore.  Several suggestions for new
      names: "message-dependent", "message-keyed", "entropy stealing",
      "entropy combining", "whitening", "keyed entropy whitening",
      "hedged", "noise".

   *  Ordering of the parameters in "dom2(F, C) || Z || prefix ||
      000... || PH(M)" in Ed25519 and similar in Ed448 and ECDSA.  There
      has also been sugestion to use a larger Z and to use several
      paddings 000....

   *  Ilari Liusvaara pointed out attacks using the context that needs
      to be considered.  Some statements "first block is composed only
      of the hashed private key and the random value" in the document
      are not true for Ed25519ctx and Ed448ctx.

   *  Jim Schaad: Is there any advantage to stealing one of the zeros
      from the end padding and using it to pad between 'Z' and 'x' in
      the construction?  I would assume that it should use the '0'/'1'
      construction between steps d and f.

   *  Jim Schaad: Is there any advantage to padding with 0x01 in step f
      rather than 0x00?

   *  Rene Stuik: MUST instead of RECOMMENDED.

7.  References

7.1.  Normative References

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   [FIPS-186-4]
              Department of Commerce, N. S., "Digital Signature Standard
              (DSS)", NIST FIPS PUB 186-4 , July 2013,
              <https://nvlpubs.nist.gov/nistpubs/FIPS/
              NIST.FIPS.186-4.pdf>.

   [KMAC]     National Institute of Standards and Technology (NIST),
              "SHA-3 Derived Functions: cSHAKE, KMAC, TupleHash and
              ParallelHash", NIST SP 800-185 , December 2016,
              <http://nvlpubs.nist.gov/nistpubs/SpecialPublications/
              NIST.SP.800-185.pdf>.

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997,
              <https://www.rfc-editor.org/info/rfc2119>.

   [RFC6979]  Pornin, T., "Deterministic Usage of the Digital Signature
              Algorithm (DSA) and Elliptic Curve Digital Signature
              Algorithm (ECDSA)", RFC 6979, DOI 10.17487/RFC6979, August
              2013, <https://www.rfc-editor.org/info/rfc6979>.

   [RFC8032]  Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital
              Signature Algorithm (EdDSA)", RFC 8032,
              DOI 10.17487/RFC8032, January 2017,
              <https://www.rfc-editor.org/info/rfc8032>.

   [RFC8174]  Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
              2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
              May 2017, <https://www.rfc-editor.org/info/rfc8174>.

   [RFC8692]  Kampanakis, P. and Q. Dang, "Internet X.509 Public Key
              Infrastructure: Additional Algorithm Identifiers for
              RSASSA-PSS and ECDSA Using SHAKEs", RFC 8692,
              DOI 10.17487/RFC8692, December 2019,
              <https://www.rfc-editor.org/info/rfc8692>.

   [SHA3]     National Institute of Standards and Technology (NIST),
              "SHA-3 Standard: Permutation-Based Hash and Extendable-
              Output Functions", NIST FIPS PUB 202 , August 2015,
              <https://nvlpubs.nist.gov/nistpubs/FIPS/
              NIST.FIPS.202.pdf>.

7.2.  Informative References

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   [ABFJLM17] Ambrose, C., Bos, J., Fay, B., Joye, M., Lochter, M., and
              B. Murray, "Differential Attacks on Deterministic
              Signatures", October 2017,
              <https://eprint.iacr.org/2017/975>.

   [AOTZ19]   Aranha, D., Orlandi, C., Takahashi, A., and G. Zaverucha,
              "Security of Hedged Fiat-Shamir Signatures under Fault
              Attacks", September 2019,
              <https://eprint.iacr.org/2019/956>.

   [BCPST14]  Batina, L., Chmielewski, L., Papachristodoulou, L.,
              Schwabe, P., and M. Tunstall, "Online Template Attacks",
              December 2014, <http://citeseerx.ist.psu.edu/viewdoc/
              download?doi=10.1.1.854.7836&rep=rep1&type=pdf>.

   [Bernstein14]
              Bernstein, D., "How to design an elliptic-curve signature
              system", March 2014,
              <https://blog.cr.yp.to/20140323-ecdsa.html>.

   [Bernstein19]
              Bernstein, D., "Why EdDSA held up better than ECDSA
              against Minerva", October 2019,
              <https://blog.cr.yp.to/20191024-eddsa.html>.

   [BP16]     Barenghi, A. and G. Pelosi, "A Note on Fault Attacks
              Against Deterministic Signature Schemes (Short Paper)",
              September 2016, <https://link.springer.com/
              chapter/10.1007/978-3-319-44524-3_11>.

   [BSI]      Bundesamt für Sicherheit in der Informationstechnik,
              "Minimum Requirements for Evaluating Side-Channel Attack
              Resistance of Elliptic Curve Implementations", November
              2016,
              <https://www.bsi.bund.de/SharedDocs/Downloads/DE/BSI/
              Zertifizierung/Interpretationen/
              AIS_46_ECCGuide_e_pdf.pdf?__blob=publicationFile>.

   [Cao20]    Weiqiong Cao, Hongsong Shi, Hua Chen, Jiazhe Chen, Limin
              Fan, and Wenling Wu, "Lattice-based Fault Attacks on
              Deterministic Signature Schemes of ECDSA and EdDSA", June
              2020, <https://eprint.iacr.org/2020/803>.

   [Comments-186-5]
              "Public Comments Received on Draft FIPS 186-5: Digital
              Signature Standards (DSS)", March 2021,
              <https://csrc.nist.gov/CSRC/media/Publications/fips/186/5/
              draft/documents/fips-186-5-draft-comments-received.pdf>.

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   [Draft-186-5]
              National Institute of Standards and Technology (NIST),
              "FIPS PUB 186-5 (Draft)", October 2019,
              <https://nvlpubs.nist.gov/nistpubs/FIPS/
              NIST.FIPS.186-5-draft.pdf>.

   [FG19]     Fischlin, M. and F. Günther, "Modeling Memory Faults in
              Signature and Encryption Schemes", September 2019,
              <https://eprint.iacr.org/2019/1053>.

   [I-D.irtf-cfrg-frost]
              Connolly, D., Komlo, C., Goldberg, I., and C. A. Wood,
              "Two-Round Threshold Schnorr Signatures with FROST", Work
              in Progress, Internet-Draft, draft-irtf-cfrg-frost-02, 12
              February 2022, <https://www.ietf.org/archive/id/draft-
              irtf-cfrg-frost-02.txt>.

   [Kampanakis16]
              Kampanakis, P., "FIPS and Deterministic ECDSA: Achieving
              robust security and conformance", December 2016,
              <https://blogs.cisco.com/security/fips-and-deterministic-
              ecdsa-achieving-robust-security-and-conformance>.

   [Langley13]
              Langley, A., "Sudden Death Entropy Failures", June 2013,
              <https://www.imperialviolet.org/2013/06/15/
              suddendeathentropy.html>.

   [libHydrogen]
              "The Hydrogen library", n.d.,
              <https://github.com/jedisct1/libhydrogen>.

   [libSodium]
              "The Sodium library", n.d.,
              <https://github.com/jedisct1/libsodium>.

   [Minerva19]
              Centre for Research on Cryptography and Security (CRoCS),
              "Minerva", October 2019,
              <https://minerva.crocs.fi.muni.cz/>.

   [Notice-186-5]
              National Institute of Standards and Technology (NIST),
              "Request for Comments on FIPS 186-5 and SP 800-186",
              October 2019, <https://www.federalregister.gov/
              documents/2019/10/31/2019-23742/request-for-comments-on-
              fips-186-5-and-sp-800-186>.

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   [OpenSSL13a]
              "Add secure DSA nonce flag", n.d.,
              <https://github.com/openssl/openssl/
              commit/8a99cb29d1f0013243a532bccc1dc70ed678eebe>.

   [OpenSSL13b]
              "Make `safe' (EC)DSA nonces the default", n.d.,
              <https://github.com/openssl/openssl/
              commit/190c615d4398cc6c8b61eb7881d7409314529a75>.

   [Plundervolt19]
              Murdock, K., Oswald, D., Garcia, F., Van Bulck, J., Gruss,
              D., and F. Piessens, "How a little bit of undervolting can
              cause a lot of problems", December 2019,
              <https://plundervolt.com/>.

   [PSSLR17]  Poddebniak, D., Somorovsky, J., Schinzel, S., Lochter, M.,
              and P. Rösler, "Attacking Deterministic Signature Schemes
              using Fault Attacks", October 2017,
              <https://eprint.iacr.org/2017/1014>.

   [RFC8037]  Liusvaara, I., "CFRG Elliptic Curve Diffie-Hellman (ECDH)
              and Signatures in JSON Object Signing and Encryption
              (JOSE)", RFC 8037, DOI 10.17487/RFC8037, January 2017,
              <https://www.rfc-editor.org/info/rfc8037>.

   [RFC8080]  Sury, O. and R. Edmonds, "Edwards-Curve Digital Security
              Algorithm (EdDSA) for DNSSEC", RFC 8080,
              DOI 10.17487/RFC8080, February 2017,
              <https://www.rfc-editor.org/info/rfc8080>.

   [RFC8152]  Schaad, J., "CBOR Object Signing and Encryption (COSE)",
              RFC 8152, DOI 10.17487/RFC8152, July 2017,
              <https://www.rfc-editor.org/info/rfc8152>.

   [RFC8208]  Turner, S. and O. Borchert, "BGPsec Algorithms, Key
              Formats, and Signature Formats", RFC 8208,
              DOI 10.17487/RFC8208, September 2017,
              <https://www.rfc-editor.org/info/rfc8208>.

   [RFC8225]  Wendt, C. and J. Peterson, "PASSporT: Personal Assertion
              Token", RFC 8225, DOI 10.17487/RFC8225, February 2018,
              <https://www.rfc-editor.org/info/rfc8225>.

   [RFC8387]  Sethi, M., Arkko, J., Keranen, A., and H. Back, "Practical
              Considerations and Implementation Experiences in Securing
              Smart Object Networks", RFC 8387, DOI 10.17487/RFC8387,
              May 2018, <https://www.rfc-editor.org/info/rfc8387>.

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   [RFC8391]  Huelsing, A., Butin, D., Gazdag, S., Rijneveld, J., and A.
              Mohaisen, "XMSS: eXtended Merkle Signature Scheme",
              RFC 8391, DOI 10.17487/RFC8391, May 2018,
              <https://www.rfc-editor.org/info/rfc8391>.

   [RFC8410]  Josefsson, S. and J. Schaad, "Algorithm Identifiers for
              Ed25519, Ed448, X25519, and X448 for Use in the Internet
              X.509 Public Key Infrastructure", RFC 8410,
              DOI 10.17487/RFC8410, August 2018,
              <https://www.rfc-editor.org/info/rfc8410>.

   [RFC8411]  Schaad, J. and R. Andrews, "IANA Registration for the
              Cryptographic Algorithm Object Identifier Range",
              RFC 8411, DOI 10.17487/RFC8411, August 2018,
              <https://www.rfc-editor.org/info/rfc8411>.

   [RFC8419]  Housley, R., "Use of Edwards-Curve Digital Signature
              Algorithm (EdDSA) Signatures in the Cryptographic Message
              Syntax (CMS)", RFC 8419, DOI 10.17487/RFC8419, August
              2018, <https://www.rfc-editor.org/info/rfc8419>.

   [RFC8420]  Nir, Y., "Using the Edwards-Curve Digital Signature
              Algorithm (EdDSA) in the Internet Key Exchange Protocol
              Version 2 (IKEv2)", RFC 8420, DOI 10.17487/RFC8420, August
              2018, <https://www.rfc-editor.org/info/rfc8420>.

   [RFC8422]  Nir, Y., Josefsson, S., and M. Pegourie-Gonnard, "Elliptic
              Curve Cryptography (ECC) Cipher Suites for Transport Layer
              Security (TLS) Versions 1.2 and Earlier", RFC 8422,
              DOI 10.17487/RFC8422, August 2018,
              <https://www.rfc-editor.org/info/rfc8422>.

   [RFC8446]  Rescorla, E., "The Transport Layer Security (TLS) Protocol
              Version 1.3", RFC 8446, DOI 10.17487/RFC8446, August 2018,
              <https://www.rfc-editor.org/info/rfc8446>.

   [RFC8463]  Levine, J., "A New Cryptographic Signature Method for
              DomainKeys Identified Mail (DKIM)", RFC 8463,
              DOI 10.17487/RFC8463, September 2018,
              <https://www.rfc-editor.org/info/rfc8463>.

   [RFC8550]  Schaad, J., Ramsdell, B., and S. Turner, "Secure/
              Multipurpose Internet Mail Extensions (S/MIME) Version 4.0
              Certificate Handling", RFC 8550, DOI 10.17487/RFC8550,
              April 2019, <https://www.rfc-editor.org/info/rfc8550>.

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   [RFC8554]  McGrew, D., Curcio, M., and S. Fluhrer, "Leighton-Micali
              Hash-Based Signatures", RFC 8554, DOI 10.17487/RFC8554,
              April 2019, <https://www.rfc-editor.org/info/rfc8554>.

   [RFC8591]  Campbell, B. and R. Housley, "SIP-Based Messaging with S/
              MIME", RFC 8591, DOI 10.17487/RFC8591, April 2019,
              <https://www.rfc-editor.org/info/rfc8591>.

   [RFC8608]  Turner, S. and O. Borchert, "BGPsec Algorithms, Key
              Formats, and Signature Formats", RFC 8608,
              DOI 10.17487/RFC8608, June 2019,
              <https://www.rfc-editor.org/info/rfc8608>.

   [RFC8624]  Wouters, P. and O. Sury, "Algorithm Implementation
              Requirements and Usage Guidance for DNSSEC", RFC 8624,
              DOI 10.17487/RFC8624, June 2019,
              <https://www.rfc-editor.org/info/rfc8624>.

   [RFC8937]  Cremers, C., Garratt, L., Smyshlyaev, S., Sullivan, N.,
              and C. Wood, "Randomness Improvements for Security
              Protocols", RFC 8937, DOI 10.17487/RFC8937, October 2020,
              <https://www.rfc-editor.org/info/rfc8937>.

   [RowHammer14]
              Kim, Y., Daly, R., Kim, J., Fallin, C., Lee, J., Lee, D.,
              Wilkerson, C., and K. Mutlu, "Flipping Bits in Memory
              Without Accessing Them: An Experimental Study of DRAM
              Disturbance Errors", June 2014,
              <https://users.ece.cmu.edu/~yoonguk/papers/kim-
              isca14.pdf>.

   [RP17]     Romailler, Y. and S. Pelissier, "Practical fault attack
              against the Ed25519 and EdDSA signature schemes",
              September 2017,
              <https://romailler.ch/ddl/10.1109_FDTC.2017.12_eddsa.pdf>.

   [SB18]     Samwel, N. and L. Batina, "Practical Fault Injection on
              Deterministic Signatures: The Case of EdDSA", April 2018,
              <https://nielssamwel.nl/papers/africacrypt2018_fault.pdf>.

   [SBBDS17]  Samwel, N., Batina, L., Bertoni, G., Daemen, J., and R.
              Susella, "Breaking Ed25519 in WolfSSL", October 2017,
              <https://eprint.iacr.org/2017/985.pdf>.

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   [SH16]     Seuschek, H., Heyszl, J., and F. De Santis, "A Cautionary
              Note: Side-Channel Leakage Implications of Deterministic
              Signature Schemes", January 2016,
              <http://www.cs2.deib.polimi.it/
              slides_16/01_Seuschek_Deterministic_Signatures.pdf>.

   [SideChannel]
              Spreitzer, R., Moonsamy, V., Korak, T., and S. Mangard,
              "Systematic Classification of Side-Channel Attacks: A Case
              Study for Mobile Devices", December 2017,
              <https://arxiv.org/pdf/1611.03748.pdf>.

   [TPM-Fail19]
              Moghimi, D., Sunar, B., Eisenbarth, T., and N. Heninge,
              "TPM-FAIL: TPM meets Timing and Lattice Attacks", October
              2019, <https://tpm.fail/>.

   [WPB19]    Weissbart, L., Picek, S., and L. Batina, "One trace is all
              it takes: Machine Learning-based Side-channel Attack on
              EdDSA", July 2019, <https://eprint.iacr.org/2019/358.pdf>.

   [XEdDSA]   Signal, "The XEdDSA and VXEdDSA Signature Schemes",
              October 2016,
              <https://signal.org/docs/specifications/xeddsa/>.

Acknowledgments

   The authors want to thank Tony Arcieri, Uri Blumenthal, Carsten
   Bormann, Phillip Hallam-Baker, Chelsea Komlo, Quynh Dang, Janos
   Follath, Ilari Liusvaara, Jim Schaad, and Ruggero Susella for their
   valuable comments and feedback.

Authors' Addresses

   John Preuß Mattsson
   Ericsson

   Email: john.mattsson@ericsson.com

   Erik Thormarker
   Ericsson

   Email: erik.thormarker@ericsson.com

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   Sini Ruohomaa
   Ericsson

   Email: sini.ruohomaa@ericsson.com

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