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Versions: 00 01 02

Network Working Group                                 J. Preuss Mattsson
Internet-Draft                                             E. Thormarker
Updates: 6979, 8032 (if approved)                            S. Ruohomaa
Intended status: Informational                                  Ericsson
Expires: September 12, 2020                               March 11, 2020


  Deterministic ECDSA and EdDSA Signatures with Additional Randomness
               draft-mattsson-cfrg-det-sigs-with-noise-02

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.

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
   working documents as Internet-Drafts.  The list of current Internet-
   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 September 12, 2020.

Copyright Notice

   Copyright (c) 2020 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents



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   (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 Simplified BSD License text as described in Section 4.e of
   the Trust Legal Provisions and are provided without warranty as
   described in the Simplified BSD License.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
   2.  Updates to RFC 8032 (EdDSA) . . . . . . . . . . . . . . . . .   4
   3.  Updates to RFC 6979 (Deterministic ECDSA) . . . . . . . . . .   5
   4.  Security Considerations . . . . . . . . . . . . . . . . . . .   5
   5.  For discussion (to be removed in the future)  . . . . . . . .   7
   6.  References  . . . . . . . . . . . . . . . . . . . . . . . . .   7
     6.1.  Normative References  . . . . . . . . . . . . . . . . . .   7
     6.2.  Informative References  . . . . . . . . . . . . . . . . .   8
   Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . .  12
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  12

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.  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]



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   [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.  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



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   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 side-channel and fault injection attacks
   recommended by [RP17] [ABFJLM17] [SBBDS17] [PSSLR17] [SB18] [AOTZ19]
   [FG19] and implemented in [XEdDSA] [libSodium] [libHydrogen] is to
   re-introduce some additional randomness to the otherwise
   deterministic generation of the per-message secret number.  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.  Deterministic ECDSA
   with additional randomness can be made compliant with [FIPS-186-4]
   but would not be compliant with the recommendations in many RFCs.
   Adding randomness to EdDSA is not compliant with [RFC8032].

   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]:







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      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



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   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.  [I-D.irtf-cfrg-randomness-improvements] 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 [I-D.irtf-cfrg-randomness-improvements] 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



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   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)

   o  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.

   o  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).

6.  References

6.1.  Normative References

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

   [I-D.irtf-cfrg-randomness-improvements]
              Cremers, C., Garratt, L., Smyshlyaev, S., Sullivan, N.,
              and C. Wood, "Randomness Improvements for Security
              Protocols", draft-irtf-cfrg-randomness-improvements-10
              (work in progress), February 2020.




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   [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>.

6.2.  Informative References

   [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>.

   [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 fuer 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>.

   [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>.





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

   [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>.

   [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. Roesler, "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>.



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   [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>.

   [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>.






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   [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>.

   [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>.

   [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>.



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   [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>.

   [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, and Quynh
   Dang 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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