1. RFC 8937
Internet Research Task Force (IRTF)                           C. Cremers
Request for Comments: 8937                                         CISPA
Category: Informational                                       L. Garratt
ISSN: 2070-1721                                             Cisco Meraki
                                                           S. Smyshlyaev
                                                             N. Sullivan
                                                                 C. Wood
                                                            October 2020

             Randomness Improvements for Security Protocols


   Randomness is a crucial ingredient for Transport Layer Security (TLS)
   and related security protocols.  Weak or predictable
   "cryptographically secure" pseudorandom number generators (CSPRNGs)
   can be abused or exploited for malicious purposes.  An initial
   entropy source that seeds a CSPRNG might be weak or broken as well,
   which can also lead to critical and systemic security problems.  This
   document describes a way for security protocol implementations to
   augment their CSPRNGs using long-term private keys.  This improves
   randomness from broken or otherwise subverted CSPRNGs.

   This document is a product of the Crypto Forum Research Group (CFRG)
   in the IRTF.

Status of This Memo

   This document is not an Internet Standards Track specification; it is
   published for informational purposes.

   This document is a product of the Internet Research Task Force
   (IRTF).  The IRTF publishes the results of Internet-related research
   and development activities.  These results might not be suitable for
   deployment.  This RFC represents the consensus of the Crypto Forum
   Research Group of the Internet Research Task Force (IRTF).  Documents
   approved for publication by the IRSG are not a candidate for any
   level of Internet Standard; see Section 2 of RFC 7841.

   Information about the current status of this document, any errata,
   and how to provide feedback on it may be obtained at

Copyright Notice

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   document authors.  All rights reserved.

   This document is subject to BCP 78 and the IETF Trust's Legal
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Table of Contents

   1.  Introduction
   2.  Conventions Used in This Document
   3.  Randomness Wrapper
   4.  Tag Generation
   5.  Application to TLS
   6.  Implementation Guidance
   7.  IANA Considerations
   8.  Security Considerations
   9.  Comparison to RFC 6979
   10. References
     10.1.  Normative References
     10.2.  Informative References
   Authors' Addresses

1.  Introduction

   Secure and properly implemented random number generators, or
   "cryptographically secure" pseudorandom number generators (CSPRNGs),
   should produce output that is indistinguishable from a random string
   of the same length.  CSPRNGs are critical building blocks for TLS and
   related transport security protocols.  TLS in particular uses CSPRNGs
   to generate several values, such as ephemeral key shares and
   ClientHello and ServerHello random values.  CSPRNG failures, such as
   the Debian bug described in [DebianBug], can lead to insecure TLS
   connections.  CSPRNGs may also be intentionally weakened to cause
   harm [DualEC].  Initial entropy sources can also be weak or broken,
   and that would lead to insecurity of all CSPRNG instances seeded with
   them.  In such cases where CSPRNGs are poorly implemented or
   insecure, an adversary, Adv, may be able to distinguish its output
   from a random string or predict its output and recover secret key
   material used to protect the connection.

   This document proposes an improvement to randomness generation in
   security protocols inspired by the "NAXOS trick" [NAXOS].
   Specifically, instead of using raw randomness where needed, e.g., in
   generating ephemeral key shares, a function of a party's long-term
   private key is mixed into the entropy pool.  In the NAXOS key
   exchange protocol, raw random value x is replaced by H(x, sk), where
   sk is the sender's private key.  Unfortunately, as private keys are
   often isolated in Hardware Security Modules (HSMs), direct access to
   compute H(x, sk) is impossible.  Moreover, some HSM APIs may only
   offer the option to sign messages using a private key, yet offer no
   other operations involving that key.  An alternate, yet functionally
   equivalent construction, is needed.

   The approach described herein replaces the NAXOS hash with a keyed
   hash, or pseudorandom function (PRF), where the key is derived from a
   raw random value and a private key signature.  Implementations SHOULD
   apply this technique a) when indirect access to a private key is
   available and CSPRNG randomness guarantees are dubious or b) to
   provide stronger guarantees about possible future issues with the
   randomness.  Roughly, the security properties provided by the
   proposed construction are as follows:

   1.  If the CSPRNG works fine (that is, in a certain adversary model,
       the CSPRNG output is indistinguishable from a truly random
       sequence), then the output of the proposed construction is also
       indistinguishable from a truly random sequence in that adversary

   2.  Adv with full control of a (potentially broken) CSPRNG and
       ability to observe all outputs of the proposed construction does
       not obtain any non-negligible advantage in leaking the private
       key (in the absence of side channel attacks).

   3.  If the CSPRNG is broken or controlled by Adv, the output of the
       proposed construction remains indistinguishable from random,
       provided that the private key remains unknown to Adv.

   This document represents the consensus of the Crypto Forum Research
   Group (CFRG).

2.  Conventions Used in This Document

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "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.  Randomness Wrapper

   The output of a properly instantiated CSPRNG should be
   indistinguishable from a random string of the same length.  However,
   as previously discussed, this is not always true.  To mitigate this
   problem, we propose an approach for wrapping the CSPRNG output with a
   construction that mixes secret data into a value that may be lacking

   Let G(n) be an algorithm that generates n random bytes, i.e., the
   output of a CSPRNG.  Define an augmented CSPRNG G' as follows.  Let
   Sig(sk, m) be a function that computes a signature of message m given
   private key sk.  Let H be a cryptographic hash function that produces
   output of length M.  Let Extract(salt, IKM) be a randomness
   extraction function, e.g., HKDF-Extract [RFC5869], which accepts a
   salt and input keying material (IKM) parameter and produces a
   pseudorandom key of L bytes suitable for cryptographic use.  It must
   be a secure PRF (for salt as a key of length M) and preserve
   uniformness of IKM (for details, see [SecAnalysis]).  L SHOULD be a
   fixed length.  Let Expand(k, info, n) be a variable-length output
   PRF, e.g., HKDF-Expand [RFC5869], that takes as input a pseudorandom
   key k of L bytes, info string, and output length n, and produces
   output of n bytes.  Finally, let tag1 be a fixed, context-dependent
   string, and let tag2 be a dynamically changing string (e.g., a
   counter) of L' bytes.  We require that L >= n - L' for each value of

   The construction works as follows.  Instead of using G(n) when
   randomness is needed, use G'(n), where

          G'(n) = Expand(Extract(H(Sig(sk, tag1)), G(L)), tag2, n)

   Functionally, this expands n random bytes from a key derived from the
   CSPRNG output and signature over a fixed string (tag1).  See
   Section 4 for details about how "tag1" and "tag2" should be generated
   and used per invocation of the randomness wrapper.  Expand()
   generates a string that is computationally indistinguishable from a
   truly random string of n bytes.  Thus, the security of this
   construction depends upon the secrecy of H(Sig(sk, tag1)) and G(L).
   If the signature is leaked, then security of G'(n) reduces to the
   scenario wherein randomness is expanded directly from G(L).

   If a private key sk is stored and used inside an HSM, then the
   signature calculation is implemented inside it, while all other
   operations (including calculation of a hash function, Extract
   function, and Expand function) can be implemented either inside or
   outside the HSM.

   Sig(sk, tag1) need only be computed once for the lifetime of the
   randomness wrapper and MUST NOT be used or exposed beyond its role in
   this computation.  Additional recommendations for tag1 are given in
   the following section.

   Sig MUST be a deterministic signature function, e.g., deterministic
   Elliptic Curve Digital Signature Algorithm (ECDSA) [RFC6979], or use
   an independent (and completely reliable) entropy source, e.g., if Sig
   is implemented in an HSM with its own internal trusted entropy source
   for signature generation.

   Because Sig(sk, tag1) can be cached, the relative cost of using G'(n)
   instead of G(n) tends to be negligible with respect to cryptographic
   operations in protocols such as TLS (the relatively inexpensive
   computational cost of HKDF-Extract and HKDF-Expand dominates when
   comparing G' to G).  A description of the performance experiments and
   their results can be found in [SecAnalysis].

   Moreover, the values of G'(n) may be precomputed and pooled.  This is
   possible since the construction depends solely upon the CSPRNG output
   and private key.

4.  Tag Generation

   Both tags MUST be generated such that they never collide with another
   contender or owner of the private key.  This can happen if, for
   example, one HSM with a private key is used from several servers or
   if virtual machines are cloned.

   The RECOMMENDED tag construction procedure is as follows:

   tag1:  Constant string bound to a specific device and protocol in
          use.  This allows caching of Sig(sk, tag1).  Device-specific
          information may include, for example, a Media Access Control
          (MAC) address.  To provide security in the cases of usage of
          CSPRNGs in virtual environments, it is RECOMMENDED to
          incorporate all available information specific to the process
          that would ensure the uniqueness of each tag1 value among
          different instances of virtual machines (including ones that
          were cloned or recovered from snapshots).  This is needed to
          address the problem of CSPRNG state cloning (see [RY2010]).
          See Section 5 for example protocol information that can be
          used in the context of TLS 1.3.  If sk could be used for other
          purposes, then selecting a value for tag1 that is different
          than the form allowed by those other uses ensures that the
          signature is not exposed.

   tag2:  A nonce, that is, a value that is unique for each use of the
          same combination of G(L), tag1, and sk values.  The tag2 value
          can be implemented using a counter or a timer, provided that
          the timer is guaranteed to be different for each invocation of

5.  Application to TLS

   The PRF randomness wrapper can be applied to any protocol wherein a
   party has a long-term private key and also generates randomness.
   This is true of most TLS servers.  Thus, to apply this construction
   to TLS, one simply replaces the "private" CSPRNG G(n), i.e., the
   CSPRNG that generates private values, such as key shares, with

   G'(n) = HKDF-Expand(HKDF-Extract(H(Sig(sk, tag1)), G(L)), tag2, n)

6.  Implementation Guidance

   Recall that the wrapper defined in Section 3 requires L >= n - L',
   where L is the Extract output length and n is the desired amount of
   randomness.  Some applications may require n to exceed this bound.
   Wrapper implementations can support this use case by invoking G'
   multiple times and concatenating the results.

7.  IANA Considerations

   This document has no IANA actions.

8.  Security Considerations

   A security analysis was performed in [SecAnalysis].  Generally
   speaking, the following security theorem has been proven: if Adv
   learns only one of the signature or the usual randomness generated on
   one particular instance, then, under the security assumptions on our
   primitives, the wrapper construction should output randomness that is
   indistinguishable from a random string.

   The main reason one might expect the signature to be exposed is via a
   side-channel attack.  It is therefore prudent when implementing this
   construction to take into consideration the extra long-term key
   operation if equipment is used in a hostile environment when such
   considerations are necessary.  Hence, it is recommended to generate a
   key specifically for the purposes of the defined construction and not
   to use it another way.

   The signature in the construction, as well as in the protocol itself,
   MUST NOT use randomness from entropy sources with dubious security
   guarantees.  Thus, the signature scheme MUST either use a reliable
   entropy source (independent from the CSPRNG that is being improved
   with the proposed construction) or be deterministic; if the
   signatures are probabilistic and use weak entropy, our construction
   does not help, and the signatures are still vulnerable due to repeat
   randomness attacks.  In such an attack, Adv might be able to recover
   the long-term key used in the signature.

   Under these conditions, applying this construction should never yield
   worse security guarantees than not applying it, assuming that
   applying the PRF does not reduce entropy.  We believe there is always
   merit in analyzing protocols specifically.  However, this
   construction is generic so the analyses of many protocols will still
   hold even if this proposed construction is incorporated.

   The proposed construction cannot provide any guarantees of security
   if the CSPRNG state is cloned due to the virtual machine snapshots or
   process forking (see [MAFS2017]).  It is RECOMMENDED that tag1
   incorporate all available information about the environment, such as
   process attributes, virtual machine user information, etc.

9.  Comparison to RFC 6979

   The construction proposed herein has similarities with that of
   [RFC6979]; both of them use private keys to seed a deterministic
   random number generator.  Section 3.3 of [RFC6979] recommends
   deterministically instantiating an instance of the HMAC_DRBG
   pseudorandom number generator, described in [SP80090A] and Annex D of
   [X962], using the private key sk as the entropy_input parameter and
   H(m) as the nonce.  The construction G'(n) provided herein is
   similar, with such difference that a key derived from G(n) and
   H(Sig(sk, tag1)) is used as the entropy input and tag2 is the nonce.

   However, the semantics and the security properties obtained by using
   these two constructions are different.  The proposed construction
   aims to improve CSPRNG usage such that certain trusted randomness
   would remain even if the CSPRNG is completely broken.  Using a
   signature scheme that requires entropy sources according to [RFC6979]
   is intended for different purposes and does not assume possession of
   any entropy source -- even an unstable one.  For example, if in a
   certain system all private key operations are performed within an
   HSM, then the differences will manifest as follows: the HMAC_DRBG
   construction described in [RFC6979] may be implemented inside the HSM
   for the sake of signature generation, while the proposed construction
   would assume calling the signature implemented in the HSM.

10.  References

10.1.  Normative References

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997,

   [RFC5869]  Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand
              Key Derivation Function (HKDF)", RFC 5869,
              DOI 10.17487/RFC5869, May 2010,

   [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>.

   [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>.

10.2.  Informative References

              Yilek, S., Rescorla, E., Shacham, H., Enright, B., and S.
              Savage, "When private keys are public: results from the
              2008 Debian OpenSSL vulnerability", ICM '09,
              DOI 10.1145/1644893.1644896, November 2009,

   [DualEC]   Bernstein, D., Lange, T., and R. Niederhagen, "Dual EC: A
              Standardized Back Door", DOI 10.1007/978-3-662-49301-4_17,
              March 2016, <https://projectbullrun.org/dual-ec/documents/

   [MAFS2017] McGrew, D., Anderson, B., Fluhrer, S., and C. Schenefiel,
              "PRNG Failures and TLS Vulnerabilities in the Wild",
              January 2017,

   [NAXOS]    LaMacchia, B., Lauter, K., and A. Mityagin, "Stronger
              Security of Authenticated Key Exchange",
              DOI 10.1007/978-3-540-75670-5_1, November 2007,

   [RY2010]   Ristenpart, T. and S. Yilek, "When Good Randomness Goes
              Bad: Virtual Machine Reset Vulnerabilities and Hedging
              Deployed Cryptography", January 2010,

              Akhmetzyanova, L., Cremers, C., Garratt, L., Smyshlyaev,
              S., and N. Sullivan, "Limiting the impact of unreliable
              randomness in deployed security protocols",
              DOI 10.1109/CSF49147.2020.00027, IEEE 33rd Computer
              Security Foundations Symposium (CSF), Boston, MA, USA, pp.
              385-393, 2020,

   [SP80090A] National Institute of Standards and Technology,
              "Recommendation for Random Number Generation Using
              Deterministic Random Bit Generators, Special Publication
              800-90A Revision 1", DOI 10.6028/NIST.SP.800-90Ar1, June
              2015, <https://doi.org/10.6028/NIST.SP.800-90Ar1>.

   [X962]     American National Standard for Financial Services (ANSI),
              "Public Key Cryptography for the Financial Services
              Industry, The Elliptic Curve Digital Signature Algorithm
              (ECDSA)", ANSI X9.62, November 2005,


   We thank Liliya Akhmetzyanova for her deep involvement in the
   security assessment in [SecAnalysis].  We thank John Mattsson, Martin
   Thomson, and Rich Salz for their careful readings and useful

Authors' Addresses

   Cas Cremers
   Saarland Informatics Campus

   Email: cremers@cispa.saarland

   Luke Garratt
   Cisco Meraki
   500 Terry A Francois Blvd
   San Francisco,
   United States of America

   Email: lgarratt@cisco.com

   Stanislav Smyshlyaev
   18, Suschevsky val
   Russian Federation

   Email: svs@cryptopro.ru

   Nick Sullivan
   101 Townsend St
   San Francisco,
   United States of America

   Email: nick@cloudflare.com

   Christopher A. Wood
   101 Townsend St
   San Francisco,
   United States of America

   Email: caw@heapingbits.net
  1. RFC 8937