



Internet Engineering Task Force (IETF)                          M. Saito
Request for Comments: 8682                                  M. Matsumoto
Category: Standards Track                           Hiroshima University
ISSN: 2070-1721                                             V. Roca, Ed.
                                                             E. Baccelli
                                                                   INRIA
                                                            January 2020


             TinyMT32 Pseudorandom Number Generator (PRNG)

Abstract

   This document describes the TinyMT32 Pseudorandom Number Generator
   (PRNG), which produces 32-bit pseudorandom unsigned integers and aims
   at having a simple-to-use and deterministic solution.  This PRNG is a
   small-sized variant of the Mersenne Twister (MT) PRNG.  The main
   advantage of TinyMT32 over MT is the use of a small internal state,
   compatible with most target platforms that include embedded devices,
   while keeping reasonably good randomness that represents a
   significant improvement compared to the Park-Miller Linear
   Congruential PRNG.  However, neither the TinyMT nor MT PRNG is meant
   to be used for cryptographic applications.

Status of This Memo

   This is an Internet Standards Track document.

   This document is a product of the Internet Engineering Task Force
   (IETF).  It represents the consensus of the IETF community.  It has
   received public review and has been approved for publication by the
   Internet Engineering Steering Group (IESG).  Further information on
   Internet Standards is available in 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
   https://www.rfc-editor.org/info/rfc8682.

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
   (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
     1.1.  Requirements Language
   2.  TinyMT32 PRNG Specification
     2.1.  TinyMT32 Source Code
     2.2.  TinyMT32 Usage
     2.3.  Specific Implementation Validation and Deterministic
           Behavior
   3.  Security Considerations
   4.  IANA Considerations
   5.  References
     5.1.  Normative References
     5.2.  Informative References
   Acknowledgments
   Authors' Addresses

1.  Introduction

   This document specifies the TinyMT32 PRNG as a specialization of the
   reference implementation version 1.1 (2015/04/24) by Mutsuo Saito and
   Makoto Matsumoto from Hiroshima University, which can be found at
   [TinyMT-web] (the TinyMT website) and [TinyMT-dev] (the GitHub site).
   This specialization aims at having a simple-to-use and deterministic
   PRNG, as explained below.  However, the TinyMT32 PRNG is not meant to
   be used for cryptographic applications.

   TinyMT is a new, small-sized variant of the Mersenne Twister (MT)
   PRNG introduced in 2011 [MT98].  This document focuses on the
   TinyMT32 variant (rather than TinyMT64) of the TinyMT PRNG, which
   outputs 32-bit unsigned integers.

   The purpose of TinyMT is not to replace the Mersenne Twister: TinyMT
   has a far shorter period (2^(127) - 1) than MT.  The merit of TinyMT
   is in the small size of the 127-bit internal state, far smaller than
   the 19937 bits of MT.  The outputs of TinyMT satisfy several
   statistical tests for non-cryptographic randomness, including
   BigCrush in TestU01 [TestU01] and AdaptiveCrush [AdaptiveCrush],
   leaving it well placed for non-cryptographic usage, especially given
   the small size of its internal state (see [TinyMT-web]).  From this
   point of view, TinyMT32 represents a major improvement with respect
   to the Park-Miller Linear Congruential PRNG (e.g., as specified in
   [RFC5170]), which suffers from several known limitations (see, for
   instance, [PTVF92], Section 7.1, p. 279 and [RFC8681], Appendix B).

   The TinyMT32 PRNG initialization depends, among other things, on a
   parameter set, namely (mat1, mat2, tmat).  In order to facilitate the
   use of this PRNG and to make the sequence of pseudorandom numbers
   depend only on the seed value, this specification requires the use of
   a specific parameter set (see Section 2.1).  This is a major
   difference with respect to the implementation version 1.1
   (2015/04/24), which leaves this parameter set unspecified.

   Finally, the determinism of this PRNG for a given seed has been
   carefully checked (see Section 2.3).  This means that the same
   sequence of pseudorandom numbers should be generated, no matter the
   target execution platform and compiler, for a given initial seed
   value.  This determinism can be a key requirement, as is the case
   with [RFC8681], which normatively depends on this specification.

1.1.  Requirements Language

   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.

2.  TinyMT32 PRNG Specification

2.1.  TinyMT32 Source Code

   The TinyMT32 PRNG must be initialized with a parameter set that needs
   to be well chosen.  In this specification, for the sake of
   simplicity, the following parameter set MUST be used:

   *  mat1 = 0x8f7011ee = 2406486510

   *  mat2 = 0xfc78ff1f = 4235788063

   *  tmat = 0x3793fdff = 932445695

   This parameter set is the first entry of the precalculated parameter
   sets in tinymt32dc/tinymt32dc.0.1048576.txt by Kenji Rikitake,
   available at [TinyMT-params].  This is also the parameter set used in
   [KR12].

   The TinyMT32 PRNG reference implementation is reproduced in Figure 1.
   This is a C language implementation written for C99 [C99].  This
   reference implementation differs from the original source code as
   follows:

   *  The original authors, who are coauthors of this document, have
      granted IETF the rights to publish this version with a license and
      copyright that are in accordance with BCP 78 and the IETF Trust's
      Legal Provisions Relating to IETF Documents
      (http://trustee.ietf.org/license-info).

   *  The source code initially spread over the tinymt32.h and
      tinymt32.c files has been merged.

   *  The unused parts of the original source code have been removed.
      This is the case of the tinymt32_init_by_array() alternative
      initialization function.  This is also the case of the
      period_certification() function after having checked it is not
      required with the chosen parameter set.

   *  The unused constants TINYMT32_MEXP and TINYMT32_MUL have been
      removed.

   *  The appropriate parameter set has been added to the initialization
      function.

   *  The function order has been changed.

   *  Certain internal variables have been renamed for compactness
      purposes.

   *  The const qualifier has been added to the constant definitions.

   *  The code that was dependent on the representation of negative
      integers by 2's complements has been replaced by a more portable
      version.

   <CODE BEGINS>
   /**
    * Tiny Mersenne Twister: only 127-bit internal state.
    * Derived from the reference implementation version 1.1 (2015/04/24)
    * by Mutsuo Saito (Hiroshima University) and Makoto Matsumoto
    * (Hiroshima University).
    */
   #include <stdint.h>

   /**
    * tinymt32 internal state vector and parameters
    */
   typedef struct {
       uint32_t status[4];
       uint32_t mat1;
       uint32_t mat2;
       uint32_t tmat;
   } tinymt32_t;

   static void tinymt32_next_state (tinymt32_t* s);
   static uint32_t tinymt32_temper (tinymt32_t* s);

   /**
    * Parameter set to use for this IETF specification. Don't change.
    * This parameter set is the first entry of the precalculated
    * parameter sets in tinymt32dc/tinymt32dc.0.1048576.txt by
    * Kenji Rikitake, available at:
    *    https://github.com/jj1bdx/tinymtdc-longbatch/.
    * It is also the parameter set used in:
    *    Rikitake, K., "TinyMT pseudo random number generator for
    *    Erlang", Proceedings of the 11th ACM SIGPLAN Erlang Workshop,
    *    September 2012.
    */
   const uint32_t  TINYMT32_MAT1_PARAM = UINT32_C(0x8f7011ee);
   const uint32_t  TINYMT32_MAT2_PARAM = UINT32_C(0xfc78ff1f);
   const uint32_t  TINYMT32_TMAT_PARAM = UINT32_C(0x3793fdff);

   /**
    * This function initializes the internal state array with a
    * 32-bit unsigned integer seed.
    * @param s     pointer to tinymt internal state.
    * @param seed  a 32-bit unsigned integer used as a seed.
    */
   void tinymt32_init (tinymt32_t* s, uint32_t seed)
   {
       const uint32_t    MIN_LOOP = 8;
       const uint32_t    PRE_LOOP = 8;
       s->status[0] = seed;
       s->status[1] = s->mat1 = TINYMT32_MAT1_PARAM;
       s->status[2] = s->mat2 = TINYMT32_MAT2_PARAM;
       s->status[3] = s->tmat = TINYMT32_TMAT_PARAM;
       for (int i = 1; i < MIN_LOOP; i++) {
           s->status[i & 3] ^= i + UINT32_C(1812433253)
               * (s->status[(i - 1) & 3]
                  ^ (s->status[(i - 1) & 3] >> 30));
       }
       /*
        * NB: The parameter set of this specification warrants
        * that none of the possible 2^^32 seeds leads to an
        * all-zero 127-bit internal state. Therefore, the
        * period_certification() function of the original
        * TinyMT32 source code has been safely removed. If
        * another parameter set is used, this function will
        * have to be reintroduced here.
        */
       for (int i = 0; i < PRE_LOOP; i++) {
           tinymt32_next_state(s);
       }
   }

   /**
    * This function outputs a 32-bit unsigned integer from
    * the internal state.
    * @param s     pointer to tinymt internal state.
    * @return      32-bit unsigned integer r (0 <= r < 2^32).
    */
   uint32_t tinymt32_generate_uint32 (tinymt32_t* s)
   {
       tinymt32_next_state(s);
       return tinymt32_temper(s);
   }

   /**
    * Internal tinymt32 constants and functions.
    * Users should not call these functions directly.
    */
   const uint32_t  TINYMT32_SH0 = 1;
   const uint32_t  TINYMT32_SH1 = 10;
   const uint32_t  TINYMT32_SH8 = 8;
   const uint32_t  TINYMT32_MASK = UINT32_C(0x7fffffff);

   /**
    * This function changes the internal state of tinymt32.
    * @param s     pointer to tinymt internal state.
    */
   static void tinymt32_next_state (tinymt32_t* s)
   {
       uint32_t x;
       uint32_t y;

       y = s->status[3];
       x = (s->status[0] & TINYMT32_MASK)
           ^ s->status[1]
           ^ s->status[2];
       x ^= (x << TINYMT32_SH0);
       y ^= (y >> TINYMT32_SH0) ^ x;
       s->status[0] = s->status[1];
       s->status[1] = s->status[2];
       s->status[2] = x ^ (y << TINYMT32_SH1);
       s->status[3] = y;
       /*
        * The if (y & 1) {...} block below replaces:
        *     s->status[1] ^= -((int32_t)(y & 1)) & s->mat1;
        *     s->status[2] ^= -((int32_t)(y & 1)) & s->mat2;
        * The adopted code is equivalent to the original code
        * but does not depend on the representation of negative
        * integers by 2's complements. It is therefore more
        * portable but includes an if branch, which may slow
        * down the generation speed.
        */
       if (y & 1) {
            s->status[1] ^= s->mat1;
            s->status[2] ^= s->mat2;
        }
   }

   /**
    * This function outputs a 32-bit unsigned integer from
    * the internal state.
    * @param s     pointer to tinymt internal state.
    * @return      32-bit unsigned pseudorandom number.
    */
   static uint32_t tinymt32_temper (tinymt32_t* s)
   {
       uint32_t t0, t1;
       t0 = s->status[3];
       t1 = s->status[0] + (s->status[2] >> TINYMT32_SH8);
       t0 ^= t1;
       /*
        * The if (t1 & 1) {...} block below replaces:
        *     t0 ^= -((int32_t)(t1 & 1)) & s->tmat;
        * The adopted code is equivalent to the original code
        * but does not depend on the representation of negative
        * integers by 2's complements. It is therefore more
        * portable but includes an if branch, which may slow
        * down the generation speed.
        */
       if (t1 & 1) {
           t0 ^= s->tmat;
       }
       return t0;
   }
   <CODE ENDS>

                Figure 1: TinyMT32 Reference Implementation

2.2.  TinyMT32 Usage

   This PRNG MUST first be initialized with the following function:

      void tinymt32_init (tinymt32_t* s, uint32_t seed);

   It takes as input a 32-bit unsigned integer used as a seed (note that
   value 0 is permitted by TinyMT32).  This function also takes as input
   a pointer to an instance of a tinymt32_t structure that needs to be
   allocated by the caller but is left uninitialized.  This structure
   will then be updated by the various TinyMT32 functions in order to
   keep the internal state of the PRNG.  The use of this structure
   admits several instances of this PRNG to be used in parallel, each of
   them having its own instance of the structure.

   Then, each time a new 32-bit pseudorandom unsigned integer between 0
   and 2^(32) - 1 inclusive is needed, the following function is used:

      uint32_t tinymt32_generate_uint32 (tinymt32_t * s);

   Of course, the tinymt32_t structure must be left unchanged by the
   caller between successive calls to this function.

2.3.  Specific Implementation Validation and Deterministic Behavior

   For a given seed, PRNG determinism can be a requirement (e.g., with
   [RFC8681]).  Consequently, any implementation of the TinyMT32 PRNG in
   line with this specification MUST have the same output as that
   provided by the reference implementation of Figure 1.  In order to
   increase the compliancy confidence, this document proposes the
   following criteria.  Using a seed value of 1, the first 50 values
   returned by tinymt32_generate_uint32(s) as 32-bit unsigned integers
   are equal to the values provided in Figure 2, which are to be read
   line by line.  Note that these values come from the tinymt/
   check32.out.txt file provided by the PRNG authors to validate
   implementations of TinyMT32 as part of the MersenneTwister-Lab/TinyMT
   GitHub repository.

   2545341989  981918433 3715302833 2387538352 3591001365
   3820442102 2114400566 2196103051 2783359912  764534509
    643179475 1822416315  881558334 4207026366 3690273640
   3240535687 2921447122 3984931427 4092394160   44209675
   2188315343 2908663843 1834519336 3774670961 3019990707
   4065554902 1239765502 4035716197 3412127188  552822483
    161364450  353727785  140085994  149132008 2547770827
   4064042525 4078297538 2057335507  622384752 2041665899
   2193913817 1080849512   33160901  662956935  642999063
   3384709977 1723175122 3866752252  521822317 2292524454

    Figure 2: First 50 decimal values (to be read per line) returned by
    tinymt32_generate_uint32(s) as 32-bit unsigned integers, with a seed
                                 value of 1

   In particular, the deterministic behavior of the Figure 1 source code
   has been checked across several platforms: high-end laptops running
   64-bit Mac OS X and Linux/Ubuntu; a board featuring a 32-bit ARM
   Cortex-A15 and running 32-bit Linux/Ubuntu; several embedded cards
   featuring either an ARM Cortex-M0+, a Cortex-M3, or a Cortex-M4
   32-bit microcontroller, all of them running RIOT [Baccelli18]; two
   low-end embedded cards featuring either a 16-bit microcontroller (TI
   MSP430) or an 8-bit microcontroller (Arduino ATMEGA2560), both of
   them running RIOT.

   This specification only outputs 32-bit unsigned pseudorandom numbers
   and does not try to map this output to a smaller integer range (e.g.,
   between 10 and 49 inclusive).  If a specific use case needs such a
   mapping, it will have to provide its own function.  In that case, if
   PRNG determinism is also required, the use of a floating point
   (single or double precision) to perform this mapping should probably
   be avoided, as these calculations may lead to different rounding
   errors across different target platforms.  Great care should also be
   taken to not introduce biases in the randomness of the mapped output
   (which may be the case with some mapping algorithms) incompatible
   with the use-case requirements.  The details of how to perform such a
   mapping are out of scope of this document.

3.  Security Considerations

   The authors do not believe the present specification generates
   specific security risks per se.  However, the TinyMT and MT PRNG must
   not be used for cryptographic applications.

4.  IANA Considerations

   This document has no IANA actions.

5.  References

5.1.  Normative References

   [C99]      International Organization for Standardization,
              "Programming languages - C: C99, correction 3:2007", ISO/
              IEC 9899:1999/Cor 3:2007, November 2007.

   [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>.

   [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>.

5.2.  Informative References

   [AdaptiveCrush]
              Haramoto, H., "Automation of Statistical Tests on
              Randomness to Obtain Clearer Conclusion", Monte Carlo and
              Quasi-Monte Carlo Methods 2008,
              DOI 10.1007/978-3-642-04107-5_26, November 2009,
              <http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/
              ADAPTIVE>.

   [Baccelli18]
              Baccelli, E., Gundogan, C., Hahm, O., Kietzmann, P.,
              Lenders, M. S., Petersen, H., Schleiser, K., Schmidt, T.
              C., and M. Wahlisch, "RIOT: An Open Source Operating
              System for Low-End Embedded Devices in the IoT", IEEE
              Internet of Things Journal, Volume 5, Issue 6,
              DOI 10.1109/JIOT.2018.2815038, December 2018,
              <https://doi.org/10.1109/JIOT.2018.2815038>.

   [KR12]     Rikitake, K., "TinyMT pseudo random number generator for
              Erlang", Proceedings of the 11th ACM SIGPLAN Erlang
              Workshop, pp. 67-72, DOI 10.1145/2364489.2364504,
              September 2012, <https://doi.org/10.1145/2364489.2364504>.

   [MT98]     Matsumoto, M. and T. Nishimura, "Mersenne twister: A
              623-dimensionally equidistributed uniform pseudo-random
              number generator", ACM Transactions on Modeling and
              Computer Simulation (TOMACS), Volume 8, Issue 1, pp. 3-30,
              DOI 10.1145/272991.272995, January 1998,
              <https://doi.org/10.1145/272991.272995>.

   [PTVF92]   Press, W., Teukolsky, S., Vetterling, W., and B. Flannery,
              "Numerical recipes in C (2nd ed.): the art of scientific
              computing", Cambridge University Press,
              ISBN 0-521-43108-5, 1992.

   [RFC5170]  Roca, V., Neumann, C., and D. Furodet, "Low Density Parity
              Check (LDPC) Staircase and Triangle Forward Error
              Correction (FEC) Schemes", RFC 5170, DOI 10.17487/RFC5170,
              June 2008, <https://www.rfc-editor.org/info/rfc5170>.

   [RFC8681]  Roca, V. and B. Teibi, "Sliding Window Random Linear Code
              (RLC) Forward Erasure Correction (FEC) Schemes for
              FECFRAME", RFC 8681, DOI 10.17487/RFC8681, January 2020,
              <https://www.rfc-editor.org/info/rfc8681>.

   [TestU01]  L'Ecuyer, P. and R. Simard, "TestU01: A C library for
              empirical testing of random number generators", ACM
              Transactions on Mathematical Software (TOMS), Volume 33,
              Issue 4, Article 22, DOI 10.1145/1268776.1268777, August
              2007, <http://simul.iro.umontreal.ca/testu01/tu01.html>.

   [TinyMT-dev]
              "Tiny Mersenne Twister (TinyMT)", commit 9d7ca3c, March
              2018, <https://github.com/MersenneTwister-Lab/TinyMT>.

   [TinyMT-params]
              "TinyMT pre-calculated parameter list", commit 30079eb,
              March 2013,
              <https://github.com/jj1bdx/tinymtdc-longbatch>.

   [TinyMT-web]
              Saito, M. and M. Matsumoto, "Tiny Mersenne Twister
              (TinyMT)",
              <http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/TINYMT/>.

Acknowledgments

   The authors would like to thank Belkacem Teibi, with whom we explored
   TinyMT32 specificities when looking to an alternative to the Park-
   Miller Linear Congruential PRNG.  The authors would also like to
   thank Carl Wallace; Stewart Bryant; Greg Skinner; Mike Heard; the
   three TSVWG chairs, Wesley Eddy (our shepherd), David Black, and
   Gorry Fairhurst; as well as Spencer Dawkins and Mirja Kuehlewind.
   Last but not least, the authors are really grateful to the IESG
   members, in particular Benjamin Kaduk, Eric Rescorla, Adam Roach,
   Roman Danyliw, Barry Leiba, Martin Vigoureux, and Eric Vyncke for
   their highly valuable feedback that greatly contributed to improving
   this specification.

Authors' Addresses

   Mutsuo Saito
   Hiroshima University
   Japan

   Email: saito@math.sci.hiroshima-u.ac.jp


   Makoto Matsumoto
   Hiroshima University
   Japan

   Email: m-mat@math.sci.hiroshima-u.ac.jp


   Vincent Roca (editor)
   INRIA
   Univ. Grenoble Alpes
   France

   Email: vincent.roca@inria.fr


   Emmanuel Baccelli
   INRIA
   France

   Email: emmanuel.baccelli@inria.fr
