TSVWG M. Saito
Internet-Draft M. Matsumoto
Intended status: Standards Track Hiroshima University
Expires: November 28, 2019 V. Roca (Ed.)
E. Baccelli
May 27, 2019

TinyMT32 Pseudo Random Number Generator (PRNG)


This document describes the TinyMT32 Pseudo Random Number Generator (PRNG) that produces 32-bit pseudo-random unsigned integers and aims at having a simple-to-use and deterministic solution. This PRNG is a small-sized variant of Mersenne Twister (MT) PRNG [MT98]. The main advantage of TinyMT32 over MT is the use of a small internal state, compatible with most target platforms including embedded devices, while keeping a reasonably good randomness.

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Table of Contents

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:

This specialisation aims at having a simple-to-use and deterministic PRNG, as explained below.

TinyMT is a new small-sized variant introduced by Mutsuo Saito and Makoto Matsumoto in 2011 of the Mersenne Twister (MT) PRNG [MT98]. This document focusses 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 Mersenne Twister: TinyMT has a far shorter period (2^^127 - 1) than MT. The merit of TinyMT is in its small size of the internal state of 127 bits, far smaller than the 19937 bits of MT. According to statistical tests (BigCrush in TestU01 and AdaptiveCrush), the quality of the outputs of TinyMT seems pretty good in terms of randomnes (in particular the uniformity of generated numbers), taking the small size of the internal state into consideration (see [TinyMT-web]). From this point of view, TinyMT32 represents a major improvement with respect to the Park-Miler Linear Congruential PRNG (e.g., as specified in [RFC5170]) that suffers several known limitations (see for instance [PTVF92], section 7.1, p. 279, and [RLC-ID], Appendix B). However, neither the TinyMT nor MT PRNG are meant to be used for cryptographic applications.

The TinyMT32 PRNG initialization depends, among other things, on a parameter set -- namely (mat1, mat2, tmat) -- that needs to be well chosen (pre-calculated values are available in the official web site). In order to facilitate the use of this PRNG and make the sequence of pseudo-random numbers depend only on the seed value, this specification requires the use of a specific parameter set (see Section 3.1). This is a first difference with respect to the implementation version 1.1 (2015/04/24) by Mutsuo Saito and Makoto Matsumoto that leaves this parameter set unspecified. A second difference is the removal of the tinymt32_init_by_array() alternative initialization function, to only keep the simple initialisation through a seed value (see Section 3.2).

Finally, the determinism of this PRNG, for a given seed, has been carefully checked (see Section 3.3). It means that the same sequence of pseudo-random 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 it the case with [RLC-ID] that normatively depends on this specification.

2. Definitions

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. TinyMT32 PRNG Specification

3.1. TinyMT32 Source Code

The TinyMT32 PRNG requires to 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: [TinyMT-params]. This is also the parameter set used in [KR12].

This parameter set is the first entry of the precalculated parameter sets in file tinymt32dc/tinymt32dc.0.1048576.txt, by Kenji Rikitake, and available at

The TinyMT32 PRNG reference implementation is reproduced in Figure 1, with the following differences with respect to the original source code:

 * 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 file tinymt32dc/tinymt32dc.0.1048576.txt, by
 * Kenji Rikitake, available at:
 *    https://github.com/jj1bdx/tinymtdc-longbatch/
 * It is also the parameter set used:
 *    Rikitake, K., "TinyMT Pseudo Random Number Generator for
 *    Erlang", ACM 11th SIGPLAN Erlang Workshop (Erlang'12),
 *    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 re-introduced here.
    for (int i = 0; i < PRE_LOOP; i++) {

 * 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)
    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 pseudo-random 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;
    t0 ^= -((int32_t)(t1 & 1)) & s->tmat;
    return t0;

Figure 1: TinyMT32 Reference Implementation

3.2. TinyMT32 Usage

This PRNG MUST first be initialized with the following function:

It takes as input a 32-bit unsigned integer used as a seed (note that value 0 is authorized 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 left uninitialized. This structure will then updated by the various TinyMT32 functions in order to keep the internal state of the PRNG. The use of this structure authorizes 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 pseudo-random unsigned integer between 0 and 2^32 - 1 inclusive is needed, the following function is used:

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

3.3. Specific Implementation Validation and Deterministic Behavior

PRNG determinism, for a given seed, can be a requirement (e.g., with [RLC-ID]). Consequently, any implementation of the TinyMT32 PRNG in line with this specification MUST comply with the following criteria. Using a seed value of 1, the first 50 values returned by tinymt32_generate_uint32(s) as 32-bit unsigned integers MUST be equal to values provided in Figure 2. 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 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-bits Mac OSX and Linux/Ubuntu; a board featuring a 32-bits 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 a 8-bit microcontroller (Arduino ATMEGA2560), both of them running RIOT.

This specification only outputs 32-bit unsigned pseudo-random 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 floating point (single or double precision) to perform this mapping should probably be avoided, these calculations leading potentially to different rounding errors across different target platforms. Great care should also be put on not introducing biases in the randomness of the mapped output (it 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.

4. Security Considerations

The authors do not believe the present specification generates specific security risks per se. However, neither the TinyMT nor MT PRNG are meant to be used for cryptographic applications.

5. IANA Considerations

This document does not require any IANA action.

6. Acknowledgments

The authors would like to thank Belkacem Teibi with whom we explored TinyMT32 specificities when looking to an alternative to the Park-Miler Linear Congruential PRNG. The authors would like to thank Carl Wallace, Stewart Bryant, Greg Skinner, the three TSVWG chairs, Wesley Eddy, our shepherd, David Black and Gorry Fairhurst, as well as Spencer Dawkins and Mirja Kuhlewind. Last but not least, the authors are really grateful to the IESG members, in particular Benjamin Kaduk, Eric Rescorla, and Adam Roach for their highly valuable feedbacks that greatly contributed to improve this specification.

7. References

7.1. 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.
[RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, May 2017.

7.2. Informative References

[Baccelli18] Baccelli, E., Gundogan, C., Hahm, O., Kietzmann, P., Lenders, M., Petersen, H., Schleiser, K., Schmidt, T. 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.
[KR12] Rikitake, K., "TinyMT Pseudo Random Number Generator for Erlang", ACM 11th SIGPLAN Erlang Workshop (Erlang'12), September 14, 2012, Copenhagen, Denmark, DOI: http://dx.doi.org/10.1145/2364489.2364504, September 2012.
[MT98] Matsumoto, M. and T. Nishimura, "Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator", ACM Transactions on Modeling and Computer Simulation (TOMACS), Volume 8 Issue 1, Jan. 1998, pp.3-30, January 1998, DOI:10.1145/272991.272995, January 1998.
[PTVF92] Press, W., Teukolsky, S., Vetterling, W. and B. Flannery, "Numerical Recipies in C; Second Edition", 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.
[RLC-ID] Roca, V. and B. Teibi, "Sliding Window Random Linear Code (RLC) Forward Erasure Correction (FEC) Scheme for FECFRAME", Work in Progress, Transport Area Working Group (TSVWG) draft-ietf-tsvwg-rlc-fec-scheme (Work in Progress), February 2019.
[TinyMT-dev] Saito, M. and M. Matsumoto, "Tiny Mersenne Twister (TinyMT) github site"
[TinyMT-params] Rikitake, K., "TinyMT pre-calculated parameter list github site"
[TinyMT-web] Saito, M. and M. Matsumoto, "Tiny Mersenne Twister (TinyMT) web site"

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 INRIA Univ. Grenoble Alpes, France EMail: vincent.roca@inria.fr
Emmanuel Baccelli INRIA France EMail: emmanuel.baccelli@inria.fr