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
December 2019
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) 2019 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, December 2019, 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
Japan
Hiroshima University
Japan
Email: saito@math.sci.hiroshima-u.ac.jp
Makoto Matsumoto
Japan
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