draft-ietf-tsvwg-tinymt32-03.txt   draft-ietf-tsvwg-tinymt32-04.txt 
TSVWG M. Saito TSVWG M. Saito
Internet-Draft M. Matsumoto Internet-Draft M. Matsumoto
Intended status: Standards Track Hiroshima University Intended status: Standards Track Hiroshima University
Expires: November 28, 2019 V. Roca (Ed.) Expires: December 14, 2019 V. Roca (Ed.)
E. Baccelli E. Baccelli
INRIA INRIA
May 27, 2019 June 12, 2019
TinyMT32 Pseudo Random Number Generator (PRNG) TinyMT32 Pseudo Random Number Generator (PRNG)
draft-ietf-tsvwg-tinymt32-03 draft-ietf-tsvwg-tinymt32-04
Abstract Abstract
This document describes the TinyMT32 Pseudo Random Number Generator This document describes the TinyMT32 Pseudo Random Number Generator
(PRNG) that produces 32-bit pseudo-random unsigned integers and aims (PRNG) that produces 32-bit pseudo-random unsigned integers and aims
at having a simple-to-use and deterministic solution. This PRNG is a at having a simple-to-use and deterministic solution. This PRNG is a
small-sized variant of Mersenne Twister (MT) PRNG [MT98]. The main small-sized variant of Mersenne Twister (MT) PRNG. The main
advantage of TinyMT32 over MT is the use of a small internal state, advantage of TinyMT32 over MT is the use of a small internal state,
compatible with most target platforms including embedded devices, compatible with most target platforms that include embedded devices,
while keeping a reasonably good randomness. while keeping a reasonably good randomness that represents a
sigificant improvement compared to the Park-Miller Linear
Congruential PRNG. However, neither the TinyMT nor MT PRNG are meant
to be used for cryptographic applications.
Status of This Memo Status of This Memo
This Internet-Draft is submitted in full conformance with the This Internet-Draft is submitted in full conformance with the
provisions of BCP 78 and BCP 79. provisions of BCP 78 and BCP 79.
Internet-Drafts are working documents of the Internet Engineering Internet-Drafts are working documents of the Internet Engineering
Task Force (IETF). Note that other groups may also distribute Task Force (IETF). Note that other groups may also distribute
working documents as Internet-Drafts. The list of current Internet- working documents as Internet-Drafts. The list of current Internet-
Drafts is at https://datatracker.ietf.org/drafts/current/. Drafts is at https://datatracker.ietf.org/drafts/current/.
Internet-Drafts are draft documents valid for a maximum of six months Internet-Drafts are draft documents valid for a maximum of six months
and may be updated, replaced, or obsoleted by other documents at any and may be updated, replaced, or obsoleted by other documents at any
time. It is inappropriate to use Internet-Drafts as reference time. It is inappropriate to use Internet-Drafts as reference
material or to cite them other than as "work in progress." material or to cite them other than as "work in progress."
This Internet-Draft will expire on November 28, 2019. This Internet-Draft will expire on December 14, 2019.
Copyright Notice Copyright Notice
Copyright (c) 2019 IETF Trust and the persons identified as the Copyright (c) 2019 IETF Trust and the persons identified as the
document authors. All rights reserved. document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal This document is subject to BCP 78 and the IETF Trust's Legal
Provisions Relating to IETF Documents Provisions Relating to IETF Documents
(https://trustee.ietf.org/license-info) in effect on the date of (https://trustee.ietf.org/license-info) in effect on the date of
publication of this document. Please review these documents publication of this document. Please review these documents
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1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2
2. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 3
3. TinyMT32 PRNG Specification . . . . . . . . . . . . . . . . . 3 3. TinyMT32 PRNG Specification . . . . . . . . . . . . . . . . . 3
3.1. TinyMT32 Source Code . . . . . . . . . . . . . . . . . . 3 3.1. TinyMT32 Source Code . . . . . . . . . . . . . . . . . . 3
3.2. TinyMT32 Usage . . . . . . . . . . . . . . . . . . . . . 7 3.2. TinyMT32 Usage . . . . . . . . . . . . . . . . . . . . . 7
3.3. Specific Implementation Validation and Deterministic 3.3. Specific Implementation Validation and Deterministic
Behavior . . . . . . . . . . . . . . . . . . . . . . . . 8 Behavior . . . . . . . . . . . . . . . . . . . . . . . . 8
4. Security Considerations . . . . . . . . . . . . . . . . . . . 9 4. Security Considerations . . . . . . . . . . . . . . . . . . . 9
5. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 9 5. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 9
6. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 9 6. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 9
7. References . . . . . . . . . . . . . . . . . . . . . . . . . 9 7. References . . . . . . . . . . . . . . . . . . . . . . . . . 10
7.1. Normative References . . . . . . . . . . . . . . . . . . 9 7.1. Normative References . . . . . . . . . . . . . . . . . . 10
7.2. Informative References . . . . . . . . . . . . . . . . . 9 7.2. Informative References . . . . . . . . . . . . . . . . . 10
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 10 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 11
1. Introduction 1. Introduction
This document specifies the TinyMT32 PRNG, as a specialization of the This document specifies the TinyMT32 PRNG, as a specialization of the
reference implementation version 1.1 (2015/04/24) by Mutsuo Saito and reference implementation version 1.1 (2015/04/24) by Mutsuo Saito and
Makoto Matsumoto, from Hiroshima University: Makoto Matsumoto, from Hiroshima University, that can be found at
[TinyMT-web] (TinyMT web site) and [TinyMT-dev] (Github site). This
o Official web site [TinyMT-web] specialisation aims at having a simple-to-use and deterministic PRNG,
o Official github site and reference implementation [TinyMT-dev] as explained below. However, the TinyMT32 PRNG is not meant to be
used for cryptographic applications.
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 TinyMT is a new small-sized variant introduced in 2011 of the
Makoto Matsumoto in 2011 of the Mersenne Twister (MT) PRNG [MT98]. Mersenne Twister (MT) PRNG [MT98]. This document focusses on the
This document focusses on the TinyMT32 variant (rather than TinyMT64) TinyMT32 variant (rather than TinyMT64) of the TinyMT PRNG, which
of the TinyMT PRNG, which outputs 32-bit unsigned integers. outputs 32-bit unsigned integers.
The purpose of TinyMT is not to replace Mersenne Twister: TinyMT has 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 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 small size of the internal state of 127 bits, far smaller than
the 19937 bits of MT. According to statistical tests (BigCrush in the 19937 bits of MT. The outputs of TinyMT satisfy several
TestU01 and AdaptiveCrush), the quality of the outputs of TinyMT statistical tests for non-cryptographic randomness, including
seems pretty good in terms of randomnes (in particular the uniformity BigCrush in TestU01 [TestU01] and AdaptiveCrush [AdaptiveCrush],
of generated numbers), taking the small size of the internal state leaving it well-placed for non-cryptographic usage, especially given
into consideration (see [TinyMT-web]). From this point of view, the small size of its internal state (see [TinyMT-web]). From this
TinyMT32 represents a major improvement with respect to the Park- point of view, TinyMT32 represents a major improvement with respect
Miler Linear Congruential PRNG (e.g., as specified in [RFC5170]) that to the Park-Miller Linear Congruential PRNG (e.g., as specified in
suffers several known limitations (see for instance [PTVF92], section [RFC5170]) that suffers several known limitations (see for instance
7.1, p. 279, and [RLC-ID], Appendix B). However, neither the TinyMT [PTVF92], section 7.1, p. 279, and [RLC-ID], Appendix B).
nor MT PRNG are meant to be used for cryptographic applications.
The TinyMT32 PRNG initialization depends, among other things, on a The TinyMT32 PRNG initialization depends, among other things, on a
parameter set -- namely (mat1, mat2, tmat) -- that needs to be well parameter set, namely (mat1, mat2, tmat). In order to facilitate the
chosen (pre-calculated values are available in the official web use of this PRNG and make the sequence of pseudo-random numbers
site). In order to facilitate the use of this PRNG and make the depend only on the seed value, this specification requires the use of
sequence of pseudo-random numbers depend only on the seed value, this a specific parameter set (see Section 3.1). This is a major
specification requires the use of a specific parameter set (see difference with respect to the implementation version 1.1
Section 3.1). This is a first difference with respect to the (2015/04/24) that leaves this parameter set unspecified.
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 Finally, the determinism of this PRNG, for a given seed, has been
carefully checked (see Section 3.3). It means that the same sequence carefully checked (see Section 3.3). It means that the same sequence
of pseudo-random numbers should be generated, no matter the target of pseudo-random numbers should be generated, no matter the target
execution platform and compiler, for a given initial seed value. execution platform and compiler, for a given initial seed value.
This determinism can be a key requirement as it the case with This determinism can be a key requirement as it the case with
[RLC-ID] that normatively depends on this specification. [RLC-ID] that normatively depends on this specification.
2. Definitions 2. Definitions
skipping to change at page 4, line 7 skipping to change at page 4, line 5
o mat1 = 0x8f7011ee = 2406486510 o mat1 = 0x8f7011ee = 2406486510
o mat2 = 0xfc78ff1f = 4235788063 o mat2 = 0xfc78ff1f = 4235788063
o tmat = 0x3793fdff = 932445695 o tmat = 0x3793fdff = 932445695
This parameter set is the first entry of the precalculated parameter This parameter set is the first entry of the precalculated parameter
sets in file tinymt32dc/tinymt32dc.0.1048576.txt, by Kenji Rikitake, sets in file tinymt32dc/tinymt32dc.0.1048576.txt, by Kenji Rikitake,
and available at [TinyMT-params]. This is also the parameter set and available at [TinyMT-params]. This is also the parameter set
used in [KR12]. used in [KR12].
The TinyMT32 PRNG reference implementation is reproduced in Figure 1, The TinyMT32 PRNG reference implementation is reproduced in Figure 1.
with the following differences with respect to the original source This is a C language implementation, compatible with the C99 (ISO/IEC
code: 9899:1999), C11 (ISO/IEC 9899:2011) and C18 (ISO/IEC 9899:2018)
versions of the C language. This reference implementation differs
from the original source code as follows:
o the original copyright and licence have been removed, in o the original copyright and license have been removed by the
original authors who are now authors of this document, in
accordance with BCP 78 and the IETF Trust's Legal Provisions accordance with BCP 78 and the IETF Trust's Legal Provisions
Relating to IETF Documents (http://trustee.ietf.org/license-info); Relating to IETF Documents (http://trustee.ietf.org/license-info);
o the source code initially spread over the tinymt32.h and o the source code initially spread over the tinymt32.h and
tinymt32.c files has been merged; tinymt32.c files has been merged;
o the unused parts of the original source code have been removed. o the unused parts of the original source code have been removed.
This is the case of the tinymt32_init_by_array() alternative This is the case of the tinymt32_init_by_array() alternative
initialisation function; initialisation function. This is also the case of the
period_certification() function after having checked it is not
required with the chosen parameter set;
o the unused constants TINYMT32_MEXP and TINYMT32_MUL have been o the unused constants TINYMT32_MEXP and TINYMT32_MUL have been
removed; removed;
o the appropriate parameter set has been added to the initialization o the appropriate parameter set has been added to the initialization
function; function;
o the function order has been changed; o the function order has been changed;
o certain internal variables have been renamed for compactness o certain internal variables have been renamed for compactness
purposes; purposes;
o the const qualifier has been added to the constant definitions. o the const qualifier has been added to the constant definitions;
o the code that was dependant on the representation of negative
integers by 2's complements has been replaced by a more portable
version;
<CODE BEGINS> <CODE BEGINS>
/** /**
* Tiny Mersenne Twister only 127 bit internal state. * Tiny Mersenne Twister only 127 bit internal state.
* Derived from the reference implementation version 1.1 (2015/04/24) * Derived from the reference implementation version 1.1 (2015/04/24)
* by Mutsuo Saito (Hiroshima University) and Makoto Matsumoto * by Mutsuo Saito (Hiroshima University) and Makoto Matsumoto
* (Hiroshima University). * (Hiroshima University).
*/ */
#include <stdint.h> #include <stdint.h>
skipping to change at page 7, line 23 skipping to change at page 7, line 27
* the internal state. * the internal state.
* @param s pointer to tinymt internal state. * @param s pointer to tinymt internal state.
* @return 32-bit unsigned pseudo-random number. * @return 32-bit unsigned pseudo-random number.
*/ */
static uint32_t tinymt32_temper (tinymt32_t* s) static uint32_t tinymt32_temper (tinymt32_t* s)
{ {
uint32_t t0, t1; uint32_t t0, t1;
t0 = s->status[3]; t0 = s->status[3];
t1 = s->status[0] + (s->status[2] >> TINYMT32_SH8); t1 = s->status[0] + (s->status[2] >> TINYMT32_SH8);
t0 ^= t1; t0 ^= t1;
t0 ^= -((int32_t)(t1 & 1)) & s->tmat; /*
* 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; return t0;
} }
<CODE ENDS> <CODE ENDS>
Figure 1: TinyMT32 Reference Implementation Figure 1: TinyMT32 Reference Implementation
3.2. TinyMT32 Usage 3.2. TinyMT32 Usage
This PRNG MUST first be initialized with the following function: This PRNG MUST first be initialized with the following function:
void tinymt32_init (tinymt32_t * s, uint32_t seed); void tinymt32_init (tinymt32_t* s, uint32_t seed);
It takes as input a 32-bit unsigned integer used as a seed (note that 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 value 0 is permitted by TinyMT32). This function also takes as input
input a pointer to an instance of a tinymt32_t structure that needs a pointer to an instance of a tinymt32_t structure that needs to be
to be allocated by the caller but left uninitialized. This structure allocated by the caller but left uninitialized. This structure will
will then updated by the various TinyMT32 functions in order to keep then be updated by the various TinyMT32 functions in order to keep
the internal state of the PRNG. The use of this structure authorizes 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 several instances of this PRNG to be used in parallel, each of them
having its own instance of the structure. having its own instance of the structure.
Then, each time a new 32-bit pseudo-random unsigned integer between 0 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: and 2^32 - 1 inclusive is needed, the following function is used:
uint32_t tinymt32_generate_uint32 (tinymt32_t * s); uint32_t tinymt32_generate_uint32 (tinymt32_t * s);
Of course, the tinymt32_t structure must be left unchanged by the Of course, the tinymt32_t structure must be left unchanged by the
caller between successive calls to this function. caller between successive calls to this function.
3.3. Specific Implementation Validation and Deterministic Behavior 3.3. Specific Implementation Validation and Deterministic Behavior
PRNG determinism, for a given seed, can be a requirement (e.g., with PRNG determinism, for a given seed, can be a requirement (e.g., with
[RLC-ID]). Consequently, any implementation of the TinyMT32 PRNG in [RLC-ID]). Consequently, any implementation of the TinyMT32 PRNG in
line with this specification MUST comply with the following criteria. line with this specification MUST have the same output as that
Using a seed value of 1, the first 50 values returned by provided by the reference implementation of Figure 1. In order to
tinymt32_generate_uint32(s) as 32-bit unsigned integers MUST be equal increase the compliancy confidence, this document proposes the
to values provided in Figure 2. Note that these values come from the following criteria. Using a seed value of 1, the first 50 values
tinymt/check32.out.txt file provided by the PRNG authors to validate returned by tinymt32_generate_uint32(s) as 32-bit unsigned integers
implementations of TinyMT32, as part of the MersenneTwister-Lab/ are equal to values provided in Figure 2, to be read line by line.
TinyMT Github repository. 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 2545341989 981918433 3715302833 2387538352 3591001365
3820442102 2114400566 2196103051 2783359912 764534509 3820442102 2114400566 2196103051 2783359912 764534509
643179475 1822416315 881558334 4207026366 3690273640 643179475 1822416315 881558334 4207026366 3690273640
3240535687 2921447122 3984931427 4092394160 44209675 3240535687 2921447122 3984931427 4092394160 44209675
2188315343 2908663843 1834519336 3774670961 3019990707 2188315343 2908663843 1834519336 3774670961 3019990707
4065554902 1239765502 4035716197 3412127188 552822483 4065554902 1239765502 4035716197 3412127188 552822483
161364450 353727785 140085994 149132008 2547770827 161364450 353727785 140085994 149132008 2547770827
4064042525 4078297538 2057335507 622384752 2041665899 4064042525 4078297538 2057335507 622384752 2041665899
2193913817 1080849512 33160901 662956935 642999063 2193913817 1080849512 33160901 662956935 642999063
3384709977 1723175122 3866752252 521822317 2292524454 3384709977 1723175122 3866752252 521822317 2292524454
Figure 2: First 50 decimal values returned by 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 tinymt32_generate_uint32(s) as 32-bit unsigned integers, with a seed
value of 1. value of 1.
In particular, the deterministic behavior of the Figure 1 source code In particular, the deterministic behavior of the Figure 1 source code
has been checked across several platforms: high-end laptops running has been checked across several platforms: high-end laptops running
64-bits Mac OSX and Linux/Ubuntu; a board featuring a 32-bits ARM 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 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 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 microcontroller, all of them running RIOT [Baccelli18]; two low-end
embedded cards featuring either a 16-bit microcontroller (TI MSP430) embedded cards featuring either a 16-bit microcontroller (TI MSP430)
skipping to change at page 9, line 21 skipping to change at page 9, line 39
PRNG are meant to be used for cryptographic applications. PRNG are meant to be used for cryptographic applications.
5. IANA Considerations 5. IANA Considerations
This document does not require any IANA action. This document does not require any IANA action.
6. Acknowledgments 6. Acknowledgments
The authors would like to thank Belkacem Teibi with whom we explored The authors would like to thank Belkacem Teibi with whom we explored
TinyMT32 specificities when looking to an alternative to the Park- TinyMT32 specificities when looking to an alternative to the Park-
Miler Linear Congruential PRNG. The authors would like to thank Carl Miller Linear Congruential PRNG. The authors would like to thank
Wallace, Stewart Bryant, Greg Skinner, the three TSVWG chairs, Wesley Carl Wallace, Stewart Bryant, Greg Skinner, Mike Heard, the three
Eddy, our shepherd, David Black and Gorry Fairhurst, as well as TSVWG chairs, Wesley Eddy, our shepherd, David Black and Gorry
Spencer Dawkins and Mirja Kuhlewind. Last but not least, the authors Fairhurst, as well as Spencer Dawkins and Mirja Kuhlewind. Last but
are really grateful to the IESG members, in particular Benjamin not least, the authors are really grateful to the IESG members, in
Kaduk, Eric Rescorla, and Adam Roach for their highly valuable particular Benjamin Kaduk, Eric Rescorla, Adam Roach, Roman Danyliw,
Barry Leiba, Martin Vigoureux, Eric Vyncke for their highly valuable
feedbacks that greatly contributed to improve this specification. feedbacks that greatly contributed to improve this specification.
7. References 7. References
7.1. Normative References 7.1. Normative References
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119, Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997, DOI 10.17487/RFC2119, March 1997,
<https://www.rfc-editor.org/info/rfc2119>. <https://www.rfc-editor.org/info/rfc2119>.
[RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC [RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
May 2017, <https://www.rfc-editor.org/info/rfc8174>. May 2017, <https://www.rfc-editor.org/info/rfc8174>.
7.2. Informative References 7.2. 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] [Baccelli18]
Baccelli, E., Gundogan, C., Hahm, O., Kietzmann, P., Baccelli, E., Gundogan, C., Hahm, O., Kietzmann, P.,
Lenders, M., Petersen, H., Schleiser, K., Schmidt, T., and Lenders, M., Petersen, H., Schleiser, K., Schmidt, T., and
M. Wahlisch, "RIOT: An Open Source Operating System for M. Wahlisch, "RIOT: An Open Source Operating System for
Low-End Embedded Devices in the IoT", IEEE Internet of Low-End Embedded Devices in the IoT", IEEE Internet of
Things Journal (Volume 5, Issue 6), DOI: Things Journal (Volume 5, Issue 6), DOI:
10.1109/JIOT.2018.2815038, December 2018. 10.1109/JIOT.2018.2815038, December 2018.
[KR12] Rikitake, K., "TinyMT Pseudo Random Number Generator for [KR12] Rikitake, K., "TinyMT Pseudo Random Number Generator for
Erlang", ACM 11th SIGPLAN Erlang Workshop (Erlang'12), Erlang", ACM 11th SIGPLAN Erlang Workshop (Erlang'12),
skipping to change at page 10, line 33 skipping to change at page 11, line 17
Correction (FEC) Schemes", RFC 5170, DOI 10.17487/RFC5170, Correction (FEC) Schemes", RFC 5170, DOI 10.17487/RFC5170,
June 2008, <https://www.rfc-editor.org/info/rfc5170>. June 2008, <https://www.rfc-editor.org/info/rfc5170>.
[RLC-ID] Roca, V. and B. Teibi, "Sliding Window Random Linear Code [RLC-ID] Roca, V. and B. Teibi, "Sliding Window Random Linear Code
(RLC) Forward Erasure Correction (FEC) Scheme for (RLC) Forward Erasure Correction (FEC) Scheme for
FECFRAME", Work in Progress, Transport Area Working Group FECFRAME", Work in Progress, Transport Area Working Group
(TSVWG) draft-ietf-tsvwg-rlc-fec-scheme (Work in (TSVWG) draft-ietf-tsvwg-rlc-fec-scheme (Work in
Progress), February 2019, <https://tools.ietf.org/html/ Progress), February 2019, <https://tools.ietf.org/html/
draft-ietf-tsvwg-rlc-fec-scheme>. draft-ietf-tsvwg-rlc-fec-scheme>.
[TestU01] L'Ecuyer, P. and R. Simard, "TestU01: A C Library for
Empirical Testing of Random Number Generators", ACM
Transactions on Mathematical Software, Vol. 33, article
22, 2007, 2007,
<http://simul.iro.umontreal.ca/testu01/tu01.html>.
[TinyMT-dev] [TinyMT-dev]
Saito, M. and M. Matsumoto, "Tiny Mersenne Twister Saito, M. and M. Matsumoto, "Tiny Mersenne Twister
(TinyMT) github site", (TinyMT) github site",
<https://github.com/MersenneTwister-Lab/TinyMT>. <https://github.com/MersenneTwister-Lab/TinyMT>.
[TinyMT-params] [TinyMT-params]
Rikitake, K., "TinyMT pre-calculated parameter list github Rikitake, K., "TinyMT pre-calculated parameter list github
site", <https://github.com/jj1bdx/tinymtdc-longbatch/>. site", <https://github.com/jj1bdx/tinymtdc-longbatch/>.
[TinyMT-web] [TinyMT-web]
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