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authorRaimo Niskanen <[email protected]>2018-09-18 16:33:47 +0200
committerGitHub <[email protected]>2018-09-18 16:33:47 +0200
commitb2c338cb8d84567204765db87c7299519f1e1ad6 (patch)
tree91c6c4225c639e50b68619c19f14146523c65e0d /lib
parent146bcbf1f4f9cefb73223a654ca5e992ebe43aa8 (diff)
parent0f79e3f3d95fd8f04e3893e50c9f27b9e04c2c7e (diff)
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Merge pull request #1857 from RaimoNiskanen/raimo/rand-crypto-xoroshiro928
OTP-14461 - New 'rand' algorithm: Xoroshiro928** also for 'crypto' Implement a new 'rand' algorithm named 'exro928ss' and a new 'crypto' plugin for 'rand' named 'crypto_aes'. Both are based on Xoroshiro928** which is derived from Xoroshiro1024** modified to use 58-bit words for performance reasons in the Erlang VM. Xoroshiro1024** has got the Xoroshiro1024 generator and the StarStar scrambler from the 2018 paper "Scrambled Linear Pseudorandom Number Generators" by David Blackman and Sebastiano Vigna. This generator and scrambler combination shows no systematic weaknesses in standard statistical tests as TestU01(BigCrush) and PractRand, unlike the previously used * and + scramblers in the 'rand' module that exhibit statistical weaknesses for the lowest bits. The 'crypto' plugin uses AES-256 as scrambler and the Xoroshiro928 as generator, which gives the same very long period and jump functions as for Xoroshiro928**, but a cryptographically secure scrambler gives absolutely no detectable statistical weaknesses regardless of how the generated numbers are used. The speed of 'exro928ss' is only about 30-50% slower than the default fast 'rand' algorithm, but the state is roughly the double and it produces about 8 times the garbage per iteration. The speed of 'crypto_aes' is about half (amortized) that of the default fast 'rand' algorithm which is fast and thanks to doing encryption in batches caching the result. Hence the state is much larger.
Diffstat (limited to 'lib')
-rw-r--r--lib/crypto/doc/src/crypto.xml110
-rw-r--r--lib/crypto/src/crypto.erl185
-rw-r--r--lib/stdlib/doc/src/rand.xml61
-rw-r--r--lib/stdlib/src/rand.erl433
-rw-r--r--lib/stdlib/test/rand_SUITE.erl373
-rw-r--r--lib/stdlib/test/rand_Xoroshiro928ss_dev.txt343
6 files changed, 1312 insertions, 193 deletions
diff --git a/lib/crypto/doc/src/crypto.xml b/lib/crypto/doc/src/crypto.xml
index 8c4dc1729a..e9ccd89911 100644
--- a/lib/crypto/doc/src/crypto.xml
+++ b/lib/crypto/doc/src/crypto.xml
@@ -1,4 +1,3 @@
-<?xml version="1.0" encoding="utf-8" ?>
<!DOCTYPE erlref SYSTEM "erlref.dtd">
<erlref>
@@ -905,7 +904,8 @@ _FloatValue = rand:uniform(). % [0.0; 1.0[</pre>
<p>
Creates state object for
<seealso marker="stdlib:rand">random number generation</seealso>,
- in order to generate cryptographically strong random numbers.
+ in order to generate cryptographically strong random numbers,
+ and saves it in the process dictionary before returning it as well.
See also
<seealso marker="stdlib:rand#seed-1">rand:seed/1</seealso> and
<seealso marker="#rand_seed_alg_s-1">rand_seed_alg_s/1</seealso>.
@@ -916,12 +916,6 @@ _FloatValue = rand:uniform(). % [0.0; 1.0[</pre>
may raise exception <c>error:low_entropy</c> in case the random generator
failed due to lack of secure "randomness".
</p>
- <p>
- The cache size can be changed from its default value using the
- <seealso marker="crypto_app">
- crypto app's
- </seealso> configuration parameter <c>rand_cache_size</c>.
- </p>
<p><em>Example</em></p>
<pre>
_ = crypto:rand_seed_alg(crypto_cache),
@@ -931,6 +925,34 @@ _FloatValue = rand:uniform(). % [0.0; 1.0[</pre>
</func>
<func>
+ <name>rand_seed_alg(Alg, Seed) -> rand:state()</name>
+ <fsummary>Strong random number generation plugin state</fsummary>
+ <type>
+ <v>Alg = crypto_aes</v>
+ </type>
+ <desc>
+ <marker id="rand_seed_alg-2" />
+ <p>
+ Creates a state object for
+ <seealso marker="stdlib:rand">random number generation</seealso>,
+ in order to generate cryptographically unpredictable random numbers,
+ and saves it in the process dictionary before returning it as well.
+ See also
+ <seealso marker="#rand_seed_alg_s-2">rand_seed_alg_s/2</seealso>.
+ </p>
+ <p><em>Example</em></p>
+ <pre>
+_ = crypto:rand_seed_alg(crypto_aes, "my seed"),
+IntegerValue = rand:uniform(42), % [1; 42]
+FloatValue = rand:uniform(), % [0.0; 1.0[
+_ = crypto:rand_seed_alg(crypto_aes, "my seed"),
+IntegerValue = rand:uniform(42), % Same values
+FloatValue = rand:uniform(). % again
+ </pre>
+ </desc>
+ </func>
+
+ <func>
<name>rand_seed_alg_s(Alg) -> rand:state()</name>
<fsummary>Strong random number generation plugin state</fsummary>
<type>
@@ -967,6 +989,12 @@ _FloatValue = rand:uniform(). % [0.0; 1.0[</pre>
crypto app's
</seealso> configuration parameter <c>rand_cache_size</c>.
</p>
+ <p>
+ When using the state object from this function the
+ <seealso marker="stdlib:rand">rand</seealso> functions using it
+ may throw exception <c>low_entropy</c> in case the random generator
+ failed due to lack of secure "randomness".
+ </p>
<note>
<p>
The state returned from this function cannot be used
@@ -989,6 +1017,72 @@ _FloatValue = rand:uniform(). % [0.0; 1.0[</pre>
</func>
<func>
+ <name>rand_seed_alg_s(Alg, Seed) -> rand:state()</name>
+ <fsummary>Strong random number generation plugin state</fsummary>
+ <type>
+ <v>Alg = crypto_aes</v>
+ </type>
+ <desc>
+ <marker id="rand_seed_alg_s-2" />
+ <p>
+ Creates a state object for
+ <seealso marker="stdlib:rand">random number generation</seealso>,
+ in order to generate cryptographically unpredictable random numbers.
+ See also
+ <seealso marker="#rand_seed_alg-1">rand_seed_alg/1</seealso>.
+ </p>
+ <p>
+ To get a long period the Xoroshiro928 generator from the
+ <seealso marker="stdlib:rand">rand</seealso>
+ module is used as a counter (with period 2^928 - 1)
+ and the generator states are scrambled through AES
+ to create 58-bit pseudo random values.
+ </p>
+ <p>
+ The result should be statistically completely unpredictable
+ random values, since the scrambling is cryptographically strong
+ and the period is ridiculously long. But the generated numbers
+ are not to be regarded as cryptographically strong since
+ there is no re-keying schedule.
+ </p>
+ <list type="bulleted">
+ <item>
+ <p>
+ If you need cryptographically strong random numbers use
+ <seealso marker="#rand_seed_alg_s-1">rand_seed_alg_s/1</seealso>
+ with <c>Alg =:= crypto</c> or <c>Alg =:= crypto_cache</c>.
+ </p>
+ </item>
+ <item>
+ <p>
+ If you need to be able to repeat the sequence use this function.
+ </p>
+ </item>
+ <item>
+ <p>
+ If you do not need the statistical quality of this function,
+ there are faster algorithms in the
+ <seealso marker="stdlib:rand">rand</seealso>
+ module.
+ </p>
+ </item>
+ </list>
+ <p>
+ Thanks to the used generator the state object supports the
+ <seealso marker="stdlib:rand#jump-0"><c>rand:jump/0,1</c></seealso>
+ function with distance 2^512.
+ </p>
+ <p>
+ Numbers are generated in batches and cached for speed reasons.
+ The cache size can be changed from its default value using the
+ <seealso marker="crypto_app">
+ crypto app's
+ </seealso> configuration parameter <c>rand_cache_size</c>.
+ </p>
+ </desc>
+ </func>
+
+ <func>
<name name="stream_init" arity="2"/>
<fsummary></fsummary>
<desc>
diff --git a/lib/crypto/src/crypto.erl b/lib/crypto/src/crypto.erl
index 2db73c4af0..68c0bcef5e 100644
--- a/lib/crypto/src/crypto.erl
+++ b/lib/crypto/src/crypto.erl
@@ -31,9 +31,10 @@
-export([cmac/3, cmac/4]).
-export([poly1305/2]).
-export([exor/2, strong_rand_bytes/1, mod_pow/3]).
--export([rand_seed/0, rand_seed_alg/1]).
--export([rand_seed_s/0, rand_seed_alg_s/1]).
+-export([rand_seed/0, rand_seed_alg/1, rand_seed_alg/2]).
+-export([rand_seed_s/0, rand_seed_alg_s/1, rand_seed_alg_s/2]).
-export([rand_plugin_next/1]).
+-export([rand_plugin_aes_next/1, rand_plugin_aes_jump/1]).
-export([rand_plugin_uniform/1]).
-export([rand_plugin_uniform/2]).
-export([rand_cache_plugin_next/1]).
@@ -92,7 +93,9 @@
]).
%% Private. For tests.
--export([packed_openssl_version/4, engine_methods_convert_to_bitmask/2, get_test_engine/0]).
+-export([packed_openssl_version/4, engine_methods_convert_to_bitmask/2,
+ get_test_engine/0]).
+-export([rand_plugin_aes_jump_2pow20/1]).
-deprecated({rand_uniform, 2, next_major_release}).
@@ -674,34 +677,73 @@ rand_seed_s() ->
rand_seed_alg(Alg) ->
rand:seed(rand_seed_alg_s(Alg)).
+-spec rand_seed_alg(Alg :: atom(), Seed :: term()) ->
+ {rand:alg_handler(),
+ atom() | rand_cache_seed()}.
+rand_seed_alg(Alg, Seed) ->
+ rand:seed(rand_seed_alg_s(Alg, Seed)).
+
-define(CRYPTO_CACHE_BITS, 56).
+-define(CRYPTO_AES_BITS, 58).
+
-spec rand_seed_alg_s(Alg :: atom()) ->
{rand:alg_handler(),
atom() | rand_cache_seed()}.
-rand_seed_alg_s(?MODULE) ->
- {#{ type => ?MODULE,
- bits => 64,
- next => fun ?MODULE:rand_plugin_next/1,
- uniform => fun ?MODULE:rand_plugin_uniform/1,
- uniform_n => fun ?MODULE:rand_plugin_uniform/2},
- no_seed};
-rand_seed_alg_s(crypto_cache) ->
+rand_seed_alg_s({AlgHandler, _AlgState} = State) when is_map(AlgHandler) ->
+ State;
+rand_seed_alg_s({Alg, AlgState}) when is_atom(Alg) ->
+ {mk_alg_handler(Alg),AlgState};
+ rand_seed_alg_s(Alg) when is_atom(Alg) ->
+ {mk_alg_handler(Alg),mk_alg_state(Alg)}.
+%%
+-spec rand_seed_alg_s(Alg :: atom(), Seed :: term()) ->
+ {rand:alg_handler(),
+ atom() | rand_cache_seed()}.
+rand_seed_alg_s(Alg, Seed) when is_atom(Alg) ->
+ {mk_alg_handler(Alg),mk_alg_state({Alg,Seed})}.
+
+mk_alg_handler(?MODULE = Alg) ->
+ #{ type => Alg,
+ bits => 64,
+ next => fun ?MODULE:rand_plugin_next/1,
+ uniform => fun ?MODULE:rand_plugin_uniform/1,
+ uniform_n => fun ?MODULE:rand_plugin_uniform/2};
+mk_alg_handler(crypto_cache = Alg) ->
+ #{ type => Alg,
+ bits => ?CRYPTO_CACHE_BITS,
+ next => fun ?MODULE:rand_cache_plugin_next/1};
+mk_alg_handler(crypto_aes = Alg) ->
+ #{ type => Alg,
+ bits => ?CRYPTO_AES_BITS,
+ next => fun ?MODULE:rand_plugin_aes_next/1,
+ jump => fun ?MODULE:rand_plugin_aes_jump/1}.
+
+mk_alg_state(?MODULE) ->
+ no_seed;
+mk_alg_state(crypto_cache) ->
CacheBits = ?CRYPTO_CACHE_BITS,
- EnvCacheSize =
- application:get_env(
- crypto, rand_cache_size, CacheBits * 16), % Cache 16 * 8 words
- Bytes = (CacheBits + 7) div 8,
+ BytesPerWord = (CacheBits + 7) div 8,
+ GenBytes =
+ ((rand_cache_size() + (2*BytesPerWord - 1)) div BytesPerWord)
+ * BytesPerWord,
+ {CacheBits, GenBytes, <<>>};
+mk_alg_state({crypto_aes,Seed}) ->
+ %% 16 byte words (128 bit crypto blocks)
+ GenWords = (rand_cache_size() + 31) div 16,
+ Key = crypto:hash(sha256, Seed),
+ {F,Count} = longcount_seed(Seed),
+ {Key,GenWords,F,Count}.
+
+rand_cache_size() ->
+ DefaultCacheSize = 1024,
CacheSize =
- case ((EnvCacheSize + (Bytes - 1)) div Bytes) * Bytes of
- Sz when is_integer(Sz), Bytes =< Sz ->
- Sz;
- _ ->
- Bytes
- end,
- {#{ type => crypto_cache,
- bits => CacheBits,
- next => fun ?MODULE:rand_cache_plugin_next/1},
- {CacheBits, CacheSize, <<>>}}.
+ application:get_env(crypto, rand_cache_size, DefaultCacheSize),
+ if
+ is_integer(CacheSize), 0 =< CacheSize ->
+ CacheSize;
+ true ->
+ DefaultCacheSize
+ end.
rand_plugin_next(Seed) ->
{bytes_to_integer(strong_rand_range(1 bsl 64)), Seed}.
@@ -712,12 +754,97 @@ rand_plugin_uniform(State) ->
rand_plugin_uniform(Max, State) ->
{bytes_to_integer(strong_rand_range(Max)) + 1, State}.
-rand_cache_plugin_next({CacheBits, CacheSize, <<>>}) ->
+
+rand_cache_plugin_next({CacheBits, GenBytes, <<>>}) ->
rand_cache_plugin_next(
- {CacheBits, CacheSize, strong_rand_bytes(CacheSize)});
-rand_cache_plugin_next({CacheBits, CacheSize, Cache}) ->
+ {CacheBits, GenBytes, strong_rand_bytes(GenBytes)});
+rand_cache_plugin_next({CacheBits, GenBytes, Cache}) ->
<<I:CacheBits, NewCache/binary>> = Cache,
- {I, {CacheBits, CacheSize, NewCache}}.
+ {I, {CacheBits, GenBytes, NewCache}}.
+
+
+%% Encrypt 128 bit counter values and use the 58 lowest
+%% encrypted bits as random numbers.
+%%
+%% The 128 bit counter is handled as 4 32 bit words
+%% to avoid bignums. Generate a bunch of numbers
+%% at the time and cache them.
+%%
+-dialyzer({no_improper_lists, rand_plugin_aes_next/1}).
+rand_plugin_aes_next([V|Cache]) ->
+ {V,Cache};
+rand_plugin_aes_next({Key,GenWords,F,Count}) ->
+ rand_plugin_aes_next(Key, GenWords, F, Count);
+rand_plugin_aes_next({Key,GenWords,F,_JumpBase,Count}) ->
+ rand_plugin_aes_next(Key, GenWords, F, Count).
+%%
+rand_plugin_aes_next(Key, GenWords, F, Count) ->
+ {Cleartext,NewCount} = aes_cleartext(<<>>, F, Count, GenWords),
+ Encrypted = crypto:block_encrypt(aes_ecb, Key, Cleartext),
+ [V|Cache] = aes_cache(Encrypted, {Key,GenWords,F,Count,NewCount}),
+ {V,Cache}.
+
+%% A jump advances the counter 2^512 steps; the jump function
+%% is applied to the jump base and then the number of used
+%% numbers from the cache has to be wasted for the jump to be correct
+%%
+rand_plugin_aes_jump({#{type := crypto_aes} = Alg, Cache}) ->
+ {Alg,rand_plugin_aes_jump(fun longcount_jump/1, 0, Cache)}.
+%% Count cached words and subtract their number from jump
+-dialyzer({no_improper_lists, rand_plugin_aes_jump/3}).
+rand_plugin_aes_jump(Jump, J, [_|Cache]) ->
+ rand_plugin_aes_jump(Jump, J + 1, Cache);
+rand_plugin_aes_jump(Jump, J, {Key,GenWords,F,JumpBase, _Count}) ->
+ rand_plugin_aes_jump(Jump, GenWords - J, Key, GenWords, F, JumpBase);
+rand_plugin_aes_jump(Jump, 0, {Key,GenWords,F,JumpBase}) ->
+ rand_plugin_aes_jump(Jump, 0, Key, GenWords, F, JumpBase).
+%%
+rand_plugin_aes_jump(Jump, Skip, Key, GenWords, F, JumpBase) ->
+ Count = longcount_next_count(Skip, Jump(JumpBase)),
+ {Key,GenWords,F,Count}.
+
+rand_plugin_aes_jump_2pow20(Cache) ->
+ rand_plugin_aes_jump(fun longcount_jump_2pow20/1, 0, Cache).
+
+
+longcount_seed(Seed) ->
+ <<X:64, _:6, F:12, S2:58, S1:58, S0:58>> =
+ crypto:hash(sha256, [Seed,<<"Xoroshiro928">>]),
+ {F,rand:exro928_seed([S0,S1,S2|rand:seed58(13, X)])}.
+
+longcount_next_count(0, Count) ->
+ Count;
+longcount_next_count(N, Count) ->
+ longcount_next_count(N - 1, rand:exro928_next_state(Count)).
+
+longcount_next(Count) ->
+ rand:exro928_next(Count).
+
+longcount_jump(Count) ->
+ rand:exro928_jump_2pow512(Count).
+
+longcount_jump_2pow20(Count) ->
+ rand:exro928_jump_2pow20(Count).
+
+
+%% Build binary with counter values to cache
+aes_cleartext(Cleartext, _F, Count, 0) ->
+ {Cleartext,Count};
+aes_cleartext(Cleartext, F, Count, GenWords) ->
+ {{S0,S1}, NewCount} = longcount_next(Count),
+ aes_cleartext(
+ <<Cleartext/binary, F:12, S1:58, S0:58>>,
+ F, NewCount, GenWords - 1).
+
+%% Parse and cache encrypted counter values aka random numbers
+-dialyzer({no_improper_lists, aes_cache/2}).
+aes_cache(<<>>, Cache) ->
+ Cache;
+aes_cache(
+ <<_:(128 - ?CRYPTO_AES_BITS), V:?CRYPTO_AES_BITS, Encrypted/binary>>,
+ Cache) ->
+ [V|aes_cache(Encrypted, Cache)].
+
strong_rand_range(Range) when is_integer(Range), Range > 0 ->
BinRange = int_to_bin(Range),
diff --git a/lib/stdlib/doc/src/rand.xml b/lib/stdlib/doc/src/rand.xml
index 21f680a0ee..25eec216ef 100644
--- a/lib/stdlib/doc/src/rand.xml
+++ b/lib/stdlib/doc/src/rand.xml
@@ -67,6 +67,26 @@
<p>Xorshift1024*, 64 bits precision and a period of 2^1024-1</p>
<p>Jump function: equivalent to 2^512 calls</p>
</item>
+ <tag><c>exro928ss</c></tag>
+ <item>
+ <p>Xoroshiro928**, 58 bits precision and a period of 2^928-1</p>
+ <p>Jump function: equivalent to 2^512 calls</p>
+ <p>
+ This is a 58 bit version of Xoroshiro1024**,
+ from the 2018 paper by David Blackman and Sebastiano Vigna:
+ <url href="http://vigna.di.unimi.it/ftp/papers/ScrambledLinear.pdf">
+ Scrambled Linear Pseudorandom Number Generators
+ </url>
+ that on a 64 bit Erlang system executes only about 30% slower than
+ the default <c>exrop</c> algorithm but with much longer period
+ and better statistical properties, and on the flip side
+ a larger state.
+ </p>
+ <p>
+ Many thanks to Sebastiano Vigna for his help with
+ the 58 bit adaption.
+ </p>
+ </item>
<tag><c>exsp</c></tag>
<item>
<p>Xorshift116+, 58 bits precision and period of 2^116-1</p>
@@ -195,8 +215,8 @@ SND0 = math:sqrt(-2 * math:log(R5)) * math:cos(math:pi() * R6)</pre>
</note>
<p>
- For all these generators the lowest bit(s) has got
- a slightly less random behaviour than all other bits.
+ For all these generators except <c>exro928ss</c> the lowest bit(s)
+ has got a slightly less random behaviour than all other bits.
1 bit for <c>exrop</c> (and <c>exsp</c>),
and 3 bits for <c>exs1024s</c>.
See for example the explanation in the
@@ -254,11 +274,32 @@ tests. We suggest to use a sign test to extract a random Boolean value.</pre>
</desc>
</datatype>
<datatype>
- <name name="exs64_state"/>
- <desc><p>Algorithm specific internal state</p></desc>
+ <name name="seed"/>
+ <desc>
+ <p>
+ A seed value for the generator.
+ </p>
+ <p>
+ A list of integers sets the generator's internal state directly,
+ after algorithm-dependent checks of the value
+ and masking to the proper word size.
+ </p>
+ <p>
+ An integer is used as the initial state for a SplitMix64 generator.
+ The output values of that is then used for setting
+ the generator's internal state
+ after masking to the proper word size
+ and if needed avoiding zero values.
+ </p>
+ <p>
+ A traditional 3-tuple of integers seed is passed through
+ algorithm-dependent hashing functions to create
+ the generator's initial state.
+ </p>
+ </desc>
</datatype>
<datatype>
- <name name="exsplus_state"/>
+ <name name="exrop_state"/>
<desc><p>Algorithm specific internal state</p></desc>
</datatype>
<datatype>
@@ -266,7 +307,15 @@ tests. We suggest to use a sign test to extract a random Boolean value.</pre>
<desc><p>Algorithm specific internal state</p></desc>
</datatype>
<datatype>
- <name name="exrop_state"/>
+ <name name="exro928_state"/>
+ <desc><p>Algorithm specific internal state</p></desc>
+ </datatype>
+ <datatype>
+ <name name="exsplus_state"/>
+ <desc><p>Algorithm specific internal state</p></desc>
+ </datatype>
+ <datatype>
+ <name name="exs64_state"/>
<desc><p>Algorithm specific internal state</p></desc>
</datatype>
</datatypes>
diff --git a/lib/stdlib/src/rand.erl b/lib/stdlib/src/rand.erl
index 4951dc727b..9854c778a1 100644
--- a/lib/stdlib/src/rand.erl
+++ b/lib/stdlib/src/rand.erl
@@ -1,7 +1,7 @@
%%
%% %CopyrightBegin%
%%
-%% Copyright Ericsson AB 2015-2017. All Rights Reserved.
+%% Copyright Ericsson AB 2015-2018. All Rights Reserved.
%%
%% Licensed under the Apache License, Version 2.0 (the "License");
%% you may not use this file except in compliance with the License.
@@ -32,14 +32,20 @@
uniform/0, uniform/1, uniform_s/1, uniform_s/2,
uniform_real/0, uniform_real_s/1,
jump/0, jump/1,
- normal/0, normal/2, normal_s/1, normal_s/3
+ normal/0, normal/2, normal_s/1, normal_s/3
]).
+%% Test, dev and internal
+-export([exro928_jump_2pow512/1, exro928_jump_2pow20/1,
+ exro928_seed/1, exro928_next/1, exro928_next_state/1,
+ format_jumpconst58/1, seed58/2]).
+
%% Debug
-export([make_float/3, float2str/1, bc64/1]).
-compile({inline, [exs64_next/1, exsplus_next/1,
exs1024_next/1, exs1024_calc/2,
+ exro928_next_state/4,
exrop_next/1, exrop_next_s/2,
get_52/1, normal_kiwi/1]}).
@@ -80,8 +86,8 @@
%% This depends on the algorithm handler function
-type alg_state() ::
- exs64_state() | exsplus_state() | exs1024_state() |
- exrop_state() | term().
+ exrop_state() | exs1024_state() | exro928_state() | exsplus_state() |
+ exs64_state() | term().
%% This is the algorithm handling definition within this module,
%% and the type to use for plugins.
@@ -124,14 +130,17 @@
%% Algorithm state
-type state() :: {alg_handler(), alg_state()}.
--type builtin_alg() :: exs64 | exsplus | exsp | exs1024 | exs1024s | exrop.
+-type builtin_alg() ::
+ exrop | exs1024s | exro928ss | exsp | exs64 | exsplus | exs1024.
-type alg() :: builtin_alg() | atom().
-type export_state() :: {alg(), alg_state()}.
+-type seed() :: [integer()] | integer() | {integer(), integer(), integer()}.
-export_type(
[builtin_alg/0, alg/0, alg_handler/0, alg_state/0,
- state/0, export_state/0]).
+ state/0, export_state/0, seed/0]).
-export_type(
- [exs64_state/0, exsplus_state/0, exs1024_state/0, exrop_state/0]).
+ [exrop_state/0, exs1024_state/0, exro928_state/0, exsplus_state/0,
+ exs64_state/0]).
%% =====================================================================
%% Range macro and helper
@@ -229,12 +238,12 @@ export_seed() ->
end.
-spec export_seed_s(State :: state()) -> export_state().
-export_seed_s({#{type:=Alg}, Seed}) -> {Alg, Seed}.
+export_seed_s({#{type:=Alg}, AlgState}) -> {Alg, AlgState}.
%% seed(Alg) seeds RNG with runtime dependent values
%% and return the NEW state
-%% seed({Alg,Seed}) setup RNG with a previously exported seed
+%% seed({Alg,AlgState}) setup RNG with a previously exported seed
%% and return the NEW state
-spec seed(
@@ -246,11 +255,11 @@ seed(Alg) ->
-spec seed_s(
AlgOrStateOrExpState :: builtin_alg() | state() | export_state()) ->
state().
-seed_s({AlgHandler, _Seed} = State) when is_map(AlgHandler) ->
+seed_s({AlgHandler, _AlgState} = State) when is_map(AlgHandler) ->
State;
-seed_s({Alg0, Seed}) ->
- {Alg,_SeedFun} = mk_alg(Alg0),
- {Alg, Seed};
+seed_s({Alg, AlgState}) when is_atom(Alg) ->
+ {AlgHandler,_SeedFun} = mk_alg(Alg),
+ {AlgHandler,AlgState};
seed_s(Alg) ->
seed_s(Alg, {erlang:phash2([{node(),self()}]),
erlang:system_time(),
@@ -259,19 +268,15 @@ seed_s(Alg) ->
%% seed/2: seeds RNG with the algorithm and given values
%% and returns the NEW state.
--spec seed(
- Alg :: builtin_alg(), Seed :: {integer(), integer(), integer()}) ->
- state().
-seed(Alg0, S0) ->
- seed_put(seed_s(Alg0, S0)).
+-spec seed(Alg :: builtin_alg(), Seed :: seed()) -> state().
+seed(Alg, Seed) ->
+ seed_put(seed_s(Alg, Seed)).
--spec seed_s(
- Alg :: builtin_alg(), Seed :: {integer(), integer(), integer()}) ->
- state().
-seed_s(Alg0, S0 = {_, _, _}) ->
- {Alg, Seed} = mk_alg(Alg0),
- AS = Seed(S0),
- {Alg, AS}.
+-spec seed_s(Alg :: builtin_alg(), Seed :: seed()) -> state().
+seed_s(Alg, Seed) ->
+ {AlgHandler,SeedFun} = mk_alg(Alg),
+ AlgState = SeedFun(Seed),
+ {AlgHandler,AlgState}.
%%% uniform/0, uniform/1, uniform_s/1, uniform_s/2 are all
%%% uniformly distributed random numbers.
@@ -281,8 +286,8 @@ seed_s(Alg0, S0 = {_, _, _}) ->
-spec uniform() -> X :: float().
uniform() ->
- {X, Seed} = uniform_s(seed_get()),
- _ = seed_put(Seed),
+ {X, State} = uniform_s(seed_get()),
+ _ = seed_put(State),
X.
%% uniform/1: given an integer N >= 1,
@@ -291,8 +296,8 @@ uniform() ->
-spec uniform(N :: pos_integer()) -> X :: pos_integer().
uniform(N) ->
- {X, Seed} = uniform_s(N, seed_get()),
- _ = seed_put(Seed),
+ {X, State} = uniform_s(N, seed_get()),
+ _ = seed_put(State),
X.
%% uniform_s/1: given a state, uniform_s/1
@@ -625,7 +630,13 @@ mk_alg(exrop) ->
{#{type=>exrop, bits=>58, weak_low_bits=>1, next=>fun exrop_next/1,
uniform=>fun exrop_uniform/1, uniform_n=>fun exrop_uniform/2,
jump=>fun exrop_jump/1},
- fun exrop_seed/1}.
+ fun exrop_seed/1};
+mk_alg(exro928ss) ->
+ {#{type=>exro928ss, bits=>58, next=>fun exro928ss_next/1,
+ uniform=>fun exro928ss_uniform/1,
+ uniform_n=>fun exro928ss_uniform/2,
+ jump=>fun exro928_jump/1},
+ fun exro928_seed/1}.
%% =====================================================================
%% exs64 PRNG: Xorshift64*
@@ -635,6 +646,14 @@ mk_alg(exrop) ->
-opaque exs64_state() :: uint64().
+exs64_seed(L) when is_list(L) ->
+ [R] = seed64_nz(1, L),
+ R;
+exs64_seed(A) when is_integer(A) ->
+ [R] = seed64(1, ?MASK(64, A)),
+ R;
+%%
+%% Traditional integer triplet seed
exs64_seed({A1, A2, A3}) ->
{V1, _} = exs64_next((?MASK(32, A1) * 4294967197 + 1)),
{V2, _} = exs64_next((?MASK(32, A2) * 4294967231 + 1)),
@@ -661,6 +680,14 @@ exs64_next(R) ->
-dialyzer({no_improper_lists, exsplus_seed/1}).
+exsplus_seed(L) when is_list(L) ->
+ [S0,S1] = seed58_nz(2, L),
+ [S0|S1];
+exsplus_seed(X) when is_integer(X) ->
+ [S0,S1] = seed58(2, ?MASK(64, X)),
+ [S0|S1];
+%%
+%% Traditional integer triplet seed
exsplus_seed({A1, A2, A3}) ->
{_, R1} = exsplus_next(
[?MASK(58, (A1 * 4294967197) + 1)|
@@ -708,7 +735,8 @@ exsp_uniform(Range, {Alg, R}) ->
-define(JUMPELEMLEN, 58).
-dialyzer({no_improper_lists, exsplus_jump/1}).
--spec exsplus_jump(state()) -> state().
+-spec exsplus_jump({alg_handler(), exsplus_state()}) ->
+ {alg_handler(), exsplus_state()}.
exsplus_jump({Alg, S}) ->
{S1, AS1} = exsplus_jump(S, [0|0], ?JUMPCONST1, ?JUMPELEMLEN),
{_, AS2} = exsplus_jump(S1, AS1, ?JUMPCONST2, ?JUMPELEMLEN),
@@ -735,6 +763,12 @@ exsplus_jump(S, [AS0|AS1], J, N) ->
-opaque exs1024_state() :: {list(uint64()), list(uint64())}.
+exs1024_seed(L) when is_list(L) ->
+ {seed64_nz(16, L), []};
+exs1024_seed(X) when is_integer(X) ->
+ {seed64(16, ?MASK(64, X)), []};
+%%
+%% Seed from traditional triple, remain backwards compatible
exs1024_seed({A1, A2, A3}) ->
B1 = ?MASK(21, (?MASK(21, A1) + 1) * 2097131),
B2 = ?MASK(21, (?MASK(21, A2) + 1) * 2097133),
@@ -806,8 +840,8 @@ exs1024_next({[H], RL}) ->
-define(JUMPTOTALLEN, 1024).
-define(RINGLEN, 16).
--spec exs1024_jump(state()) -> state().
-
+-spec exs1024_jump({alg_handler(), exs1024_state()}) ->
+ {alg_handler(), exs1024_state()}.
exs1024_jump({Alg, {L, RL}}) ->
P = length(RL),
AS = exs1024_jump({L, RL},
@@ -832,6 +866,194 @@ exs1024_jump({L, RL}, AS, JL, J, N, TN) ->
end.
%% =====================================================================
+%% exro928ss PRNG: Xoroshiro928**
+%%
+%% Reference URL: http://vigna.di.unimi.it/ftp/papers/ScrambledLinear.pdf
+%% i.e the Xoroshiro1024 generator with ** scrambler
+%% with {S, R, T} = {5, 7, 9} as recommended in the paper.
+%%
+%% {A, B, C} were tried out and selected as {44, 9, 45}
+%% and the jump coefficients calculated.
+%%
+%% Standard jump function pseudocode:
+%%
+%% Jump constant j = 0xb10773cb...44085302f77130ca
+%% Generator state: s
+%% New generator state: t = 0
+%% foreach bit in j, low to high:
+%% if the bit is one:
+%% t ^= s
+%% next s
+%% s = t
+%%
+%% Generator used for reference value calculation:
+%%
+%% #include <stdint.h>
+%% #include <stdio.h>
+%%
+%% int p = 0;
+%% uint64_t s[16];
+%%
+%% #define MASK(x) ((x) & ((UINT64_C(1) << 58) - 1))
+%% static __inline uint64_t rotl(uint64_t x, int n) {
+%% return MASK(x << n) | (x >> (58 - n));
+%% }
+%%
+%% uint64_t next() {
+%% const int q = p;
+%% const uint64_t s0 = s[p = (p + 1) & 15];
+%% uint64_t s15 = s[q];
+%%
+%% const uint64_t result_starstar = MASK(rotl(MASK(s0 * 5), 7) * 9);
+%%
+%% s15 ^= s0;
+%% s[q] = rotl(s0, 44) ^ s15 ^ MASK(s15 << 9);
+%% s[p] = rotl(s15, 45);
+%%
+%% return result_starstar;
+%% }
+%%
+%% static const uint64_t jump_2pow512[15] =
+%% { 0x44085302f77130ca, 0xba05381fdfd14902, 0x10a1de1d7d6813d2,
+%% 0xb83fe51a1eb3be19, 0xa81b0090567fd9f0, 0x5ac26d5d20f9b49f,
+%% 0x4ddd98ee4be41e01, 0x0657e19f00d4b358, 0xf02f778573cf0f0a,
+%% 0xb45a3a8a3cef3cc0, 0x6e62a33cc2323831, 0xbcb3b7c4cc049c53,
+%% 0x83f240c6007e76ce, 0xe19f5fc1a1504acd, 0x00000000b10773cb };
+%%
+%% static const uint64_t jump_2pow20[15] =
+%% { 0xbdb966a3daf905e6, 0x644807a56270cf78, 0xda90f4a806c17e9e,
+%% 0x4a426866bfad3c77, 0xaf699c306d8e7566, 0x8ebc73c700b8b091,
+%% 0xc081a7bf148531fb, 0xdc4d3af15f8a4dfd, 0x90627c014098f4b6,
+%% 0x06df2eb1feaf0fb6, 0x5bdeb1a5a90f2e6b, 0xa480c5878c3549bd,
+%% 0xff45ef33c82f3d48, 0xa30bebc15fefcc78, 0x00000000cb3d181c };
+%%
+%% void jump(const uint64_t *jump) {
+%% uint64_t j, t[16] = {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0};
+%% int m, n, k;
+%% for (m = 0; m < 15; m++, jump++) {
+%% for (n = 0, j = *jump; n < 64; n++, j >>= 1) {
+%% if ((j & 1) != 0) {
+%% for (k = 0; k < 16; k++) {
+%% t[k] ^= s[(p + k) & 15];
+%% }
+%% }
+%% next();
+%% }
+%% }
+%% for (k = 0; k < 16; k++) {
+%% s[(p + k) & 15] = t[k];
+%% }
+%% }
+%%
+%% =====================================================================
+
+-opaque exro928_state() :: {list(uint58()), list(uint58())}.
+
+-spec exro928_seed(
+ list(uint58()) | integer() | {integer(), integer(), integer()}) ->
+ exro928_state().
+exro928_seed(L) when is_list(L) ->
+ {seed58_nz(16, L), []};
+exro928_seed(X) when is_integer(X) ->
+ {seed58(16, ?MASK(64, X)), []};
+%%
+%% Seed from traditional integer triple - mix into splitmix
+exro928_seed({A1, A2, A3}) ->
+ {S0, X0} = seed58(?MASK(64, A1)),
+ {S1, X1} = seed58(?MASK(64, A2) bxor X0),
+ {S2, X2} = seed58(?MASK(64, A3) bxor X1),
+ {[S0,S1,S2|seed58(13, X2)], []}.
+
+
+%% Update the state and calculate output word
+-spec exro928ss_next(exro928_state()) -> {uint58(), exro928_state()}.
+exro928ss_next({[S15,S0|Ss], Rs}) ->
+ SR = exro928_next_state(Ss, Rs, S15, S0),
+ %%
+ %% {S, R, T} = {5, 7, 9}
+ %% const uint64_t result_starstar = rotl(s0 * S, R) * T;
+ %%
+ %% The multiply by add shifted trick avoids creating bignums
+ %% which improves performance significantly
+ %%
+ V0 = ?MASK(58, S0 + ?BSL(58, S0, 2)), % V0 = S0 * 5
+ V1 = ?ROTL(58, V0, 7),
+ V = ?MASK(58, V1 + ?BSL(58, V1, 3)), % V = V1 * 9
+ {V, SR};
+exro928ss_next({[S15], Rs}) ->
+ exro928ss_next({[S15|lists:reverse(Rs)], []}).
+
+-spec exro928_next(exro928_state()) -> {{uint58(),uint58()}, exro928_state()}.
+exro928_next({[S15,S0|Ss], Rs}) ->
+ SR = exro928_next_state(Ss, Rs, S15, S0),
+ {{S15,S0}, SR};
+exro928_next({[S15], Rs}) ->
+ exro928_next({[S15|lists:reverse(Rs)], []}).
+
+%% Just update the state
+-spec exro928_next_state(exro928_state()) -> exro928_state().
+exro928_next_state({[S15,S0|Ss], Rs}) ->
+ exro928_next_state(Ss, Rs, S15, S0);
+exro928_next_state({[S15], Rs}) ->
+ [S0|Ss] = lists:reverse(Rs),
+ exro928_next_state(Ss, [], S15, S0).
+
+exro928_next_state(Ss, Rs, S15, S0) ->
+ %% {A, B, C} = {44, 9, 45},
+ %% s15 ^= s0;
+ %% NewS15: s[q] = rotl(s0, A) ^ s15 ^ (s15 << B);
+ %% NewS0: s[p] = rotl(s15, C);
+ %%
+ Q = S15 bxor S0,
+ NewS15 = ?ROTL(58, S0, 44) bxor Q bxor ?BSL(58, Q, 9),
+ NewS0 = ?ROTL(58, Q, 45),
+ {[NewS0|Ss], [NewS15|Rs]}.
+
+
+exro928ss_uniform({Alg, SR}) ->
+ {V, NewSR} = exro928ss_next(SR),
+ {(V bsr (58-53)) * ?TWO_POW_MINUS53, {Alg, NewSR}}.
+
+exro928ss_uniform(Range, {Alg, SR}) ->
+ {V, NewSR} = exro928ss_next(SR),
+ MaxMinusRange = ?BIT(58) - Range,
+ ?uniform_range(Range, Alg, NewSR, V, MaxMinusRange, I).
+
+
+-spec exro928_jump({alg_handler(), exro928_state()}) ->
+ {alg_handler(), exro928_state()}.
+exro928_jump({Alg, SR}) ->
+ {Alg,exro928_jump_2pow512(SR)}.
+
+-spec exro928_jump_2pow512(exro928_state()) -> exro928_state().
+exro928_jump_2pow512(SR) ->
+ polyjump(
+ SR, fun exro928_next_state/1,
+ %% 2^512
+ [16#4085302F77130CA, 16#54E07F7F4524091,
+ 16#5E1D7D6813D2BA0, 16#4687ACEF8644287,
+ 16#4567FD9F0B83FE5, 16#43E6D27EA06C024,
+ 16#641E015AC26D5D2, 16#6CD61377663B92F,
+ 16#70A0657E19F00D4, 16#43C0BDDE15CF3C3,
+ 16#745A3A8A3CEF3CC, 16#58A8CF308C8E0C6,
+ 16#7B7C4CC049C536E, 16#431801F9DB3AF2C,
+ 16#41A1504ACD83F24, 16#6C41DCF2F867D7F]).
+
+-spec exro928_jump_2pow20(exro928_state()) -> exro928_state().
+exro928_jump_2pow20(SR) ->
+ polyjump(
+ SR, fun exro928_next_state/1,
+ %% 2^20
+ [16#5B966A3DAF905E6, 16#601E9589C33DE2F,
+ 16#74A806C17E9E644, 16#59AFEB4F1DF6A43,
+ 16#46D8E75664A4268, 16#42E2C246BDA670C,
+ 16#4531FB8EBC73C70, 16#537F702069EFC52,
+ 16#4B6DC4D3AF15F8A, 16#5A4189F0050263D,
+ 16#46DF2EB1FEAF0FB, 16#77AC696A43CB9AC,
+ 16#4C5878C3549BD5B, 16#7CCF20BCF522920,
+ 16#415FEFCC78FF45E, 16#72CF460728C2FAF]).
+
+%% =====================================================================
%% exrop PRNG: Xoroshiro116+
%%
%% Reference URL: http://xorshift.di.unimi.it/
@@ -899,6 +1121,15 @@ exs1024_jump({L, RL}, AS, JL, J, N, TN) ->
-opaque exrop_state() :: nonempty_improper_list(uint58(), uint58()).
-dialyzer({no_improper_lists, exrop_seed/1}).
+
+exrop_seed(L) when is_list(L) ->
+ [S0,S1] = seed58_nz(2, L),
+ [S0|S1];
+exrop_seed(X) when is_integer(X) ->
+ [S0,S1] = seed58(2, ?MASK(64, X)),
+ [S0|S1];
+%%
+%% Traditional integer triplet seed
exrop_seed({A1, A2, A3}) ->
[_|S1] =
exrop_next_s(
@@ -962,6 +1193,142 @@ exrop_jump([S__0|S__1] = _S, S0, S1, J, Js) ->
end.
%% =====================================================================
+%% Mask and fill state list, ensure not all zeros
+%% =====================================================================
+
+seed58_nz(N, Ss) ->
+ seed_nz(N, Ss, 58, false).
+
+seed64_nz(N, Ss) ->
+ seed_nz(N, Ss, 64, false).
+
+seed_nz(_N, [], _M, false) ->
+ erlang:error(zero_seed);
+seed_nz(0, [_|_], _M, _NZ) ->
+ erlang:error(too_many_seed_integers);
+seed_nz(0, [], _M, _NZ) ->
+ [];
+seed_nz(N, [], M, true) ->
+ [0|seed_nz(N - 1, [], M, true)];
+seed_nz(N, [S|Ss], M, NZ) ->
+ if
+ is_integer(S) ->
+ R = ?MASK(M, S),
+ [R|seed_nz(N - 1, Ss, M, NZ orelse R =/= 0)];
+ true ->
+ erlang:error(non_integer_seed)
+ end.
+
+%% =====================================================================
+%% Splitmix seeders, lowest bits of SplitMix64, zeros skipped
+%% =====================================================================
+
+-spec seed58(non_neg_integer(), uint64()) -> list(uint58()).
+seed58(0, _X) ->
+ [];
+seed58(N, X) ->
+ {Z,NewX} = seed58(X),
+ [Z|seed58(N - 1, NewX)].
+%%
+seed58(X_0) ->
+ {Z0,X} = splitmix64_next(X_0),
+ case ?MASK(58, Z0) of
+ 0 ->
+ seed58(X);
+ Z ->
+ {Z,X}
+ end.
+
+-spec seed64(non_neg_integer(), uint64()) -> list(uint64()).
+seed64(0, _X) ->
+ [];
+seed64(N, X) ->
+ {Z,NewX} = seed64(X),
+ [Z|seed64(N - 1, NewX)].
+%%
+seed64(X_0) ->
+ {Z,X} = ZX = splitmix64_next(X_0),
+ if
+ Z =:= 0 ->
+ seed64(X);
+ true ->
+ ZX
+ end.
+
+%% The SplitMix64 generator:
+%%
+%% uint64_t splitmix64_next() {
+%% uint64_t z = (x += 0x9e3779b97f4a7c15);
+%% z = (z ^ (z >> 30)) * 0xbf58476d1ce4e5b9;
+%% z = (z ^ (z >> 27)) * 0x94d049bb133111eb;
+%% return z ^ (z >> 31);
+%% }
+%%
+splitmix64_next(X_0) ->
+ X = ?MASK(64, X_0 + 16#9e3779b97f4a7c15),
+ Z_0 = ?MASK(64, (X bxor (X bsr 30)) * 16#bf58476d1ce4e5b9),
+ Z_1 = ?MASK(64, (Z_0 bxor (Z_0 bsr 27)) * 16#94d049bb133111eb),
+ {?MASK(64, Z_1 bxor (Z_1 bsr 31)),X}.
+
+%% =====================================================================
+%% Polynomial jump with a jump constant word list,
+%% high bit in each word marking top of word,
+%% SR is a {Forward, Reverse} queue tuple with Forward never empty
+%% =====================================================================
+
+polyjump({Ss, Rs} = SR, NextState, JumpConst) ->
+ %% Create new state accumulator T
+ Ts = lists:duplicate(length(Ss) + length(Rs), 0),
+ polyjump(SR, NextState, JumpConst, Ts).
+%%
+%% Foreach jump word
+polyjump(_SR, _NextState, [], Ts) ->
+ %% Return new calculated state
+ {Ts, []};
+polyjump(SR, NextState, [J|Js], Ts) ->
+ polyjump(SR, NextState, Js, Ts, J).
+%%
+%% Foreach bit in jump word until top bit
+polyjump(SR, NextState, Js, Ts, 1) ->
+ polyjump(SR, NextState, Js, Ts);
+polyjump({Ss, Rs} = SR, NextState, Js, Ts, J) when J =/= 0 ->
+ NewSR = NextState(SR),
+ NewJ = J bsr 1,
+ case ?MASK(1, J) of
+ 0 ->
+ polyjump(NewSR, NextState, Js, Ts, NewJ);
+ 1 ->
+ %% Xor this state onto T
+ polyjump(NewSR, NextState, Js, xorzip_sr(Ts, Ss, Rs), NewJ)
+ end.
+
+xorzip_sr([], [], undefined) ->
+ [];
+xorzip_sr(Ts, [], Rs) ->
+ xorzip_sr(Ts, lists:reverse(Rs), undefined);
+xorzip_sr([T|Ts], [S|Ss], Rs) ->
+ [T bxor S|xorzip_sr(Ts, Ss, Rs)].
+
+%% =====================================================================
+
+format_jumpconst58(String) ->
+ ReOpts = [{newline,any},{capture,all_but_first,binary},global],
+ {match,Matches} = re:run(String, "0x([a-zA-Z0-9]+)", ReOpts),
+ format_jumcons58_matches(lists:reverse(Matches), 0).
+
+format_jumcons58_matches([], J) ->
+ format_jumpconst58_value(J);
+format_jumcons58_matches([[Bin]|Matches], J) ->
+ NewJ = (J bsl 64) bor binary_to_integer(Bin, 16),
+ format_jumcons58_matches(Matches, NewJ).
+
+format_jumpconst58_value(0) ->
+ ok;
+format_jumpconst58_value(J) ->
+ io:format("16#~s,~n", [integer_to_list(?MASK(58, J) bor ?BIT(58), 16)]),
+ format_jumpconst58_value(J bsr 58).
+
+%% =====================================================================
%% Ziggurat cont
%% =====================================================================
-define(NOR_R, 3.6541528853610087963519472518).
diff --git a/lib/stdlib/test/rand_SUITE.erl b/lib/stdlib/test/rand_SUITE.erl
index d753d929f5..4cb1c0b13d 100644
--- a/lib/stdlib/test/rand_SUITE.erl
+++ b/lib/stdlib/test/rand_SUITE.erl
@@ -21,24 +21,7 @@
-compile({nowarn_deprecated_function,[{random,seed,1},
{random,uniform_s,1},
{random,uniform_s,2}]}).
-
--export([all/0, suite/0, groups/0, group/1]).
-
--export([interval_int/1, interval_float/1, seed/1,
- api_eq/1, reference/1,
- basic_stats_uniform_1/1, basic_stats_uniform_2/1,
- basic_stats_standard_normal/1,
- basic_stats_normal/1,
- stats_standard_normal_box_muller/1,
- stats_standard_normal_box_muller_2/1,
- stats_standard_normal/1,
- uniform_real_conv/1,
- plugin/1, measure/1,
- reference_jump_state/1, reference_jump_procdict/1]).
-
--export([test/0, gen/1]).
-
--export([uniform_real_gen/1, uniform_gen/2]).
+-compile([export_all, nowarn_export_all]).
-include_lib("common_test/include/ct.hrl").
@@ -56,7 +39,8 @@ all() ->
{group, distr_stats},
uniform_real_conv,
plugin, measure,
- {group, reference_jump}
+ {group, reference_jump},
+ short_jump
].
groups() ->
@@ -95,7 +79,7 @@ test() ->
end, Tests).
algs() ->
- [exrop, exsp, exs1024s, exs64, exsplus, exs1024].
+ [exrop, exsp, exs1024s, exs64, exsplus, exs1024, exro928ss].
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -125,7 +109,7 @@ seed_1(Alg) ->
S0 = get(rand_seed),
S0 = rand:seed_s(Alg, {0, 0, 0}),
%% Check that process_dict should not be used for seed_s functionality
- _ = rand:seed_s(Alg, {1, 0, 0}),
+ _ = rand:seed_s(Alg, 4711),
S0 = get(rand_seed),
%% Test export
ES0 = rand:export_seed(),
@@ -262,31 +246,43 @@ reference(Config) when is_list(Config) ->
ok.
reference_1(Alg) ->
- Refval = reference_val(Alg),
- Testval = gen(Alg),
- case Refval =:= Testval of
- true -> ok;
- false when Refval =:= not_implemented ->
- exit({not_implemented,Alg});
- false ->
- io:format("Failed: ~p~n",[Alg]),
- io:format("Length ~p ~p~n",[length(Refval), length(Testval)]),
- io:format("Head ~p ~p~n",[hd(Refval), hd(Testval)]),
- exit(wrong_value)
+ Refval = reference_val(Alg),
+ if
+ Refval =:= not_implemented -> Refval;
+ true ->
+ case gen(Alg) of
+ Refval ->
+ io:format("Ok: ~p~n",[Alg]),
+ ok;
+ Testval ->
+ io:format("Failed: ~p~n",[Alg]),
+ io:format("Length ~p ~p~n",[length(Refval), length(Testval)]),
+ io:format("Head ~p ~p~n",[hd(Refval), hd(Testval)]),
+ show_wrong(Refval, Testval),
+ exit(wrong_value)
+ end
end.
+show_wrong([], []) ->
+ ok;
+show_wrong([H|T1], [H|T2]) ->
+ show_wrong(T1, T2);
+show_wrong([H1|_], [H2|_]) ->
+ io:format("Wrong ~p ~p~n",[H1,H2]).
+
+
gen(Algo) ->
State =
- case Algo of
- exs64 -> %% Printed with orig 'C' code and this seed
- rand:seed_s({exs64, 12345678});
- _ when Algo =:= exsplus; Algo =:= exsp; Algo =:= exrop ->
+ if
+ Algo =:= exs64 -> %% Printed with orig 'C' code and this seed
+ rand:seed_s(exs64, [12345678]);
+ Algo =:= exsplus; Algo =:= exsp; Algo =:= exrop ->
%% Printed with orig 'C' code and this seed
- rand:seed_s({Algo, [12345678|12345678]});
- _ when Algo =:= exs1024; Algo =:= exs1024s ->
+ rand:seed_s(Algo, [12345678,12345678]);
+ Algo =:= exs1024; Algo =:= exs1024s; Algo =:= exro928ss ->
%% Printed with orig 'C' code and this seed
- rand:seed_s({Algo, {lists:duplicate(16, 12345678), []}});
- _ ->
+ rand:seed_s(Algo, lists:duplicate(16, 12345678));
+ true ->
rand:seed(Algo, {100, 200, 300})
end,
Max = range(State),
@@ -852,7 +848,8 @@ do_measure(_Config) ->
Algs =
algs() ++
try crypto:strong_rand_bytes(1) of
- <<_>> -> [crypto64, crypto_cache, crypto]
+ <<_>> ->
+ [crypto64, crypto_cache, crypto_aes, crypto]
catch
error:low_entropy -> [];
error:undef -> []
@@ -1101,6 +1098,10 @@ measure_1(RangeFun, Fun, Alg, TMark) ->
{rand, crypto:rand_seed_alg(crypto_cache)};
crypto ->
{rand, crypto:rand_seed_s()};
+ crypto_aes ->
+ {rand,
+ crypto:rand_seed_alg(
+ crypto_aes, crypto:strong_rand_bytes(256))};
random ->
{random, random:seed(os:timestamp()), get(random_seed)};
_ ->
@@ -1116,7 +1117,7 @@ measure_1(RangeFun, Fun, Alg, TMark) ->
_ -> (Time * 100 + 50) div TMark
end,
io:format(
- "~.12w: ~p ns ~p% [16#~.16b]~n",
+ "~.20w: ~p ns ~p% [16#~.16b]~n",
[Alg, (Time * 1000 + 500) div ?LOOP_MEASURE,
Percent, Range]),
Parent ! {self(), Time},
@@ -1141,104 +1142,156 @@ reference_jump_state(Config) when is_list(Config) ->
ok.
reference_jump_1(Alg) ->
- Refval = reference_jump_val(Alg),
- Testval = gen_jump_1(Alg),
- case Refval =:= Testval of
- true -> ok;
- false ->
- io:format("Failed: ~p~n",[Alg]),
- io:format("Length ~p ~p~n",[length(Refval), length(Testval)]),
- io:format("Head ~p ~p~n",[hd(Refval), hd(Testval)]),
- io:format("Vals ~p ~p~n",[Refval, Testval]),
- exit(wrong_value)
+ Refval = reference_jump_val(Alg),
+ if
+ Refval =:= not_implemented -> Refval;
+ true ->
+ case gen_jump_1(Alg) of
+ Refval -> ok;
+ Testval ->
+ io:format(
+ "Failed: ~p~n",[Alg]),
+ io:format(
+ "Length ~p ~p~n",
+ [length(Refval), length(Testval)]),
+ io:format(
+ "Head ~p ~p~n",[hd(Refval), hd(Testval)]),
+ io:format(
+ "Vals ~p ~p~n",[Refval, Testval]),
+ exit(wrong_value)
+ end
end.
gen_jump_1(Algo) ->
- State =
- case Algo of
- exs64 -> %% Test exception of not_implemented notice
- try rand:jump(rand:seed_s(exs64))
- catch
- error:not_implemented -> not_implemented
- end;
- _ when Algo =:= exsplus; Algo =:= exsp; Algo =:= exrop ->
- %% Printed with orig 'C' code and this seed
- rand:seed_s({Algo, [12345678|12345678]});
- _ when Algo =:= exs1024; Algo =:= exs1024s ->
- %% Printed with orig 'C' code and this seed
- rand:seed_s({Algo, {lists:duplicate(16, 12345678), []}});
- _ -> % unimplemented
- not_implemented
- end,
- case State of
- not_implemented -> [not_implemented];
- _ ->
- Max = range(State),
- gen_jump_1(?LOOP_JUMP, State, Max, [])
+ case Algo of
+ exs64 -> %% Test exception of not_implemented notice
+ try rand:jump(rand:seed_s(exs64))
+ catch
+ error:not_implemented -> [error_not_implemented]
+ end;
+ _ when Algo =:= exsplus; Algo =:= exsp; Algo =:= exrop ->
+ %% Printed with orig 'C' code and this seed
+ gen_jump_2(
+ rand:seed_s(Algo, [12345678,12345678]));
+ _ when Algo =:= exs1024; Algo =:= exs1024s; Algo =:= exro928ss ->
+ %% Printed with orig 'C' code and this seed
+ gen_jump_2(
+ rand:seed_s(Algo, lists:duplicate(16, 12345678)))
end.
-gen_jump_1(N, State0, Max, Acc) when N > 0 ->
+gen_jump_2(State) ->
+ Max = range(State),
+ gen_jump_3(?LOOP_JUMP, State, Max, []).
+
+gen_jump_3(N, State0, Max, Acc) when N > 0 ->
{_, State1} = rand:uniform_s(Max, State0),
{Random, State2} = rand:uniform_s(Max, rand:jump(State1)),
case N rem (?LOOP_JUMP div 100) of
- 0 -> gen_jump_1(N-1, State2, Max, [Random|Acc]);
- _ -> gen_jump_1(N-1, State2, Max, Acc)
+ 0 -> gen_jump_3(N-1, State2, Max, [Random|Acc]);
+ _ -> gen_jump_3(N-1, State2, Max, Acc)
end;
-gen_jump_1(_, _, _, Acc) -> lists:reverse(Acc).
+gen_jump_3(_, _, _, Acc) -> lists:reverse(Acc).
%% Check if each algorithm generates the proper jump sequence
%% with the internal state in the process dictionary.
reference_jump_procdict(Config) when is_list(Config) ->
- [reference_jump_0(Alg) || Alg <- algs()],
+ [reference_jump_p1(Alg) || Alg <- algs()],
ok.
-reference_jump_0(Alg) ->
+reference_jump_p1(Alg) ->
Refval = reference_jump_val(Alg),
- Testval = gen_jump_0(Alg),
- case Refval =:= Testval of
- true -> ok;
- false ->
- io:format("Failed: ~p~n",[Alg]),
- io:format("Length ~p ~p~n",[length(Refval), length(Testval)]),
- io:format("Head ~p ~p~n",[hd(Refval), hd(Testval)]),
- exit(wrong_value)
+ if
+ Refval =:= not_implemented -> Refval;
+ true ->
+ case gen_jump_p1(Alg) of
+ Refval -> ok;
+ Testval ->
+ io:format("Failed: ~p~n",[Alg]),
+ io:format("Length ~p ~p~n",[length(Refval), length(Testval)]),
+ io:format("Head ~p ~p~n",[hd(Refval), hd(Testval)]),
+ exit(wrong_value)
+ end
end.
-gen_jump_0(Algo) ->
- Seed = case Algo of
- exs64 -> %% Test exception of not_implemented notice
- try
- _ = rand:seed(exs64),
- rand:jump()
- catch
- error:not_implemented -> not_implemented
- end;
- _ when Algo =:= exsplus; Algo =:= exsp; Algo =:= exrop ->
- %% Printed with orig 'C' code and this seed
- rand:seed({Algo, [12345678|12345678]});
- _ when Algo =:= exs1024; Algo =:= exs1024s ->
- %% Printed with orig 'C' code and this seed
- rand:seed({Algo, {lists:duplicate(16, 12345678), []}});
- _ -> % unimplemented
- not_implemented
- end,
- case Seed of
- not_implemented -> [not_implemented];
- _ ->
- Max = range(Seed),
- gen_jump_0(?LOOP_JUMP, Max, [])
+gen_jump_p1(Algo) ->
+ case Algo of
+ exs64 -> %% Test exception of not_implemented notice
+ try
+ _ = rand:seed(exs64),
+ rand:jump()
+ catch
+ error:not_implemented -> [error_not_implemented]
+ end;
+ _ when Algo =:= exsplus; Algo =:= exsp; Algo =:= exrop ->
+ %% Printed with orig 'C' code and this seed
+ gen_jump_p2(
+ rand:seed(Algo, [12345678,12345678]));
+ _ when Algo =:= exs1024; Algo =:= exs1024s; Algo =:= exro928ss ->
+ %% Printed with orig 'C' code and this seed
+ gen_jump_p2(
+ rand:seed(Algo, lists:duplicate(16, 12345678)))
end.
-gen_jump_0(N, Max, Acc) when N > 0 ->
+gen_jump_p2(Seed) ->
+ Max = range(Seed),
+ gen_jump_p3(?LOOP_JUMP, Max, []).
+
+gen_jump_p3(N, Max, Acc) when N > 0 ->
_ = rand:uniform(Max),
_ = rand:jump(),
Random = rand:uniform(Max),
case N rem (?LOOP_JUMP div 100) of
- 0 -> gen_jump_0(N-1, Max, [Random|Acc]);
- _ -> gen_jump_0(N-1, Max, Acc)
+ 0 -> gen_jump_p3(N-1, Max, [Random|Acc]);
+ _ -> gen_jump_p3(N-1, Max, Acc)
end;
-gen_jump_0(_, _, Acc) -> lists:reverse(Acc).
+gen_jump_p3(_, _, Acc) -> lists:reverse(Acc).
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+short_jump(Config) when is_list(Config) ->
+ Seed = erlang:system_time(),
+ short_jump(
+ rand:seed_s(exro928ss, Seed),
+ fun ({Alg,AlgState}) ->
+ {Alg,rand:exro928_jump_2pow20(AlgState)}
+ end),
+ short_jump(
+ crypto:rand_seed_alg_s(crypto_aes, integer_to_list(Seed)),
+ fun ({Alg,AlgState}) ->
+ {Alg,crypto:rand_plugin_aes_jump_2pow20(AlgState)}
+ end),
+ ok.
+
+short_jump({#{bits := Bits},_} = State_0, Jump2Pow20) ->
+ Range = 1 bsl Bits,
+ State_1 = repeat(7, Range, State_0),
+ %%
+ State_2a = repeat(1 bsl 20, Range, State_1),
+ State_2b = Jump2Pow20(State_1),
+ check(17, Range, State_2a, State_2b),
+ %%
+ {_,State_3a} = rand:uniform_s(Range, State_2a),
+ State_4a = Jump2Pow20(State_3a),
+ State_4b = repeat((1 bsl 20) + 1, Range, State_2b),
+ check(17, Range, State_4a, State_4b).
+
+repeat(0, _Range, State) ->
+ State;
+repeat(N, Range, State) ->
+ {_, NewState} = rand:uniform_s(Range, State),
+ repeat(N - 1, Range, NewState).
+
+check(0, _Range, _StateA, _StateB) ->
+ ok;
+check(N, Range, StateA, StateB) ->
+ {V,NewStateA} = rand:uniform_s(Range, StateA),
+ case rand:uniform_s(Range, StateB) of
+ {V,NewStateB} ->
+ check(N - 1, Range, NewStateA, NewStateB);
+ {Wrong,_} ->
+ ct:fail({Wrong,neq,V,for,N})
+ end.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Data
@@ -1389,7 +1442,50 @@ reference_val(exrop) ->
250789092615679985,78848633178610658,72059442721196128,
98223942961505519,191144652663779840,
102425686803727694,89058927716079076,80721467542933080,
- 8462479817391645,2774921106204163].
+ 8462479817391645,2774921106204163];
+
+reference_val(exro928ss) ->
+%% Same as for exrop, but this state init:
+%% for (n = 0; n < 16; n++) {
+%% s[n] = 12345678;
+ [16#000000108e8d5b01,16#03604028f2769dff,16#007f92f60bc7170c,
+ 16#035ea81a9898a5e2,16#0104c90c5a0c8178,16#0313514025cca717,
+ 16#03c5506b2a2e98cf,16#0098a5405961552e,16#004ad29eabb785a0,
+ 16#033ea8ec4efb8058,16#00b21545e62bef1c,16#0333fc5320703482,
+ 16#02c3c650e51a8d47,16#03a3b7fc848c9cda,16#03775adea6cddff5,
+ 16#01ae5499c9049973,16#03d3c90e5504e16b,16#0383cd6b6cb852e6,
+ 16#009c8d0996ef543a,16#0059cf671371af60,16#03dfd68ed980b719,
+ 16#0290f2a0acf2c5b0,16#029061df18d63b55,16#02e702ea4b45137b,
+ 16#029a0ccca604d848,16#01664c7cd31f0fa6,16#00dced83e60ccddc,
+ 16#008764d2c9a05f3e,16#02b9ca5f6a80c4ba,16#02daf93d2c566750,
+ 16#0147d326ead18ace,16#014b452efc19297f,16#0242d3f7a7237eca,
+ 16#0141bb68c2abce39,16#02d798e1230baf45,16#0216bf8f25c1ec2d,
+ 16#003a43ea733f1e1f,16#036c75390db736f3,16#028cca5f5f48c6f9,
+ 16#0186e4a17174d6cf,16#02152679dfa4c25c,16#01429b9f15e3b9d6,
+ 16#0134a61411d22bb0,16#01593f7d970d1c94,16#0205a7d8a305490f,
+ 16#01dd092272595a9c,16#0028c95208aad2d4,16#016347c25cc24162,
+ 16#025306acfb891309,16#0207a07e2bebef2f,16#024ee78d86ff5288,
+ 16#030b53192db97613,16#03f765cb9e98e611,16#025ec35a1e237377,
+ 16#03d81fd73102ef6f,16#0242dc8fea9a68b2,16#00abb876c1d4ea1b,
+ 16#00871ffd2b7e45fb,16#03593ff73c9be08d,16#00b96b2b8aca3688,
+ 16#0174aba957b7cf7b,16#012b7a5d4cf4a5b7,16#032a5260f2123db8,
+ 16#00f9374d88ee0080,16#030df39bec2ad657,16#00dce0cb81d006c4,
+ 16#038213b806303c76,16#03940aafdbfabf84,16#0398dbb26aeba037,
+ 16#01eb28d61951587f,16#00fed3d2aacfeef4,16#03499587547d6e40,
+ 16#01b192fe6e979e3c,16#00e974bf5f0a26d0,16#012ed94f76459c83,
+ 16#02d76859e7a82587,16#00d1d2c7b791f51b,16#03988058017a031b,
+ 16#00bbcf4b59d8e86d,16#015ed8b73a1b767c,16#0277283ea6a5ee74,
+ 16#002211460dd6d422,16#001ad62761ee9fbd,16#037311b44518b067,
+ 16#02b5ed61bf70904e,16#011862a05c1929fa,16#014be68683c3bab4,
+ 16#025c29aa5c508b07,16#00895c6106f97378,16#026ce91a3d671c7f,
+ 16#02591f4c74784293,16#02f0ed2a70bc1853,16#00a2762ff614bfbc,
+ 16#008f4e354f0c20d4,16#038b66fb587ed430,16#00636296e188de89,
+ 16#0278fadd143e74f5,16#029697ccf1b3a4c2,16#011eccb273404458,
+ 16#03f204064a9fe0c0];
+
+reference_val(_) ->
+ not_implemented.
+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -1451,7 +1547,7 @@ reference_jump_val(exsp) ->
reference_jump_val(exsplus);
reference_jump_val(exs1024s) ->
reference_jump_val(exs1024);
-reference_jump_val(exs64) -> [not_implemented];
+reference_jump_val(exs64) -> [error_not_implemented];
reference_jump_val(exrop) ->
%% #include <stdint.h>
%% #include <stdio.h>
@@ -1516,7 +1612,50 @@ reference_jump_val(exrop) ->
250227633882474729,171181147785250210,55437891969696407,
241227318715885854,77323084015890802,
1663590009695191,234064400749487599,222983191707424780,
- 254956809144783896,203898972156838252].
+ 254956809144783896,203898972156838252];
+
+reference_jump_val(exro928ss) ->
+%% Same as for exrop, but this state init:
+%% for (n = 0; n < 16; n++) {
+%% s[n] = 12345678;
+ [16#031ee449e53b6689,16#001afeee12813137,16#005e2172711df36b,
+ 16#02850aea3a595d36,16#0029705187e891c7,16#001794badd489667,
+ 16#00ab621be15be56c,16#024b663a6924786b,16#03cab70b8ab854bf,
+ 16#01daa37601285320,16#02db955a53c40e89,16#01fbef51d5c65891,
+ 16#02fecf4116ed5f77,16#0349c2057246ac5d,16#01217f257c4fa148,
+ 16#0367ee84d020697d,16#01d5cf647fe23335,16#020941838adfb750,
+ 16#02c2da26b1d7b3e5,16#00d1583d34cea6c0,16#038be9cb5b527f50,
+ 16#00bfa93c1d7f4864,16#03778912a4f56b14,16#037fcabc483fa5c5,
+ 16#00a3c9de6aaf5fc7,16#03600b883b2f2b42,16#03797a99ffddfdfb,
+ 16#0189fead429945b7,16#0103ac90cd912508,16#03e3d872fd950d64,
+ 16#0214fc3e77dc2f02,16#02a084f4f0e580ca,16#035d2fe72266a7f3,
+ 16#02887c49ae7e41a4,16#0011dc026af83c51,16#02d28bfd32c2c517,
+ 16#022e4165c33ad4f3,16#01f053cf0687b052,16#035315e6e53c8918,
+ 16#01255312da07b572,16#0237f1da11ec9221,16#02faf2e282fb1fb1,
+ 16#0227423ec1787ebc,16#011fa5eb1505571c,16#0275ff9eaaa1abdd,
+ 16#03e2d032c3981cb4,16#0181bb32d51d3072,16#01b1d3939b9f16ec,
+ 16#0259f09f55d1112f,16#0396464a2767e428,16#039777c0368bdb9e,
+ 16#0320925f35f36c5f,16#02a35289e0af1248,16#02e80bd4bc72254b,
+ 16#00a8b11af1674d68,16#027735036100a69e,16#03c8c268ded7f254,
+ 16#03de80aa57c65217,16#00f2247754d24000,16#005582a42b467f89,
+ 16#0031906569729477,16#00fd523f2ca4fefe,16#00ad223113d1e336,
+ 16#0238ddf026cbfca9,16#028b98211cfed876,16#0354353ebcc0de9a,
+ 16#009ee370c1e154f4,16#033131af3b8a7f88,16#032291baa45801e3,
+ 16#00941fc2b45eb217,16#035d6a61fa101647,16#03fdb51f736f1bbc,
+ 16#0232f7b98539faa0,16#0311b35319e3a61e,16#0048356b17860eb5,
+ 16#01a205b2554ce71e,16#03f873ea136e29d6,16#003c67d5c3df5ffd,
+ 16#00cd19e7a8641648,16#0149a8c54e4ba45e,16#0329498d134d2f6a,
+ 16#03b69421ae65ee2b,16#01a8d20b59447429,16#006b2292571032a2,
+ 16#00c193b17da22ba5,16#01faa7ab62181249,16#00acd401cd596a00,
+ 16#005b5086c3531402,16#0259113d5d3d058d,16#00bef3f3ce4a43b2,
+ 16#014837a4070b893c,16#00460a26ac2eeec1,16#026219a8b8c63d7e,
+ 16#03c7b8ed032cf5a6,16#004da912a1fff131,16#0297de3716215741,
+ 16#0079fb9b4c715466,16#00a73bad4ae5a356,16#0072e606c0d4ab86,
+ 16#02374382d5f9bd2e];
+
+reference_jump_val(_) ->
+ not_implemented.
+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
diff --git a/lib/stdlib/test/rand_Xoroshiro928ss_dev.txt b/lib/stdlib/test/rand_Xoroshiro928ss_dev.txt
new file mode 100644
index 0000000000..150f37fcfa
--- /dev/null
+++ b/lib/stdlib/test/rand_Xoroshiro928ss_dev.txt
@@ -0,0 +1,343 @@
+%CopyrightBegin%
+
+Copyright Ericsson AB 2015-2017. All Rights Reserved.
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+ http://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+
+%CopyrightEnd%
+
+
+Memorable facts from designing the Xoroshiro928** generator
+===========================================================
+AKA: exro928ss in the rand module
+
+Author: Raimo Niskanen, for the Erlang/OTP team @ Ericsson.
+
+Reference URL: http://vigna.di.unimi.it/ftp/papers/ScrambledLinear.pdf
+i.e the Xoroshiro1024 generator with ** scrambler:
+
+ int p;
+ uint64_t s[16];
+
+ const int q = p;
+ const uint64_t s0 = s[p = (p + 1) & 15];
+
+ uint64_t s15 = s[q];
+
+ const uint64_t result_starstar = rotl(s0 * S, R) * T;
+
+ s15 ^= s0;
+ s[q] = rotl(s0, A) ^ s15 ^ (s15 << B);
+ s[p] = rotl(s15, C);
+
+Where {S, R, T} = {5, 7, 9} as recommended in the paper.
+
+We want to scale down to 58 bit words (16 of them)
+so we get a generator with period 2^928 - 1.
+
+{A, B, C} were deduced as follows
+---------------------------------
+
+First, to find which triplets that give a full period generator
+one have to factor 2^928 - 1.
+
+https://www.alpertron.com.ar/ECM.HTM actually could do that
+and gave the result:
+
+ Value
+ 2^928 - 1
+
+ 2269 007733 883335 972287 082669 296112 915239 349672 942191 252221 331572
+ 442536 403137 824056 312817 862695 551072 066953 619064 625508 194663
+ 368599 769448 406663 254670 871573 830845 597595 897613 333042 429214
+ 224697 474472 410882 236254 024057 110212 260250 671521 235807 709272
+ 244389 361641 091086 035023 229622 419455 (280 digits) = 3 × 5 × 17 × 59 ×
+ 233 × 257 × 929 × 1103 × 2089 × 5569 × 8353 × 59393 × 65537 × 3 033169 ×
+ 39 594977 × 107 367629 × 536 903681 × 748 264961 × 2245 984577 × 239
+ 686663 718401 × 15 929619 591127 520827 829953 × 82280 195167 144119
+ 832390 568177 × 6033 312171 721035 031651 315652 130497 (34 digits) × 18
+ 774318 450142 955120 650303 957350 521748 903233 (44 digits) × 15 694604
+ 006012 505869 851221 169365 594050 637743 819041 (50 digits)
+
+Sebastiano Vigna from that calculated all full period triplets, at the
+end of this document, and the ones with the highest degree were:
+
+ 23-25-12 411
+ 36-2-35 411
+ 55-5-54 411
+ 14-19-11 415
+ 34-37-5 415
+ 37-3-56 415
+ 55-11-54 417
+ 30-3-41 419
+ 11-45-50 423
+ 50-19-47 423
+ 52-27-13 427
+ 54-9-25 433
+ 56-43-35 433
+ 44-9-45 441
+
+All these candidates were tested with TestU01-1.2.3:
+
+ http://simul.iro.umontreal.ca/testu01/tu01.html
+
+A plugin was created with parameters for {A, B, C} and, since
+TestU01 is a 32-bit test tool, with parameters to reverse
+the generated bits or not, and to take the 32 highest or lowest
+bits from the reversed or non-reversed 58 bit output.
+
+The generators were seeded with a SplitMix64 generator like the
+one used for seeding this generator in the rand module,
+taking the 58 lowest bits and wasting all zero values.
+
+3 runs were made with all candidates and all four bit selection variants.
+For these runs the seeder was initialized with 12345678, 876543212345678
+and 1234567890.
+
+After all these runs the candidate with the highest degree: 44-9-45
+had not gotten any suspicious p-value at all. All the other
+got p-values around 5.0e-4 worst 9.8e-6 suggesting only random
+failures, so they would probably have worked about as well.
+
+Finally 44-9-45 was run through PractRand-0.93:
+
+ http://pracrand.sourceforge.net/
+
+Again, all 4 bit selection variants of 32 bits were run.
+Random failures with p-values around e-3..e-4 for the "smaller"
+tests, but for 8, 16 and 32 TB tests no anomalies were found
+(with the internal seeder masked to 58 bits):
+
+ Xoroshiro928High seed: 0x3178ec5d
+ Xoroshiro928Low seed: 0xa9a04fb9
+ Xoroshiro928ReverseHigh seed: 0xfa0bdbab
+ Xoroshiro928ReverseLow seed: 0xada51705
+
+
+Then, S. Vigna calculated the 2^512 jump coefficient as well
+as a 2^20 jump coefficient (for testing purposes) for 44-9-45.
+
+2^512:
+ { 0x44085302f77130ca, 0xba05381fdfd14902, 0x10a1de1d7d6813d2,
+ 0xb83fe51a1eb3be19, 0xa81b0090567fd9f0, 0x5ac26d5d20f9b49f,
+ 0x4ddd98ee4be41e01, 0x0657e19f00d4b358, 0xf02f778573cf0f0a,
+ 0xb45a3a8a3cef3cc0, 0x6e62a33cc2323831, 0xbcb3b7c4cc049c53,
+ 0x83f240c6007e76ce, 0xe19f5fc1a1504acd, 0x00000000b10773cb }
+
+2^20:
+ { 0xbdb966a3daf905e6, 0x644807a56270cf78, 0xda90f4a806c17e9e,
+ 0x4a426866bfad3c77, 0xaf699c306d8e7566, 0x8ebc73c700b8b091,
+ 0xc081a7bf148531fb, 0xdc4d3af15f8a4dfd, 0x90627c014098f4b6,
+ 0x06df2eb1feaf0fb6, 0x5bdeb1a5a90f2e6b, 0xa480c5878c3549bd,
+ 0xff45ef33c82f3d48, 0xa30bebc15fefcc78, 0x00000000cb3d181c }
+
+Standard jump function pseudocode:
+
+ Jump constant j = 0xb10773cb...44085302f77130ca
+ Generator state: s
+ New generator state: t = 0
+ foreach bit in j, low to high:
+ if the bit is one:
+ t ^= s
+ next s
+ s = t
+
+
+The complete list of full period constants
+------------------------------------------
+
+29-48-54 27 x^928 + x^874 + x^870 + x^840 + x^814 + x^784 + x^759 + x^750 + x^724 + x^720 + x^634 + x^630 + x^600 + x^574 + x^544 + x^519 + x^510 + x^484 + x^480 + x^390 + x^360 + x^270 + x^240 + x^150 + x^120 + x^30 + 1
+
+29-18-56 65 x^928 + x^870 + x^850 + x^840 + x^832 + x^821 + x^802 + x^791 + x^761 + x^750 + x^734 + x^731 + x^720 + x^712 + x^701 + x^686 + x^674 + x^671 + x^663 + x^641 + x^630 + x^611 + x^610 + x^603 + x^600 + x^596 + x^577 + x^566 + x^547 + x^524 + x^517 + x^510 + x^487 + x^480 + x^476 + x^464 + x^457 + x^446 + x^427 + x^423 + x^397 + x^390 + x^367 + x^363 + x^360 + x^356 + x^352 + x^341 + x^326 + x^322 + x^311 + x^281 + x^270 + x^251 + x^240 + x^236 + x^232 + x^221 + x^191 + x^161 + x^150 + x^131 + x^120 + x^30 + 1
+
+29-32-36 81 x^928 + x^874 + x^870 + x^840 + x^820 + x^819 + x^794 + x^790 + x^770 + x^765 + x^764 + x^760 + x^754 + x^750 + x^740 + x^739 + x^735 + x^734 + x^730 + x^709 + x^704 + x^674 + x^670 + x^665 + x^660 + x^650 + x^649 + x^635 + x^634 + x^619 + x^614 + x^605 + x^595 + x^585 + x^584 + x^579 + x^564 + x^555 + x^545 + x^540 + x^524 + x^515 + x^514 + x^495 + x^490 + x^485 + x^460 + x^455 + x^435 + x^425 + x^400 + x^395 + x^375 + x^370 + x^365 + x^340 + x^315 + x^310 + x^305 + x^300 + x^290 + x^285 + x^275 + x^260 + x^250 + x^245 + x^240 + x^215 + x^190 + x^180 + x^170 + x^150 + x^130 + x^120 + x^115 + x^105 + x^100 + x^75 + x^40 + x^30 + 1
+
+39-10-20 105 x^928 + x^898 + x^870 + x^841 + x^840 + x^808 + x^783 + x^782 + x^781 + x^780 + x^778 + x^754 + x^752 + x^724 + x^721 + x^720 + x^696 + x^692 + x^688 + x^667 + x^666 + x^661 + x^660 + x^658 + x^638 + x^636 + x^632 + x^609 + x^607 + x^606 + x^605 + x^604 + x^601 + x^600 + x^580 + x^578 + x^576 + x^572 + x^568 + x^549 + x^548 + x^541 + x^540 + x^538 + x^516 + x^512 + x^489 + x^485 + x^484 + x^481 + x^480 + x^456 + x^452 + x^448 + x^429 + x^428 + x^421 + x^420 + x^418 + x^400 + x^398 + x^396 + x^392 + x^368 + x^365 + x^364 + x^336 + x^328 + x^319 + x^318 + x^315 + x^314 + x^298 + x^286 + x^278 + x^276 + x^274 + x^255 + x^254 + x^249 + x^245 + x^228 + x^226 + x^220 + x^208 + x^199 + x^198 + x^189 + x^188 + x^178 + x^168 + x^166 + x^160 + x^129 + x^128 + x^108 + x^100 + x^88 + x^79 + x^78 + x^69 + x^68 + x^58 + x^40 + 1
+
+4-4-9 149 x^928 + x^898 + x^886 + x^870 + x^856 + x^826 + x^823 + x^808 + x^796 + x^793 + x^784 + x^778 + x^772 + x^766 + x^760 + x^754 + x^751 + x^742 + x^736 + x^734 + x^724 + x^713 + x^709 + x^706 + x^703 + x^694 + x^691 + x^688 + x^676 + x^674 + x^673 + x^664 + x^662 + x^658 + x^653 + x^641 + x^640 + x^634 + x^632 + x^631 + x^620 + x^614 + x^611 + x^608 + x^604 + x^602 + x^599 + x^590 + x^583 + x^581 + x^578 + x^574 + x^572 + x^571 + x^569 + x^568 + x^561 + x^554 + x^553 + x^551 + x^544 + x^542 + x^540 + x^538 + x^532 + x^531 + x^518 + x^514 + x^512 + x^502 + x^498 + x^489 + x^488 + x^484 + x^482 + x^480 + x^479 + x^468 + x^463 + x^456 + x^454 + x^452 + x^449 + x^448 + x^438 + x^433 + x^429 + x^424 + x^422 + x^420 + x^418 + x^408 + x^406 + x^401 + x^396 + x^394 + x^392 + x^380 + x^378 + x^376 + x^371 + x^367 + x^364 + x^362 + x^360 + x^350 + x^348 + x^346 + x^341 + x^334 + x^332 + x^328 + x^318 + x^311 + x^307 + x^304 + x^302 + x^300 + x^298 + x^288 + x^274 + x^272 + x^254 + x^249 + x^242 + x^240 + x^233 + x^212 + x^208 + x^189 + x^180 + x^178 + x^166 + x^152 + x^143 + x^136 + x^127 + x^122 + x^120 + x^113 + x^106 + x^88 + x^83 + x^81 + x^67 + x^58 + x^51 + x^14 + 1
+
+44-44-45 157 x^928 + x^898 + x^870 + x^850 + x^830 + x^820 + x^808 + x^800 + x^790 + x^782 + x^778 + x^777 + x^760 + x^757 + x^752 + x^747 + x^742 + x^727 + x^702 + x^694 + x^689 + x^688 + x^679 + x^674 + x^659 + x^658 + x^657 + x^644 + x^642 + x^637 + x^634 + x^627 + x^619 + x^612 + x^607 + x^606 + x^601 + x^596 + x^594 + x^591 + x^590 + x^584 + x^576 + x^568 + x^566 + x^561 + x^560 + x^556 + x^554 + x^546 + x^541 + x^539 + x^538 + x^518 + x^516 + x^511 + x^509 + x^503 + x^502 + x^499 + x^496 + x^488 + x^476 + x^471 + x^468 + x^466 + x^464 + x^448 + x^446 + x^441 + x^436 + x^421 + x^420 + x^415 + x^411 + x^408 + x^404 + x^400 + x^388 + x^383 + x^381 + x^379 + x^376 + x^363 + x^358 + x^356 + x^354 + x^346 + x^344 + x^340 + x^333 + x^330 + x^328 + x^325 + x^316 + x^312 + x^307 + x^305 + x^302 + x^284 + x^282 + x^277 + x^270 + x^268 + x^267 + x^256 + x^252 + x^249 + x^245 + x^244 + x^239 + x^238 + x^236 + x^231 + x^228 + x^226 + x^224 + x^222 + x^215 + x^208 + x^207 + x^205 + x^203 + x^201 + x^199 + x^196 + x^194 + x^192 + x^189 + x^185 + x^184 + x^182 + x^180 + x^175 + x^171 + x^168 + x^166 + x^162 + x^161 + x^155 + x^152 + x^150 + x^141 + x^134 + x^132 + x^131 + x^127 + x^122 + x^111 + x^104 + x^92 + x^90 + x^83 + x^81 + x^58 + x^37 + 1
+
+40-40-7 167 x^928 + x^898 + x^870 + x^866 + x^838 + x^836 + x^807 + x^806 + x^804 + x^778 + x^777 + x^774 + x^745 + x^744 + x^742 + x^718 + x^714 + x^713 + x^712 + x^687 + x^658 + x^657 + x^656 + x^655 + x^654 + x^626 + x^620 + x^619 + x^598 + x^596 + x^592 + x^591 + x^589 + x^566 + x^564 + x^563 + x^562 + x^560 + x^558 + x^538 + x^534 + x^530 + x^529 + x^504 + x^503 + x^502 + x^500 + x^499 + x^498 + x^496 + x^478 + x^474 + x^470 + x^469 + x^443 + x^442 + x^441 + x^439 + x^438 + x^436 + x^434 + x^432 + x^418 + x^416 + x^414 + x^410 + x^409 + x^407 + x^406 + x^405 + x^402 + x^386 + x^383 + x^380 + x^374 + x^373 + x^372 + x^370 + x^356 + x^350 + x^349 + x^348 + x^347 + x^346 + x^345 + x^340 + x^328 + x^326 + x^324 + x^323 + x^321 + x^316 + x^314 + x^308 + x^298 + x^296 + x^294 + x^289 + x^285 + x^283 + x^282 + x^281 + x^280 + x^278 + x^263 + x^262 + x^259 + x^257 + x^255 + x^251 + x^249 + x^248 + x^246 + x^233 + x^232 + x^229 + x^220 + x^218 + x^217 + x^216 + x^208 + x^202 + x^201 + x^199 + x^195 + x^190 + x^188 + x^187 + x^186 + x^178 + x^172 + x^169 + x^167 + x^162 + x^161 + x^159 + x^158 + x^157 + x^146 + x^142 + x^139 + x^135 + x^125 + x^123 + x^116 + x^112 + x^107 + x^105 + x^94 + x^93 + x^88 + x^86 + x^84 + x^82 + x^75 + x^73 + x^69 + x^64 + x^58 + x^56 + x^54 + x^52 + x^43 + x^41 + x^39 + x^32 + 1
+
+47-47-18 217 x^928 + x^898 + x^871 + x^870 + x^843 + x^842 + x^841 + x^813 + x^812 + x^811 + x^808 + x^787 + x^786 + x^785 + x^783 + x^781 + x^778 + x^759 + x^757 + x^754 + x^753 + x^751 + x^729 + x^727 + x^726 + x^723 + x^721 + x^701 + x^700 + x^699 + x^697 + x^693 + x^691 + x^688 + x^673 + x^672 + x^670 + x^667 + x^663 + x^661 + x^658 + x^647 + x^644 + x^641 + x^637 + x^633 + x^631 + x^618 + x^617 + x^614 + x^613 + x^612 + x^611 + x^610 + x^607 + x^603 + x^601 + x^586 + x^585 + x^582 + x^581 + x^580 + x^577 + x^573 + x^571 + x^568 + x^562 + x^560 + x^556 + x^555 + x^554 + x^553 + x^551 + x^547 + x^543 + x^541 + x^538 + x^532 + x^530 + x^528 + x^524 + x^522 + x^521 + x^517 + x^513 + x^511 + x^506 + x^504 + x^502 + x^500 + x^497 + x^495 + x^491 + x^487 + x^483 + x^481 + x^478 + x^474 + x^472 + x^467 + x^466 + x^465 + x^461 + x^457 + x^453 + x^451 + x^444 + x^438 + x^437 + x^436 + x^435 + x^431 + x^427 + x^423 + x^412 + x^408 + x^407 + x^405 + x^401 + x^397 + x^390 + x^384 + x^377 + x^375 + x^371 + x^367 + x^366 + x^365 + x^363 + x^362 + x^360 + x^358 + x^356 + x^352 + x^350 + x^347 + x^345 + x^341 + x^337 + x^336 + x^334 + x^332 + x^324 + x^317 + x^315 + x^311 + x^307 + x^306 + x^305 + x^304 + x^302 + x^300 + x^298 + x^296 + x^287 + x^285 + x^281 + x^279 + x^277 + x^276 + x^274 + x^268 + x^266 + x^257 + x^255 + x^247 + x^244 + x^242 + x^240 + x^238 + x^234 + x^227 + x^221 + x^219 + x^216 + x^214 + x^206 + x^197 + x^195 + x^193 + x^182 + x^176 + x^161 + x^156 + x^152 + x^146 + x^139 + x^133 + x^116 + x^111 + x^109 + x^107 + x^96 + x^94 + x^92 + x^90 + x^88 + x^85 + x^81 + x^79 + x^77 + x^73 + x^66 + x^64 + x^62 + x^60 + x^53 + x^51 + x^47 + x^45 + x^36 + x^34 + x^32 + x^19 + x^17 + x^15 + 1
+
+31-52-54 249 x^928 + x^898 + x^872 + x^870 + x^842 + x^841 + x^838 + x^836 + x^812 + x^810 + x^782 + x^781 + x^779 + x^778 + x^774 + x^753 + x^752 + x^750 + x^748 + x^723 + x^722 + x^721 + x^719 + x^717 + x^714 + x^712 + x^694 + x^690 + x^689 + x^688 + x^686 + x^684 + x^664 + x^663 + x^662 + x^661 + x^660 + x^658 + x^657 + x^656 + x^655 + x^654 + x^653 + x^652 + x^650 + x^634 + x^633 + x^631 + x^630 + x^629 + x^628 + x^622 + x^604 + x^599 + x^597 + x^596 + x^593 + x^592 + x^590 + x^588 + x^570 + x^567 + x^566 + x^562 + x^560 + x^536 + x^534 + x^533 + x^531 + x^530 + x^526 + x^513 + x^512 + x^510 + x^509 + x^508 + x^507 + x^506 + x^505 + x^504 + x^502 + x^498 + x^483 + x^482 + x^481 + x^479 + x^478 + x^476 + x^466 + x^464 + x^454 + x^445 + x^438 + x^424 + x^423 + x^422 + x^421 + x^419 + x^418 + x^411 + x^404 + x^402 + x^394 + x^393 + x^392 + x^391 + x^390 + x^387 + x^386 + x^383 + x^380 + x^379 + x^378 + x^377 + x^376 + x^374 + x^364 + x^362 + x^361 + x^359 + x^357 + x^356 + x^355 + x^354 + x^351 + x^349 + x^348 + x^347 + x^345 + x^342 + x^340 + x^332 + x^330 + x^329 + x^328 + x^327 + x^325 + x^324 + x^323 + x^321 + x^319 + x^318 + x^316 + x^314 + x^312 + x^302 + x^301 + x^300 + x^296 + x^295 + x^294 + x^291 + x^289 + x^287 + x^286 + x^283 + x^278 + x^266 + x^265 + x^264 + x^263 + x^262 + x^260 + x^259 + x^255 + x^250 + x^234 + x^232 + x^229 + x^227 + x^226 + x^225 + x^222 + x^218 + x^216 + x^208 + x^204 + x^200 + x^198 + x^197 + x^195 + x^194 + x^193 + x^190 + x^188 + x^178 + x^174 + x^173 + x^172 + x^171 + x^170 + x^169 + x^168 + x^166 + x^165 + x^164 + x^162 + x^161 + x^160 + x^159 + x^158 + x^157 + x^156 + x^154 + x^142 + x^141 + x^139 + x^138 + x^135 + x^131 + x^128 + x^126 + x^116 + x^112 + x^110 + x^107 + x^106 + x^105 + x^104 + x^102 + x^100 + x^98 + x^96 + x^95 + x^94 + x^92 + x^88 + x^80 + x^79 + x^76 + x^74 + x^73 + x^71 + x^70 + x^69 + x^66 + x^65 + x^64 + x^56 + x^54 + x^48 + x^46 + x^45 + x^43 + x^38 + x^35 + x^33 + x^30 + 1
+
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+
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