diff options
| -rw-r--r-- | lib/crypto/doc/src/crypto.xml | 110 | ||||
| -rw-r--r-- | lib/crypto/src/crypto.erl | 185 | ||||
| -rw-r--r-- | lib/stdlib/doc/src/rand.xml | 61 | ||||
| -rw-r--r-- | lib/stdlib/src/rand.erl | 433 | ||||
| -rw-r--r-- | lib/stdlib/test/rand_SUITE.erl | 373 | ||||
| -rw-r--r-- | lib/stdlib/test/rand_Xoroshiro928ss_dev.txt | 343 |
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 + 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+ 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 + 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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 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