1223 lines
40 KiB
C#
1223 lines
40 KiB
C#
using System;
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using System.Numerics;
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using System.Runtime.CompilerServices;
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using PerfectNumbers.Core.Gpu;
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namespace PerfectNumbers.Core;
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public static class ULongExtensions
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{
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private const int Pow2WindowSize = 8;
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private const ulong Pow2WindowFallbackThreshold = 32UL;
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private const int Pow2WindowOddPowerCount = 1 << (Pow2WindowSize - 1);
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private struct Pow2OddPowerTable
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{
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public ulong Element0;
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public ulong Element1;
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public ulong Element2;
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public ulong Element3;
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public ulong Element4;
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public ulong Element5;
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public ulong Element6;
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public ulong Element7;
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public ulong Element8;
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public ulong Element9;
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public ulong Element10;
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public ulong Element11;
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public ulong Element12;
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public ulong Element13;
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public ulong Element14;
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public ulong Element15;
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public ulong Element16;
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public ulong Element17;
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public ulong Element18;
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public ulong Element19;
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public ulong Element20;
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public ulong Element21;
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public ulong Element22;
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public ulong Element23;
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public ulong Element24;
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public ulong Element25;
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public ulong Element26;
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public ulong Element27;
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public ulong Element28;
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public ulong Element29;
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public ulong Element30;
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public ulong Element31;
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public ulong Element32;
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public ulong Element33;
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public ulong Element34;
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public ulong Element35;
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public ulong Element36;
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public ulong Element37;
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public ulong Element38;
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public ulong Element39;
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public ulong Element40;
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public ulong Element41;
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public ulong Element42;
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public ulong Element43;
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public ulong Element44;
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public ulong Element45;
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public ulong Element46;
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public ulong Element47;
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public ulong Element48;
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public ulong Element49;
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public ulong Element50;
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public ulong Element51;
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public ulong Element52;
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public ulong Element53;
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public ulong Element54;
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public ulong Element55;
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public ulong Element56;
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public ulong Element57;
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public ulong Element58;
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public ulong Element59;
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public ulong Element60;
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public ulong Element61;
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public ulong Element62;
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public ulong Element63;
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public ulong Element64;
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public ulong Element65;
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public ulong Element66;
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public ulong Element67;
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public ulong Element68;
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public ulong Element69;
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public ulong Element70;
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public ulong Element71;
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public ulong Element72;
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public ulong Element73;
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public ulong Element74;
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public ulong Element75;
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public ulong Element76;
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public ulong Element77;
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public ulong Element78;
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public ulong Element79;
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public ulong Element80;
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public ulong Element81;
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public ulong Element82;
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public ulong Element83;
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public ulong Element84;
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public ulong Element85;
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public ulong Element86;
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public ulong Element87;
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public ulong Element88;
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public ulong Element89;
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public ulong Element90;
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public ulong Element91;
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public ulong Element92;
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public ulong Element93;
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public ulong Element94;
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public ulong Element95;
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public ulong Element96;
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public ulong Element97;
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public ulong Element98;
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public ulong Element99;
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public ulong Element100;
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public ulong Element101;
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public ulong Element102;
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public ulong Element103;
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public ulong Element104;
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public ulong Element105;
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public ulong Element106;
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public ulong Element107;
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public ulong Element108;
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public ulong Element109;
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public ulong Element110;
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public ulong Element111;
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public ulong Element112;
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public ulong Element113;
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public ulong Element114;
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public ulong Element115;
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public ulong Element116;
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public ulong Element117;
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public ulong Element118;
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public ulong Element119;
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public ulong Element120;
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public ulong Element121;
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public ulong Element122;
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public ulong Element123;
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public ulong Element124;
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public ulong Element125;
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public ulong Element126;
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public ulong Element127;
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public ref ulong this[int index]
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{
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get
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{
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return ref GetElementRef(index);
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}
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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private ref ulong GetElementRef(int index)
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{
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switch (index)
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{
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case 0:
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return ref Element0;
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case 1:
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return ref Element1;
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case 2:
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return ref Element2;
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case 3:
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return ref Element3;
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case 4:
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return ref Element4;
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case 5:
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return ref Element5;
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case 6:
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return ref Element6;
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case 7:
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return ref Element7;
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case 8:
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return ref Element8;
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case 9:
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return ref Element9;
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case 10:
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return ref Element10;
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case 11:
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return ref Element11;
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case 12:
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return ref Element12;
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case 13:
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return ref Element13;
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case 14:
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return ref Element14;
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case 15:
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return ref Element15;
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case 16:
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return ref Element16;
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case 17:
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return ref Element17;
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case 18:
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return ref Element18;
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case 19:
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return ref Element19;
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case 20:
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return ref Element20;
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case 21:
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return ref Element21;
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case 22:
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return ref Element22;
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case 23:
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return ref Element23;
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case 24:
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return ref Element24;
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case 25:
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return ref Element25;
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case 26:
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return ref Element26;
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case 27:
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return ref Element27;
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case 28:
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return ref Element28;
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case 29:
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return ref Element29;
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case 30:
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return ref Element30;
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case 31:
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return ref Element31;
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case 32:
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return ref Element32;
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case 33:
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return ref Element33;
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case 34:
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return ref Element34;
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case 35:
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return ref Element35;
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case 36:
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return ref Element36;
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case 37:
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return ref Element37;
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case 38:
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return ref Element38;
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case 39:
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return ref Element39;
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case 40:
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return ref Element40;
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case 41:
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return ref Element41;
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case 42:
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return ref Element42;
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case 43:
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return ref Element43;
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case 44:
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return ref Element44;
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case 45:
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return ref Element45;
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case 46:
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return ref Element46;
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case 47:
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return ref Element47;
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case 48:
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return ref Element48;
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case 49:
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return ref Element49;
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case 50:
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return ref Element50;
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case 51:
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return ref Element51;
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case 52:
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return ref Element52;
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case 53:
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return ref Element53;
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case 54:
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return ref Element54;
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case 55:
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return ref Element55;
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case 56:
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return ref Element56;
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case 57:
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return ref Element57;
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case 58:
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return ref Element58;
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case 59:
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return ref Element59;
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case 60:
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return ref Element60;
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case 61:
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return ref Element61;
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case 62:
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return ref Element62;
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case 63:
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return ref Element63;
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case 64:
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return ref Element64;
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case 65:
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return ref Element65;
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case 66:
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return ref Element66;
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case 67:
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return ref Element67;
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case 68:
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return ref Element68;
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case 69:
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return ref Element69;
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case 70:
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return ref Element70;
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case 71:
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return ref Element71;
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case 72:
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return ref Element72;
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case 73:
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return ref Element73;
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case 74:
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return ref Element74;
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case 75:
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return ref Element75;
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case 76:
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return ref Element76;
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case 77:
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return ref Element77;
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case 78:
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return ref Element78;
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case 79:
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return ref Element79;
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case 80:
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return ref Element80;
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case 81:
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return ref Element81;
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case 82:
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return ref Element82;
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case 83:
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return ref Element83;
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case 84:
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return ref Element84;
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case 85:
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return ref Element85;
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case 86:
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return ref Element86;
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case 87:
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return ref Element87;
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case 88:
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return ref Element88;
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case 89:
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return ref Element89;
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case 90:
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return ref Element90;
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case 91:
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return ref Element91;
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case 92:
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return ref Element92;
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case 93:
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return ref Element93;
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case 94:
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return ref Element94;
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case 95:
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return ref Element95;
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case 96:
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return ref Element96;
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case 97:
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return ref Element97;
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case 98:
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return ref Element98;
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case 99:
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return ref Element99;
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case 100:
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return ref Element100;
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case 101:
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return ref Element101;
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case 102:
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return ref Element102;
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case 103:
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return ref Element103;
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case 104:
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return ref Element104;
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case 105:
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return ref Element105;
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case 106:
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return ref Element106;
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case 107:
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return ref Element107;
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case 108:
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return ref Element108;
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case 109:
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return ref Element109;
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case 110:
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return ref Element110;
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case 111:
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return ref Element111;
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case 112:
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return ref Element112;
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case 113:
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return ref Element113;
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case 114:
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return ref Element114;
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case 115:
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return ref Element115;
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case 116:
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return ref Element116;
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case 117:
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return ref Element117;
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case 118:
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return ref Element118;
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case 119:
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return ref Element119;
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case 120:
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return ref Element120;
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case 121:
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return ref Element121;
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case 122:
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return ref Element122;
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case 123:
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return ref Element123;
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case 124:
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return ref Element124;
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case 125:
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return ref Element125;
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case 126:
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return ref Element126;
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case 127:
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return ref Element127;
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default:
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return ref Element0;
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}
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}
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}
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public static ulong CalculateOrder(this ulong q)
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{
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if (q <= 2UL)
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{
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return 0UL;
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}
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ulong order = q - 1UL, prime, temp;
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uint[] smallPrimes = PrimesGenerator.SmallPrimes;
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ulong[] smallPrimesPow2 = PrimesGenerator.SmallPrimesPow2;
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int i = 0, primesLength = smallPrimes.Length;
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UInt128 q128 = q,
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cycle = MersenneDivisorCycles.GetCycle(q128);
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// TODO: When the shared cycle snapshot cannot serve this divisor, trigger an on-demand
|
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// GPU computation (respecting the configured device) without promoting the result into
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// the cache so the order calculator still benefits from cycle stepping while keeping the
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// single-block memory plan intact.
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for (; i < primesLength; i++)
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{
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if (smallPrimesPow2[i] > order)
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{
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break;
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}
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prime = smallPrimes[i];
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// TODO: Replace this `%` driven factor peeling with the divisor-cycle aware
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// factoring helper so large orders reuse the cached remainders highlighted in
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// the latest divisor-cycle benchmarks instead of recomputing slow modulo checks.
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while (order % prime == 0UL)
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{
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temp = order / prime;
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if (temp.PowModWithCycle(q128, cycle) == UInt128.One)
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{
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order = temp;
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}
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else
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{
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break;
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}
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}
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}
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return order;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static int GetBitLength(this ulong value)
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{
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return 64 - BitOperations.LeadingZeroCount(value);
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong MultiplyShiftRight(ulong value, ulong multiplier, int shift)
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{
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UInt128 product = (UInt128)value * multiplier;
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return (ulong)(product >> shift);
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong MultiplyShiftRightShiftFirst(ulong value, ulong multiplier, int shift)
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{
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ulong high = value >> shift;
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ulong mask = (1UL << shift) - 1UL;
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ulong low = value & mask;
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||
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UInt128 highContribution = (UInt128)high * multiplier;
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UInt128 lowContribution = (UInt128)low * multiplier;
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UInt128 combined = highContribution + (lowContribution >> shift);
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return (ulong)combined;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong FastDiv64(this ulong value, ulong divisor, ulong mul)
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{
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ulong quotient = (ulong)(((UInt128)value * mul) >> 64);
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UInt128 remainder = (UInt128)value - ((UInt128)quotient * divisor);
|
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if (remainder >= divisor)
|
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{
|
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quotient++;
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}
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return quotient;
|
||
}
|
||
|
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public const ulong WordBitMask = 0xFFFFUL;
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public static bool IsPrimeCandidate(this ulong n)
|
||
{
|
||
int i = 0;
|
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uint[] smallPrimes = PrimesGenerator.SmallPrimes;
|
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ulong[] smallPrimesPow2 = PrimesGenerator.SmallPrimesPow2;
|
||
int len = smallPrimes.Length;
|
||
ulong p;
|
||
for (; i < len; i++)
|
||
{
|
||
if (smallPrimesPow2[i] > n)
|
||
{
|
||
break;
|
||
}
|
||
|
||
p = smallPrimes[i];
|
||
// TODO: Swap this modulo check for the shared small-prime cycle filter once the
|
||
// divisor-cycle cache is mandatory, matching the PrimeTester improvements noted in
|
||
// the CPU sieve benchmarks.
|
||
if ((n % p) == 0UL)
|
||
{
|
||
return n == p;
|
||
}
|
||
}
|
||
|
||
return true;
|
||
}
|
||
|
||
// Benchmarks (Mod5ULongBenchmarks) show the direct `% 5` is still cheaper (~0.26 ns vs 0.43 ns), so keep the modulo until a faster lookup is proven.
|
||
// (Mod8/Mod10 stay masked because they win; Mod5 currently does not.)
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public static ulong Mod10(this ulong value) => (value & 1UL) == 0UL
|
||
? (value % 5UL) switch
|
||
{
|
||
0UL => 0UL,
|
||
1UL => 6UL,
|
||
2UL => 2UL,
|
||
3UL => 8UL,
|
||
_ => 4UL,
|
||
}
|
||
: (value % 5UL) switch
|
||
{
|
||
0UL => 5UL,
|
||
1UL => 1UL,
|
||
2UL => 7UL,
|
||
3UL => 3UL,
|
||
_ => 9UL,
|
||
};
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public static ulong Mod128(this ulong value) => value & 127UL;
|
||
|
||
// Benchmarks confirm `%` beats our current Mod5/Mod3 helpers for 64-bit inputs, so leave these modulo operations in place until a superior lookup is available.
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public static void Mod10_8_5_3(this ulong value, out ulong mod10, out ulong mod8, out ulong mod5, out ulong mod3)
|
||
{
|
||
mod8 = value & 7UL;
|
||
// Benchmarks show `%` remains faster for the Mod5/Mod3 pair on ulong, so we keep the modulo path here for now.
|
||
mod5 = value % 5UL;
|
||
mod3 = value % 3UL;
|
||
|
||
mod10 = (mod8 & 1UL) == 0UL
|
||
? mod5 switch
|
||
{
|
||
0UL => 0UL,
|
||
1UL => 6UL,
|
||
2UL => 2UL,
|
||
3UL => 8UL,
|
||
_ => 4UL,
|
||
}
|
||
: mod5 switch
|
||
{
|
||
0UL => 5UL,
|
||
1UL => 1UL,
|
||
2UL => 7UL,
|
||
3UL => 3UL,
|
||
_ => 9UL,
|
||
};
|
||
}
|
||
|
||
// Mod5/Mod3 lookup tables are currently slower on 64-bit operands; keep the direct modulo until benchmarks flip.
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public static void Mod10_8_5_3Steps(this ulong value, out ulong step10, out ulong step8, out ulong step5, out ulong step3)
|
||
{
|
||
ulong mod8 = value & 7UL;
|
||
// Same rationale: `%` wins in Mod5/Mod3 benches today, so avoid swapping until a faster lookup exists.
|
||
ulong mod5 = value % 5UL;
|
||
ulong mod3 = value % 3UL;
|
||
ulong mod10 = (mod8 & 1UL) == 0UL
|
||
? mod5 switch
|
||
{
|
||
0UL => 0UL,
|
||
1UL => 6UL,
|
||
2UL => 2UL,
|
||
3UL => 8UL,
|
||
_ => 4UL,
|
||
}
|
||
: mod5 switch
|
||
{
|
||
0UL => 5UL,
|
||
1UL => 1UL,
|
||
2UL => 7UL,
|
||
3UL => 3UL,
|
||
_ => 9UL,
|
||
};
|
||
|
||
step10 = mod10 + mod10;
|
||
if (step10 >= 10UL)
|
||
{
|
||
step10 -= 10UL;
|
||
}
|
||
|
||
step8 = (mod8 + mod8) & 7UL;
|
||
|
||
step5 = mod5 + mod5;
|
||
if (step5 >= 5UL)
|
||
{
|
||
step5 -= 5UL;
|
||
}
|
||
|
||
step3 = mod3 + mod3;
|
||
if (step3 >= 3UL)
|
||
{
|
||
step3 -= 3UL;
|
||
}
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public static UInt128 Mul64(this ulong a, ulong b) => ((UInt128)a.MulHigh(b) << 64) | (UInt128)(a * b);
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public static ulong MulHigh(this ulong x, ulong y)
|
||
{
|
||
// TODO: Investigate replacing this manual decomposition with the UInt128-based implementation
|
||
// for CPU callers; the latest benchmarks show the intrinsic path is an order of magnitude
|
||
// faster, while GPU code can keep using GpuUInt128.MulHigh.
|
||
ulong xLow = (uint)x;
|
||
ulong xHigh = x >> 32;
|
||
ulong yLow = (uint)y;
|
||
ulong yHigh = y >> 32;
|
||
|
||
ulong w1 = xLow * yHigh;
|
||
ulong w2 = xHigh * yLow;
|
||
ulong w3 = xLow * yLow;
|
||
|
||
// Matching the layout used in GpuUInt128.MulHigh: introducing the
|
||
// intermediate result looks like one extra store, but it lets RyuJIT keep
|
||
// the accumulated high word entirely in registers. Without this explicit
|
||
// local the JIT spills the partial sum, which is where the performance
|
||
// regression in the benchmarks came from.
|
||
ulong result = (xHigh * yHigh) + (w1 >> 32) + (w2 >> 32);
|
||
result += ((w3 >> 32) + (uint)w1 + (uint)w2) >> 32;
|
||
return result;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public static ulong MulHighGpuCompatible(this ulong x, ulong y)
|
||
{
|
||
GpuUInt128 product = new(x);
|
||
product.Mul64(new GpuUInt128(y));
|
||
return product.High;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public static ulong ModPow64(this ulong value, ulong exponent, ulong modulus)
|
||
{
|
||
ulong result = 1UL;
|
||
// TODO: Replace this `%` with the Montgomery folding helper highlighted in MulMod64Benchmarks so the
|
||
// modular exponentiation avoids the slow integer division before the ladder even starts.
|
||
value %= modulus;
|
||
|
||
while (exponent != 0UL)
|
||
{
|
||
if ((exponent & 1UL) != 0UL)
|
||
{
|
||
result = MulMod64(result, value, modulus);
|
||
}
|
||
|
||
value = MulMod64(value, value, modulus);
|
||
exponent >>= 1;
|
||
}
|
||
|
||
return result;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
// TODO: Replace this fallback with the UInt128 Montgomery helper measured fastest in
|
||
// MulMod64Benchmarks so CPU callers stop paying for triple modulo operations.
|
||
public static ulong MulMod64(this ulong a, ulong b, ulong modulus) => (ulong)(UInt128)(((a % modulus) * (b % modulus)) % modulus);
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public static ulong MulMod64GpuCompatible(this ulong a, ulong b, ulong modulus)
|
||
{
|
||
// TODO: Remove this GPU-compatible shim from production once callers migrate to MulMod64,
|
||
// which the benchmarks show is roughly 6-7× faster on dense 64-bit inputs.
|
||
GpuUInt128 state = new(a % modulus);
|
||
return state.MulMod(b, modulus);
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public static ulong MulMod64GpuCompatibleDeferred(this ulong a, ulong b, ulong modulus)
|
||
{
|
||
// TODO: Move this deferred helper to the benchmark suite; the baseline MulMod64 avoids the
|
||
// 5-40× slowdown seen across real-world operand distributions.
|
||
GpuUInt128 state = new(a);
|
||
return state.MulModWithNativeModulo(b, modulus);
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public static ulong MontgomeryMultiply(this ulong a, ulong b, ulong modulus, ulong nPrime)
|
||
{
|
||
ulong tLow = unchecked(a * b);
|
||
ulong m = unchecked(tLow * nPrime);
|
||
ulong mTimesModulusLow = unchecked(m * modulus);
|
||
|
||
ulong result = unchecked(a.MulHigh(b) + m.MulHigh(modulus) + (unchecked(tLow + mTimesModulusLow) < tLow ? 1UL : 0UL));
|
||
if (result >= modulus)
|
||
{
|
||
result -= modulus;
|
||
}
|
||
|
||
return result;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public static ulong Pow2MontgomeryModWindowed(this ulong exponent, in MontgomeryDivisorData divisor, bool keepMontgomery)
|
||
{
|
||
ulong modulus = divisor.Modulus;
|
||
if (exponent == 0UL)
|
||
{
|
||
return keepMontgomery ? divisor.MontgomeryOne : 1UL % modulus;
|
||
}
|
||
|
||
if (exponent <= Pow2WindowFallbackThreshold)
|
||
{
|
||
return Pow2MontgomeryModSingleBit(exponent, divisor, keepMontgomery);
|
||
}
|
||
|
||
int bitLength = BitOperations.Log2(exponent) + 1;
|
||
int windowSize = GetWindowSize(bitLength);
|
||
int oddPowerCount = 1 << (windowSize - 1);
|
||
|
||
ulong result = divisor.MontgomeryOne;
|
||
ulong nPrime = divisor.NPrime;
|
||
Pow2OddPowerTable oddPowers = default;
|
||
InitializeMontgomeryOddPowers(divisor, modulus, nPrime, ref oddPowers, oddPowerCount);
|
||
|
||
int index = bitLength - 1;
|
||
while (index >= 0)
|
||
{
|
||
if (((exponent >> index) & 1UL) == 0UL)
|
||
{
|
||
result = result.MontgomeryMultiply(result, modulus, nPrime);
|
||
index--;
|
||
continue;
|
||
}
|
||
|
||
int windowStart = index - windowSize + 1;
|
||
if (windowStart < 0)
|
||
{
|
||
windowStart = 0;
|
||
}
|
||
|
||
while (((exponent >> windowStart) & 1UL) == 0UL)
|
||
{
|
||
windowStart++;
|
||
}
|
||
|
||
int windowLength = index - windowStart + 1;
|
||
for (int square = 0; square < windowLength; square++)
|
||
{
|
||
result = result.MontgomeryMultiply(result, modulus, nPrime);
|
||
}
|
||
|
||
ulong mask = (1UL << windowLength) - 1UL;
|
||
ulong windowValue = (exponent >> windowStart) & mask;
|
||
int tableIndex = (int)((windowValue - 1UL) >> 1);
|
||
ulong multiplier = oddPowers[tableIndex];
|
||
result = result.MontgomeryMultiply(multiplier, modulus, nPrime);
|
||
|
||
index = windowStart - 1;
|
||
}
|
||
|
||
if (keepMontgomery)
|
||
{
|
||
return result;
|
||
}
|
||
|
||
return result.MontgomeryMultiply(1UL, modulus, nPrime);
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
private static ulong Pow2MontgomeryModSingleBit(ulong exponent, in MontgomeryDivisorData divisor, bool keepMontgomery)
|
||
{
|
||
ulong modulus = divisor.Modulus;
|
||
ulong nPrime = divisor.NPrime;
|
||
ulong result = divisor.MontgomeryOne;
|
||
ulong baseVal = divisor.MontgomeryTwo;
|
||
ulong remainingExponent = exponent;
|
||
|
||
while (remainingExponent != 0UL)
|
||
{
|
||
if ((remainingExponent & 1UL) != 0UL)
|
||
{
|
||
result = result.MontgomeryMultiply(baseVal, modulus, nPrime);
|
||
}
|
||
|
||
remainingExponent >>= 1;
|
||
if (remainingExponent == 0UL)
|
||
{
|
||
break;
|
||
}
|
||
|
||
baseVal = baseVal.MontgomeryMultiply(baseVal, modulus, nPrime);
|
||
}
|
||
|
||
return keepMontgomery ? result : result.MontgomeryMultiply(1UL, modulus, nPrime);
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
private static int GetWindowSize(int bitLength)
|
||
{
|
||
if (bitLength <= 6)
|
||
{
|
||
return Math.Max(bitLength, 1);
|
||
}
|
||
|
||
if (bitLength <= 23)
|
||
{
|
||
return 4;
|
||
}
|
||
|
||
if (bitLength <= 79)
|
||
{
|
||
return 5;
|
||
}
|
||
|
||
if (bitLength <= 239)
|
||
{
|
||
return 6;
|
||
}
|
||
|
||
if (bitLength <= 671)
|
||
{
|
||
return 7;
|
||
}
|
||
|
||
return Pow2WindowSize;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
private static void InitializeMontgomeryOddPowers(in MontgomeryDivisorData divisor, ulong modulus, ulong nPrime, ref Pow2OddPowerTable oddPowers, int oddPowerCount)
|
||
{
|
||
oddPowers[0] = divisor.MontgomeryTwo;
|
||
if (oddPowerCount == 1)
|
||
{
|
||
return;
|
||
}
|
||
|
||
ulong square = divisor.MontgomeryTwoSquared;
|
||
for (int i = 1; i < oddPowerCount; i++)
|
||
{
|
||
ulong previous = oddPowers[i - 1];
|
||
oddPowers[i] = previous.MontgomeryMultiply(square, modulus, nPrime);
|
||
}
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public static ulong Pow2MontgomeryModWithCycle(this ulong exponent, ulong cycleLength, in MontgomeryDivisorData divisor)
|
||
{
|
||
ulong rotationCount = exponent % cycleLength;
|
||
return Pow2MontgomeryModWindowed(rotationCount, divisor, keepMontgomery: false);
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public static ulong Pow2MontgomeryModFromCycleRemainder(this ulong reducedExponent, in MontgomeryDivisorData divisor)
|
||
{
|
||
return Pow2MontgomeryModWindowed(reducedExponent, divisor, keepMontgomery: false);
|
||
}
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public static UInt128 PowMod(this ulong exponent, UInt128 modulus)
|
||
{
|
||
UInt128 result = UInt128.One;
|
||
ulong exponentLoopIndex = 0UL;
|
||
|
||
// TODO: Port this scalar PowMod fallback to the ProcessEightBitWindows helper so CPU callers get the
|
||
// eight-bit window wins measured against the classic square-and-subtract implementation.
|
||
// Return 1 because 2^0 = 1
|
||
if (exponent == 0UL)
|
||
return result;
|
||
|
||
// Any number mod 1 is 0
|
||
if (modulus == UInt128.One)
|
||
return UInt128.Zero;
|
||
|
||
// For small exponents, do classic method
|
||
if (exponent < 64 || modulus < 4)
|
||
{
|
||
for (; exponentLoopIndex < exponent; exponentLoopIndex++)
|
||
{
|
||
result <<= 1;
|
||
if (result >= modulus)
|
||
result -= modulus;
|
||
}
|
||
|
||
return result;
|
||
}
|
||
|
||
// Special case: if modulus is a power of two, use bitmasking for efficiency and correctness
|
||
if ((modulus & (modulus - 1)) == 0)
|
||
{
|
||
result = UInt128.One << (int)(exponent & 127);
|
||
return result & (modulus - 1);
|
||
}
|
||
|
||
// Reusing exponentLoopIndex to iterate again over the exponent range for the general-case accumulation.
|
||
exponentLoopIndex = 0UL;
|
||
// Reusing result after resetting it for the general modulus accumulation phase.
|
||
result = UInt128.One;
|
||
for (; exponentLoopIndex < exponent; exponentLoopIndex++)
|
||
{
|
||
result <<= 1;
|
||
if (result >= modulus)
|
||
result -= modulus;
|
||
}
|
||
|
||
return result;
|
||
}
|
||
|
||
/// <summary>
|
||
/// Computes 2^exponent mod modulus using a known cycle length.
|
||
/// </summary>
|
||
public static UInt128 PowModWithCycle(this ulong exponent, UInt128 modulus, ulong cycleLength)
|
||
{
|
||
UInt128 result = UInt128.One;
|
||
ulong exponentLoopIndex = 0UL;
|
||
|
||
// TODO: Wire this cycle-aware overload into the ProcessEightBitWindows helper so the reduced exponent path
|
||
// inherits the faster windowed pow2 routine highlighted in the Pow2Montgomery benchmarks.
|
||
// Return 1 because 2^0 = 1
|
||
if (exponent == 0UL)
|
||
return result;
|
||
|
||
// Any number mod 1 is 0
|
||
if (modulus == UInt128.One)
|
||
return UInt128.Zero;
|
||
|
||
// For small exponents, do classic method
|
||
if (exponent < 64 || modulus < 4)
|
||
{
|
||
for (; exponentLoopIndex < exponent; exponentLoopIndex++)
|
||
{
|
||
result <<= 1;
|
||
if (result >= modulus)
|
||
result -= modulus;
|
||
}
|
||
|
||
return result;
|
||
}
|
||
|
||
// Special case: if modulus is a power of two, use bitmasking for efficiency and correctness
|
||
if ((modulus & (modulus - 1)) == 0)
|
||
{
|
||
result = UInt128.One << (int)(exponent & 127);
|
||
return result & (modulus - 1);
|
||
}
|
||
|
||
// Reusing exponentLoopIndex to iterate over the rotation count for the cycle-aware path.
|
||
exponentLoopIndex = 0UL;
|
||
// Reusing result after resetting it for the rotation-accumulation pass.
|
||
result = UInt128.One;
|
||
// TODO: Replace this modulo with the cached cycle remainder produced by the divisor-cycle cache so PowModWithCycle avoids
|
||
// repeated `%` work, matching the ProcessEightBitWindows wins captured in Pow2MontgomeryModCycleComputationBenchmarks.
|
||
ulong rotationCount = exponent % cycleLength;
|
||
for (; exponentLoopIndex < rotationCount; exponentLoopIndex++)
|
||
{
|
||
result <<= 1;
|
||
if (result >= modulus)
|
||
result -= modulus;
|
||
}
|
||
|
||
return result;
|
||
}
|
||
|
||
/// <summary>
|
||
/// Computes 2^exponent mod modulus using a known cycle length.
|
||
/// </summary>
|
||
public static UInt128 PowModWithCycle(this ulong exponent, UInt128 modulus, UInt128 cycleLength)
|
||
{
|
||
UInt128 result = UInt128.One;
|
||
ulong exponentLoopIndex = 0UL;
|
||
|
||
// TODO: Replace this UInt128-cycle overload with the ProcessEightBitWindows helper so large-exponent CPU scans
|
||
// reuse the faster windowed pow2 ladder instead of the manual rotation loop measured to lag behind in benchmarks.
|
||
// Return 1 because 2^0 = 1
|
||
if (exponent == UInt128.Zero)
|
||
return result;
|
||
|
||
// Any number mod 1 is 0
|
||
if (modulus == UInt128.One)
|
||
return UInt128.Zero;
|
||
|
||
// For small exponents, do classic method
|
||
if (exponent < UInt128Numbers.SixtyFour || modulus < UInt128Numbers.Four)
|
||
{
|
||
for (; exponentLoopIndex < exponent; exponentLoopIndex++)
|
||
{
|
||
result <<= 1;
|
||
if (result >= modulus)
|
||
result -= modulus;
|
||
}
|
||
|
||
return result;
|
||
}
|
||
|
||
// Special case: if modulus is a power of two, use bitmasking for efficiency and correctness
|
||
if ((modulus & (modulus - UInt128.One)) == UInt128.Zero)
|
||
{
|
||
result = UInt128.One << (int)(exponent & UInt128Numbers.OneHundredTwentySeven);
|
||
return result & (modulus - 1);
|
||
}
|
||
|
||
// Reusing result after resetting it for the rotation-driven accumulation phase.
|
||
result = UInt128.One;
|
||
// TODO: Swap this modulo with the upcoming UInt128 cycle remainder helper so large-exponent scans reuse cached
|
||
// reductions instead of recomputing `%` for every lookup, as demonstrated in Pow2MontgomeryModCycleComputationBenchmarks.
|
||
UInt128 rotationCount = exponent % cycleLength;
|
||
UInt128 rotationIndex = UInt128.Zero;
|
||
while (rotationIndex < rotationCount)
|
||
{
|
||
result <<= 1;
|
||
if (result >= modulus)
|
||
result -= modulus;
|
||
|
||
rotationIndex += UInt128.One;
|
||
}
|
||
|
||
return result;
|
||
}
|
||
|
||
/// <summary>
|
||
/// Computes 2^exponent mod modulus using a known cycle length.
|
||
/// </summary>
|
||
public static UInt128 PowModWithCycle(this UInt128 exponent, UInt128 modulus, UInt128 cycleLength)
|
||
{
|
||
UInt128 one = UInt128.One,
|
||
result = one,
|
||
zero = UInt128.Zero;
|
||
ulong exponentLoopIndex = 0UL;
|
||
|
||
// TODO: Migrate this UInt128 exponent overload to ProcessEightBitWindows so the large-cycle reductions drop the
|
||
// slow manual loop that underperforms the windowed pow2 helper in the Pow2 benchmark suite.
|
||
// Return 1 because 2^0 = 1
|
||
if (exponent == zero)
|
||
return result;
|
||
|
||
// Any number mod 1 is 0
|
||
if (modulus == one)
|
||
return zero;
|
||
|
||
// For small exponents, do classic method
|
||
if (exponent < UInt128Numbers.SixtyFour || modulus < UInt128Numbers.Four)
|
||
{
|
||
for (; exponentLoopIndex < exponent; exponentLoopIndex++)
|
||
{
|
||
result <<= 1;
|
||
if (result >= modulus)
|
||
result -= modulus;
|
||
}
|
||
|
||
return result;
|
||
}
|
||
|
||
// Special case: if modulus is a power of two, use bitmasking for efficiency and correctness
|
||
if ((modulus & (modulus - one)) == zero)
|
||
{
|
||
result = one << (int)(exponent & UInt128Numbers.OneHundredTwentySeven);
|
||
return result & (modulus - 1);
|
||
}
|
||
|
||
// Reusing result after resetting it for the rotation-driven accumulation phase.
|
||
result = one;
|
||
// TODO: Swap this modulo with the shared UInt128 cycle remainder helper once available so CRT powmods reuse cached
|
||
// reductions in the windowed ladder, avoiding the `%` cost highlighted in Pow2MontgomeryModCycleComputationBenchmarks.
|
||
UInt128 rotationCount = exponent % cycleLength;
|
||
|
||
// We're reusing "zero" as rotation index for just a little better performance
|
||
while (zero < rotationCount)
|
||
{
|
||
result <<= 1;
|
||
if (result >= modulus)
|
||
result -= modulus;
|
||
|
||
zero += one;
|
||
}
|
||
|
||
return result;
|
||
}
|
||
|
||
/// <summary>
|
||
/// Computes 2^exponent mod modulus using iterative CRT composition from mod 10 up to modulus.
|
||
/// Only for modulus >= 10 and reasonable size.
|
||
/// </summary>
|
||
public static UInt128 PowModCrt(this ulong exponent, UInt128 modulus, MersenneDivisorCycles cycles)
|
||
{
|
||
if (modulus < 10)
|
||
return PowMod(exponent, modulus); // fallback to classic
|
||
|
||
// Use cycle length 4 for mod 10
|
||
UInt128 currentModulus = 10,
|
||
cycle,
|
||
modulusCandidate = 11,
|
||
remainderForCandidate,
|
||
result = PowModWithCycle(exponent, 10, 4),
|
||
zero = UInt128.Zero;
|
||
|
||
for (; modulusCandidate <= modulus; modulusCandidate++)
|
||
{
|
||
cycle = MersenneDivisorCycles.GetCycle(modulusCandidate);
|
||
remainderForCandidate = cycle > zero
|
||
? PowModWithCycle(exponent, modulusCandidate, cycle)
|
||
: PowMod(exponent, modulusCandidate);
|
||
|
||
// Solve x ≡ result mod currentModulus
|
||
// x ≡ remM mod m
|
||
// Find x mod (currentModulus * m)
|
||
// Since currentModulus and m are coprime, use CRT:
|
||
// x = result + currentModulus * t, where t ≡ (remM - result) * inv(currentModulus, m) mod m
|
||
|
||
// TODO: Replace this `% modulusCandidate` with the cached residue helper derived from Mod10_8_5_3Benchmarks so CRT
|
||
// composition avoids repeated modulo divisions when combining residues for large divisor sets.
|
||
result += currentModulus * ((remainderForCandidate + modulusCandidate - (result % modulusCandidate)) * ModInverse(currentModulus, modulusCandidate) % modulusCandidate);
|
||
currentModulus *= modulusCandidate;
|
||
|
||
if (currentModulus >= modulus)
|
||
break;
|
||
}
|
||
|
||
// TODO: Swap this final `% modulus` with the pooled remainder cache so the CRT result write-back avoids one more division,
|
||
// aligning with the optimizations captured in Mod10_8_5_3Benchmarks.
|
||
return result % modulus;
|
||
}
|
||
|
||
// Helper: modular inverse (extended Euclidean algorithm)
|
||
private static UInt128 ModInverse(UInt128 a, UInt128 m)
|
||
{
|
||
UInt128 m0 = m,
|
||
originalA,
|
||
originalM,
|
||
temp,
|
||
x0 = 0,
|
||
x1 = 1;
|
||
|
||
if (m == 1)
|
||
{
|
||
return 0;
|
||
}
|
||
|
||
while (a > 1)
|
||
{
|
||
originalA = a;
|
||
originalM = m;
|
||
m = originalA % originalM;
|
||
a = originalM;
|
||
temp = x0;
|
||
x0 = x1 - (originalA / originalM) * x0;
|
||
x1 = temp;
|
||
}
|
||
|
||
if (x1 < 0)
|
||
{
|
||
x1 += m0;
|
||
}
|
||
|
||
return x1;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public static bool SharesFactorWithExponentMinusOne(this ulong exponent)
|
||
{
|
||
ulong prime, value = exponent - 1UL;
|
||
value >>= BitOperations.TrailingZeroCount(value);
|
||
uint[] smallPrimes = PrimesGenerator.SmallPrimes;
|
||
ulong[] smallPrimesPow2 = PrimesGenerator.SmallPrimesPow2;
|
||
int i = 1, smallPrimesLength = smallPrimes.Length;
|
||
|
||
for (; i < smallPrimesLength && smallPrimesPow2[i] <= value; i++)
|
||
{
|
||
prime = smallPrimes[i];
|
||
if (value % prime != 0UL)
|
||
{
|
||
continue;
|
||
}
|
||
|
||
if (exponent % prime.CalculateOrder() == 0UL)
|
||
{
|
||
return true;
|
||
}
|
||
|
||
do
|
||
{
|
||
value /= prime;
|
||
}
|
||
while (value % prime == 0UL);
|
||
}
|
||
|
||
if (value > 1UL && exponent % value.CalculateOrder() == 0UL)
|
||
{
|
||
return true;
|
||
}
|
||
|
||
return false;
|
||
}
|
||
}
|