1414 lines
41 KiB
C#
1414 lines
41 KiB
C#
using System;
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using System.Numerics;
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using System.Runtime.CompilerServices;
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namespace PerfectNumbers.Core.Gpu;
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public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
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{
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public ulong High;
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public ulong Low;
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private const int NativeModuloChunkBits = 8;
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private const int NativeModuloChunkBitsMinusOne = NativeModuloChunkBits - 1;
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private const int NativeModuloBitMaskTableSize = 1024;
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private const ulong NativeModuloChunkMask = (1UL << NativeModuloChunkBits) - 1UL;
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private static readonly ulong[] NativeModuloBitMasks = CreateNativeModuloBitMasks();
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private const int Pow2WindowSizeBits = 8;
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private const int Pow2WindowOddPowerCount = 1 << (Pow2WindowSizeBits - 1);
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private const int Pow2WindowMaxExponent = (1 << Pow2WindowSizeBits) - 1;
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public static readonly GpuUInt128 Zero = new();
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public static readonly GpuUInt128 One = new(1UL);
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public GpuUInt128()
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{
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public GpuUInt128(ulong high, ulong low)
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{
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High = high;
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Low = low;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public GpuUInt128(ulong low)
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{
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Low = low;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public GpuUInt128(UInt128 value)
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{
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High = (ulong)(value >> 64);
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Low = (ulong)value;
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}
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public readonly bool IsZero
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{
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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get => High == 0UL && Low == 0UL;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static implicit operator GpuUInt128(UInt128 value) => new(value);
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static implicit operator GpuUInt128(ulong value) => new(value);
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static implicit operator UInt128(GpuUInt128 value) =>
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((UInt128)value.High << 64) | value.Low;
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static GpuUInt128 operator +(GpuUInt128 left, GpuUInt128 right)
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{
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left.Add(right);
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return left;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static GpuUInt128 operator -(GpuUInt128 left, GpuUInt128 right)
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{
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left.Sub(right);
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return left;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static GpuUInt128 operator ^(GpuUInt128 left, GpuUInt128 right) =>
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new(left.High ^ right.High, left.Low ^ right.Low);
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static GpuUInt128 operator *(GpuUInt128 left, GpuUInt128 right)
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{
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Multiply(left, right, out var high, out var low);
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left.High = high;
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left.Low = low;
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return left;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static GpuUInt128 operator <<(GpuUInt128 value, int shift)
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{
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shift &= 127;
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if (shift == 0)
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{
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return value;
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}
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ulong high = value.High;
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ulong low = value.Low;
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if (shift >= 64)
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{
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int longShift = shift - 64;
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value.High = low << longShift;
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value.Low = 0UL;
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return value;
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}
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int inverseShift = 64 - shift;
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value.High = (high << shift) | (low >> inverseShift);
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value.Low = low << shift;
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return value;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static GpuUInt128 operator >>(GpuUInt128 value, int shift)
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{
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shift &= 127;
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if (shift == 0)
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{
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return value;
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}
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ulong high = value.High;
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ulong low = value.Low;
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if (shift >= 64)
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{
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int longShift = shift - 64;
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value.Low = longShift == 0 ? high : high >> longShift;
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value.High = 0UL;
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return value;
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}
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int inverseShift = 64 - shift;
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value.Low = (low >> shift) | (high << inverseShift);
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value.High = high >> shift;
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return value;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static bool operator ==(GpuUInt128 left, GpuUInt128 right) => left.Equals(right);
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static bool operator !=(GpuUInt128 left, GpuUInt128 right) => !left.Equals(right);
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static bool operator <(GpuUInt128 left, GpuUInt128 right) => left.CompareTo(right) < 0;
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static bool operator >(GpuUInt128 left, GpuUInt128 right) => left.CompareTo(right) > 0;
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static bool operator <=(GpuUInt128 left, GpuUInt128 right) => left.CompareTo(right) <= 0;
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static bool operator >=(GpuUInt128 left, GpuUInt128 right) => left.CompareTo(right) >= 0;
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static GpuUInt128 operator ++(GpuUInt128 value)
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{
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value.Add(1UL);
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return value;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public readonly int CompareTo(GpuUInt128 other)
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{
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if (High < other.High)
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{
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return -1;
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}
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if (High > other.High)
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{
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return 1;
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}
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if (Low < other.Low)
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{
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return -1;
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}
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if (Low > other.Low)
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{
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return 1;
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}
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return 0;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public readonly bool Equals(GpuUInt128 other) => High == other.High && Low == other.Low;
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public override readonly bool Equals(object? obj) => obj is GpuUInt128 value && Equals(value);
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public override readonly int GetHashCode() => HashCode.Combine(High, Low);
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public void Add(GpuUInt128 other)
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{
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ulong originalLow = Low;
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ulong low = originalLow + other.Low;
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ulong carry = low < originalLow ? 1UL : 0UL;
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High = High + other.High + carry;
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Low = low;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public void Add(ulong value)
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{
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ulong low = Low + value;
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High += (low < Low ? 1UL : 0UL);
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Low = low;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public void AddMod(GpuUInt128 value, GpuUInt128 modulus)
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{
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if (modulus.High == 0UL)
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{
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AddMod(value, modulus.Low);
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return;
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}
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Add(value);
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if (CompareTo(modulus) >= 0)
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{
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Sub(modulus);
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}
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public void AddMod(ulong value, GpuUInt128 modulus)
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{
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if (modulus.High == 0UL)
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{
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AddMod(value, modulus.Low);
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return;
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}
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Add(value);
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if (CompareTo(modulus) >= 0)
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{
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Sub(modulus);
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}
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static GpuUInt128 Pow2Minus1Mod(ulong exponent, GpuUInt128 modulus)
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{
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if (modulus.IsZero)
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{
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return Zero;
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}
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GpuUInt128 pow = Pow2Mod(exponent, modulus);
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if (pow.IsZero)
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{
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pow = modulus;
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}
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pow.Sub(One);
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return pow;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static void Pow2Minus1ModBatch(GpuUInt128 modulus, ReadOnlySpan<ulong> exponents, Span<GpuUInt128> results)
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{
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int count = exponents.Length;
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for (int i = 0; i < count; i++)
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{
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results[i] = Pow2Minus1Mod(exponents[i], modulus);
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}
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public void AddMod(GpuUInt128 value, ulong modulus)
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{
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// TODO: Pre-reduce the operands via the Montgomery ladder used in MulMod64Benchmarks so the GPU
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// compatible shim stops paying for `%` on every call; the InlineUInt128 helper ran 6–82× faster on
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// large 64-bit workloads while preserving compatibility with the CPU scanner.
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ulong a = Low % modulus;
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ulong b = value.Low % modulus;
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ulong sum = a + b;
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if (sum >= modulus || sum < a)
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{
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sum -= modulus;
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}
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High = 0UL;
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Low = sum;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public void AddMod(ulong value, ulong modulus)
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{
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// TODO: Fold these operands with the ImmediateModulo helper once the GPU shim exposes it, avoiding
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// repeated `%` reductions that the benchmarks showed are far slower than the Montgomery-based path
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// for dense operands.
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ulong a = Low % modulus;
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ulong b = value % modulus;
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ulong sum = a + b;
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if (sum >= modulus || sum < a)
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{
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sum -= modulus;
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}
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High = 0UL;
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Low = sum;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public void Sub(GpuUInt128 other)
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{
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ulong borrow = Low < other.Low ? 1UL : 0UL;
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High = High - other.High - borrow;
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Low -= other.Low;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong MulHigh(ulong x, ulong y)
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{
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ulong xLow = (uint)x;
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ulong yLow = (uint)y;
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ulong lowProduct = xLow * yLow;
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ulong yHigh = y >> 32;
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ulong cross = xLow * yHigh;
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ulong xHigh = x >> 32;
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ulong result = xHigh * yHigh;
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ulong temp = xHigh * yLow;
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// Keeping the wide partial sum in a dedicated local prevents the JIT from
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// materialising it on the stack before the final carry propagation. The
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// additional store looks redundant in C#, but it shortens the generated
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// instruction sequence by avoiding an extra temporary and results in a
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// measurable throughput win in the MulHigh benchmarks.
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result += cross >> 32;
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result += temp >> 32;
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lowProduct >>= 32;
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lowProduct += (uint)cross;
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lowProduct += (uint)temp;
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result += lowProduct >> 32;
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return result;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public void Mul64(GpuUInt128 other)
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{
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// Multiply this.Low (assumed 64-bit value) by full 128-bit other
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ulong operand = Low;
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Low = operand * other.Low;
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ulong highProduct = operand * other.High;
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High = highProduct + MulHigh(operand, other.Low);
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public void MulMod(in GpuUInt128 value, in GpuUInt128 modulus)
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{
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GpuUInt128 multiplicand = this;
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GpuUInt128 multiplier = value;
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High = 0UL;
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Low = 0UL;
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while (!multiplier.IsZero)
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{
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if ((multiplier.Low & 1UL) != 0UL)
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{
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AddMod(multiplicand, modulus);
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}
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multiplicand.ShiftLeft(1);
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if (multiplicand.CompareTo(modulus) >= 0)
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{
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multiplicand.Sub(modulus);
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}
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multiplier.ShiftRight(1);
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}
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static GpuUInt128 Pow2Mod(ulong exponent, GpuUInt128 modulus)
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{
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if (modulus.IsZero || modulus == One)
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{
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return new();
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}
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if (exponent == 0UL)
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{
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return new GpuUInt128(1UL);
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}
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Span<GpuUInt128> oddPowers = stackalloc GpuUInt128[Pow2WindowOddPowerCount];
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GpuUInt128 power = new GpuUInt128(1UL);
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int oddIndex = 0;
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for (int bit = 1; bit <= Pow2WindowMaxExponent; bit++)
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{
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power += power;
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if (power.CompareTo(modulus) >= 0)
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{
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power.Sub(modulus);
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}
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if ((bit & 1) != 0)
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{
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oddPowers[oddIndex++] = power;
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}
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}
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GpuUInt128 result = new GpuUInt128(1UL);
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int bitLength = BitOperations.Log2(exponent) + 1;
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int index = bitLength - 1;
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while (index >= 0)
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{
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if (((exponent >> index) & 1UL) == 0UL)
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{
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result.SquareMod(modulus);
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index--;
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continue;
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}
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int windowStart = index - Pow2WindowSizeBits + 1;
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if (windowStart < 0)
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{
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windowStart = 0;
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}
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while (((exponent >> windowStart) & 1UL) == 0UL)
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{
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windowStart++;
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}
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int windowBitCount = index - windowStart + 1;
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for (int square = 0; square < windowBitCount; square++)
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{
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result.SquareMod(modulus);
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}
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ulong mask = (1UL << windowBitCount) - 1UL;
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ulong windowValue = (exponent >> windowStart) & mask;
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int tableIndex = (int)((windowValue - 1UL) >> 1);
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GpuUInt128 factor = oddPowers[tableIndex];
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result.MulMod(factor, modulus);
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index = windowStart - 1;
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}
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return result;
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}
|
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
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public static int TrailingZeroCount(GpuUInt128 value)
|
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{
|
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if (value.Low != 0UL)
|
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{
|
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return BitOperations.TrailingZeroCount(value.Low);
|
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}
|
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return 64 + BitOperations.TrailingZeroCount(value.High);
|
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}
|
||
|
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public static GpuUInt128 BinaryGcd(GpuUInt128 u, GpuUInt128 v)
|
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{
|
||
// TODO: Replace this scalar binary GCD with the branchless reduction from
|
||
// GpuUInt128BinaryGcdBenchmarks so CPU fallbacks stay aligned with the GPU kernel
|
||
// performance when resolving large divisor residues.
|
||
if (u.IsZero)
|
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{
|
||
return v;
|
||
}
|
||
|
||
if (v.IsZero)
|
||
{
|
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return u;
|
||
}
|
||
|
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int shift = TrailingZeroCount(new GpuUInt128(u.High | v.High, u.Low | v.Low));
|
||
int zu = TrailingZeroCount(u);
|
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u >>= zu;
|
||
|
||
do
|
||
{
|
||
int zv = TrailingZeroCount(v);
|
||
v >>= zv;
|
||
if (u > v)
|
||
{
|
||
(u, v) = (v, u);
|
||
}
|
||
|
||
v -= u;
|
||
}
|
||
while (!v.IsZero);
|
||
|
||
return u << shift;
|
||
}
|
||
|
||
|
||
public void Sub(ulong value)
|
||
{
|
||
ulong borrow = Low < value ? 1UL : 0UL;
|
||
High -= borrow;
|
||
Low -= value;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public void SubMod(GpuUInt128 value, GpuUInt128 modulus)
|
||
{
|
||
if (CompareTo(value) >= 0)
|
||
{
|
||
Sub(value);
|
||
}
|
||
else
|
||
{
|
||
Add(modulus);
|
||
Sub(value);
|
||
}
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public void SubMod(ulong value, GpuUInt128 modulus)
|
||
{
|
||
ulong low = Low;
|
||
ulong high = High;
|
||
if (high == 0UL && low < value)
|
||
{
|
||
low += modulus.Low;
|
||
high = modulus.High + (low < modulus.Low ? 1UL : 0UL);
|
||
}
|
||
|
||
ulong borrow = low < value ? 1UL : 0UL;
|
||
High = high - borrow;
|
||
Low = low - value;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public void Mul(GpuUInt128 other)
|
||
{
|
||
Multiply(this, other, out var high, out var low);
|
||
High = high;
|
||
Low = low;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public void Mul(ulong value)
|
||
{
|
||
var (highPart, lowPart) = Mul64(Low, value);
|
||
High = highPart + High * value;
|
||
Low = lowPart;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public void Xor(GpuUInt128 other)
|
||
{
|
||
High ^= other.High;
|
||
Low ^= other.Low;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public void Xor(ulong value)
|
||
{
|
||
Low ^= value;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public void ShiftLeft(int shift)
|
||
{
|
||
shift &= 127;
|
||
if (shift == 0)
|
||
{
|
||
return;
|
||
}
|
||
|
||
ulong high = High;
|
||
ulong low = Low;
|
||
|
||
if (shift >= 64)
|
||
{
|
||
High = low << (shift - 64);
|
||
Low = 0UL;
|
||
return;
|
||
}
|
||
|
||
int inverseShift = 64 - shift;
|
||
High = (high << shift) | (low >> inverseShift);
|
||
Low = low << shift;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public void ShiftRight(int shift)
|
||
{
|
||
shift &= 127;
|
||
if (shift == 0)
|
||
{
|
||
return;
|
||
}
|
||
|
||
ulong high = High;
|
||
ulong low = Low;
|
||
|
||
if (shift >= 64)
|
||
{
|
||
Low = shift == 64 ? high : high >> (shift - 64);
|
||
High = 0UL;
|
||
return;
|
||
}
|
||
|
||
int inverseShift = 64 - shift;
|
||
Low = (low >> shift) | (high << inverseShift);
|
||
High = high >> shift;
|
||
}
|
||
|
||
public void MulMod(GpuUInt128 other, GpuUInt128 modulus)
|
||
{
|
||
if (modulus.High == 0UL)
|
||
{
|
||
Low = MulMod(other, modulus.Low);
|
||
High = 0UL;
|
||
return;
|
||
}
|
||
|
||
MultiplyFull(this, other, out var p3, out var p2, out var p1, out var p0);
|
||
ReduceProductBitwise(p3, p2, p1, p0, modulus);
|
||
// TODO(MOD-OPT): Replace this bitwise long-division style reduction
|
||
// with a faster algorithm suitable for GPU kernels without '%' support.
|
||
// Options:
|
||
// - Montgomery reduction for 128-bit moduli (R=2^128), requires pre-
|
||
// computed modulus-dependent constants (n' and R^2 mod n). This is
|
||
// ideal for NTT primes (odd modulus). Cache per-modulus constants.
|
||
// - Barrett reduction with a 256/128 quotient approximation using only
|
||
// multiplies and shifts. Cache mu = floor(2^k / n) for k=256.
|
||
// Implement a fast path for common 64-bit NTT moduli and a separate
|
||
// path for 128-bit moduli. This will significantly reduce the cost of
|
||
// each butterfly in NTT and LL steps.
|
||
// TODO(MOD-OPT): Plumb constants through caches in NttGpuMath.SquareCacheEntry
|
||
// (e.g., Montgomery n', R2) and provide device-friendly accessors.
|
||
}
|
||
|
||
/// <summary>
|
||
/// Modular multiplication using <see cref="BigInteger"/> reduction. This
|
||
/// method is intended for validation only and should not be used inside
|
||
/// GPU kernels.
|
||
/// </summary>
|
||
internal void MulModBigInteger(GpuUInt128 other, GpuUInt128 modulus)
|
||
{
|
||
var left = (BigInteger)(UInt128)this;
|
||
var right = (BigInteger)(UInt128)other;
|
||
var mod = (BigInteger)(UInt128)modulus;
|
||
var reduced = (UInt128)((left * right) % mod);
|
||
High = (ulong)(reduced >> 64);
|
||
Low = (ulong)reduced;
|
||
}
|
||
|
||
/// <summary>
|
||
/// Experimental limb-based reduction. The current implementation performs
|
||
/// repeated subtractions and becomes extremely slow for large remainders.
|
||
/// Kept for future optimization work.
|
||
/// </summary>
|
||
internal void MulModByLimb(GpuUInt128 other, GpuUInt128 modulus)
|
||
{
|
||
// TODO: Relocate this limb-based reducer to the benchmark project once the production
|
||
// pipeline switches to the faster allocating legacy path demonstrated in the benchmarks.
|
||
MultiplyFull(this, other, out var p3, out var p2, out var p1, out var p0);
|
||
|
||
High = p3;
|
||
Low = p2;
|
||
while (CompareTo(modulus) >= 0)
|
||
{
|
||
Sub(modulus);
|
||
}
|
||
|
||
ulong limb = p1;
|
||
for (int i = 0; i < 2; i++)
|
||
{
|
||
ShiftLeft(64);
|
||
Low = limb;
|
||
while (CompareTo(modulus) >= 0)
|
||
{
|
||
Sub(modulus);
|
||
}
|
||
|
||
limb = p0;
|
||
}
|
||
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public void MulMod(ulong value, GpuUInt128 modulus)
|
||
{
|
||
if (modulus.High == 0UL)
|
||
{
|
||
Low = MulMod(value, modulus.Low);
|
||
High = 0UL;
|
||
return;
|
||
}
|
||
|
||
MultiplyFull(this, value, out var p3, out var p2, out var p1, out var p0);
|
||
ReduceProductBitwise(p3, p2, p1, p0, modulus);
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public readonly ulong MulMod(GpuUInt128 other, ulong modulus)
|
||
{
|
||
ulong modulusLocal = modulus;
|
||
ulong a = Low % modulusLocal;
|
||
ulong b = other.Low % modulusLocal;
|
||
|
||
if (a == 0UL || b == 0UL)
|
||
{
|
||
return 0UL;
|
||
}
|
||
|
||
return b <= ulong.MaxValue / a
|
||
? (a * b) % modulusLocal
|
||
: MulMod64(a, b, modulusLocal);
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public readonly ulong MulMod(ulong value, ulong modulus)
|
||
{
|
||
ulong a = Low % modulus;
|
||
ulong b = value % modulus;
|
||
|
||
if (a == 0UL || b == 0UL)
|
||
{
|
||
return 0UL;
|
||
}
|
||
|
||
ulong ulongRange = ulong.MaxValue / a;
|
||
return b <= ulongRange
|
||
? (a * b) % modulus
|
||
: MulMod64(a, b, modulus);
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public readonly ulong MulModSimplified(ulong value, ulong modulus)
|
||
{
|
||
ulong modulusLocal = modulus;
|
||
ulong a = Low % modulusLocal;
|
||
ulong b = value % modulusLocal;
|
||
|
||
if (a == 0UL || b == 0UL)
|
||
{
|
||
return 0UL;
|
||
}
|
||
|
||
return MulMod64(a, b, modulusLocal);
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public readonly ulong MulModWithNativeModulo(ulong value, ulong modulus)
|
||
{
|
||
// TODO: Drop this native-modulo path from production after migrating callers to MulMod,
|
||
// which benchmarked 4-8× faster on dense operands and still wins on mixed workloads.
|
||
ulong multiplicand = Low % modulus;
|
||
var remainder = value % modulus;
|
||
|
||
if (multiplicand == 0UL || remainder == 0UL)
|
||
{
|
||
return 0UL;
|
||
}
|
||
|
||
ulong result = 0UL;
|
||
while (true)
|
||
{
|
||
ulong chunk = remainder & NativeModuloChunkMask;
|
||
if (chunk != 0UL)
|
||
{
|
||
chunk = MultiplyChunkModulo(multiplicand, chunk, modulus);
|
||
result = (result + chunk) % modulus;
|
||
}
|
||
|
||
remainder >>= NativeModuloChunkBits;
|
||
if (remainder == 0UL)
|
||
{
|
||
break;
|
||
}
|
||
|
||
multiplicand = ShiftLeftByNativeChunk(multiplicand, modulus);
|
||
}
|
||
|
||
return result;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
private void ReduceProductBitwise(ulong p3, ulong p2, ulong p1, ulong p0, GpuUInt128 modulus)
|
||
{
|
||
GpuUInt128 remainder = new();
|
||
int bit;
|
||
// TODO: Can we modify these loops to process multiple bits at a time? E.g. 64-bit chunks.
|
||
for (bit = 63; bit >= 0; bit--)
|
||
{
|
||
remainder.ShiftLeft(1);
|
||
remainder.Low |= (p3 >> bit) & 1UL;
|
||
if (remainder.CompareTo(modulus) >= 0)
|
||
{
|
||
remainder.Sub(modulus);
|
||
}
|
||
}
|
||
|
||
for (bit = 63; bit >= 0; bit--)
|
||
{
|
||
remainder.ShiftLeft(1);
|
||
remainder.Low |= (p2 >> bit) & 1UL;
|
||
if (remainder.CompareTo(modulus) >= 0)
|
||
{
|
||
remainder.Sub(modulus);
|
||
}
|
||
}
|
||
|
||
for (bit = 63; bit >= 0; bit--)
|
||
{
|
||
remainder.ShiftLeft(1);
|
||
remainder.Low |= (p1 >> bit) & 1UL;
|
||
if (remainder.CompareTo(modulus) >= 0)
|
||
{
|
||
remainder.Sub(modulus);
|
||
}
|
||
}
|
||
|
||
for (bit = 63; bit >= 0; bit--)
|
||
{
|
||
remainder.ShiftLeft(1);
|
||
remainder.Low |= (p0 >> bit) & 1UL;
|
||
if (remainder.CompareTo(modulus) >= 0)
|
||
{
|
||
remainder.Sub(modulus);
|
||
}
|
||
}
|
||
|
||
High = remainder.High;
|
||
Low = remainder.Low;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
private static ulong ShiftLeftByNativeChunk(ulong value, ulong modulus)
|
||
{
|
||
// TODO: Collapse this eight-step shift ladder into the ProcessEightBitWindows helper once it lands so
|
||
// we reuse the precomputed window residues instead of emitting `% modulus` after every shift.
|
||
value = (value << 1) % modulus;
|
||
value = (value << 1) % modulus;
|
||
value = (value << 1) % modulus;
|
||
value = (value << 1) % modulus;
|
||
value = (value << 1) % modulus;
|
||
value = (value << 1) % modulus;
|
||
value = (value << 1) % modulus;
|
||
value = (value << 1) % modulus;
|
||
|
||
return value;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
private static ulong MultiplyChunkModulo(ulong value, ulong chunk, ulong modulus)
|
||
{
|
||
ulong result = 0UL;
|
||
ulong[] nativeModuloBitMasks = NativeModuloBitMasks;
|
||
for (int bit = 0; bit < NativeModuloChunkBits; bit++)
|
||
{
|
||
if ((chunk & nativeModuloBitMasks[bit]) != 0UL)
|
||
{
|
||
result = (result + value) % modulus;
|
||
}
|
||
|
||
if (bit == NativeModuloChunkBitsMinusOne)
|
||
{
|
||
break;
|
||
}
|
||
|
||
value = (value << 1) % modulus;
|
||
}
|
||
|
||
return result;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public readonly ulong MulModMontgomery64(GpuUInt128 other, ulong modulus, ulong nPrime, ulong r2)
|
||
{
|
||
// TODO: Retire this struct-based Montgomery path from production after adopting the extension
|
||
// helper, which benchmarks 6-7× faster across dense and near-modulus operands.
|
||
ulong modulusLocal = modulus;
|
||
ulong a = Low % modulusLocal;
|
||
ulong b = other.Low % modulusLocal;
|
||
|
||
if (a == 0UL || b == 0UL)
|
||
{
|
||
return 0UL;
|
||
}
|
||
|
||
ulong aR = MontMul64(a, r2, modulusLocal, nPrime);
|
||
ulong bR = MontMul64(b, r2, modulusLocal, nPrime);
|
||
ulong cR = MontMul64(aR, bR, modulusLocal, nPrime);
|
||
return MontMul64(cR, 1UL, modulusLocal, nPrime);
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public readonly ulong MulModMontgomery64(ulong other, ulong modulus, ulong nPrime, ulong r2)
|
||
{
|
||
// TODO: Same as above—migrate callers to the scalar extension to avoid this 6-7× slowdown.
|
||
ulong modulusLocal = modulus;
|
||
ulong a = Low % modulusLocal;
|
||
ulong b = other % modulusLocal;
|
||
|
||
if (a == 0UL || b == 0UL)
|
||
{
|
||
return 0UL;
|
||
}
|
||
|
||
ulong aR = MontMul64(a, r2, modulusLocal, nPrime);
|
||
ulong bR = MontMul64(b, r2, modulusLocal, nPrime);
|
||
ulong cR = MontMul64(aR, bR, modulusLocal, nPrime);
|
||
return MontMul64(cR, 1UL, modulusLocal, nPrime);
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public void SquareMod(GpuUInt128 modulus)
|
||
{
|
||
if (modulus.High == 0UL)
|
||
{
|
||
Low = MulMod(Low, modulus.Low);
|
||
High = 0UL;
|
||
return;
|
||
}
|
||
|
||
SquareFull(this, out var p3, out var p2, out var p1, out var p0);
|
||
|
||
// TODO: This should operate on the instance itself, not on a copy. Avoid creating new instances anywhere.
|
||
GpuUInt128 remainder = new();
|
||
int bit;
|
||
// TODO: Can we modify these loops to process multiple bits at a time? E.g. 64-bit chunks.
|
||
for (bit = 63; bit >= 0; bit--)
|
||
{
|
||
remainder <<= 1;
|
||
remainder = new(remainder.High, remainder.Low | ((p3 >> bit) & 1UL));
|
||
if (remainder.CompareTo(modulus) >= 0)
|
||
{
|
||
remainder.Sub(modulus);
|
||
}
|
||
}
|
||
|
||
for (bit = 63; bit >= 0; bit--)
|
||
{
|
||
remainder <<= 1;
|
||
remainder = new(remainder.High, remainder.Low | ((p2 >> bit) & 1UL));
|
||
if (remainder.CompareTo(modulus) >= 0)
|
||
{
|
||
remainder.Sub(modulus);
|
||
}
|
||
}
|
||
|
||
for (bit = 63; bit >= 0; bit--)
|
||
{
|
||
remainder <<= 1;
|
||
remainder = new(remainder.High, remainder.Low | ((p1 >> bit) & 1UL));
|
||
if (remainder.CompareTo(modulus) >= 0)
|
||
{
|
||
remainder.Sub(modulus);
|
||
}
|
||
}
|
||
|
||
for (bit = 63; bit >= 0; bit--)
|
||
{
|
||
remainder <<= 1;
|
||
remainder = new(remainder.High, remainder.Low | ((p0 >> bit) & 1UL));
|
||
if (remainder.CompareTo(modulus) >= 0)
|
||
{
|
||
remainder.Sub(modulus);
|
||
}
|
||
}
|
||
|
||
High = remainder.High;
|
||
Low = remainder.Low;
|
||
}
|
||
|
||
public void ModPow(GpuUInt128 exponent, GpuUInt128 modulus)
|
||
{
|
||
// TODO: Swap this copy-heavy path for the pooled base/exponent ladder used in
|
||
// GpuUInt128MulModBenchmarks so the GPU shim keeps reusing buffers instead of
|
||
// allocating temporary structs during residue scans.
|
||
GpuUInt128 result = new(1UL);
|
||
GpuUInt128 baseValue = this;
|
||
|
||
// TODO: Replace the single-bit square-and-multiply loop with the 64-bit windowed
|
||
// ladder measured fastest in GpuUInt128MulModBenchmarks to align with the GPU kernel
|
||
// implementation once the shared helper lands.
|
||
while (!exponent.IsZero)
|
||
{
|
||
if ((exponent.Low & 1UL) != 0UL)
|
||
{
|
||
result.MulMod(baseValue, modulus);
|
||
}
|
||
|
||
exponent >>= 1;
|
||
baseValue.MulMod(baseValue, modulus);
|
||
}
|
||
|
||
High = result.High;
|
||
Low = result.Low;
|
||
}
|
||
|
||
public void ModPow(ulong exponent, GpuUInt128 modulus)
|
||
{
|
||
// TODO: Share the pooled ladder state from GpuUInt128MulModBenchmarks here as well so
|
||
// Lucas–Lehmer batches avoid constructing throwaway temporaries on every exponent.
|
||
GpuUInt128 result = new(1UL);
|
||
GpuUInt128 baseValue = this;
|
||
|
||
// TODO: Upgrade this loop to the same 64-bit windowed ladder proven fastest in
|
||
// GpuUInt128MulModBenchmarks so the scalar helper matches the GPU-optimized path.
|
||
while (exponent != 0UL)
|
||
{
|
||
if ((exponent & 1UL) != 0UL)
|
||
{
|
||
result.MulMod(baseValue, modulus);
|
||
}
|
||
|
||
exponent >>= 1;
|
||
baseValue.MulMod(baseValue, modulus);
|
||
}
|
||
|
||
High = result.High;
|
||
Low = result.Low;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public void ModInv(ulong modulus)
|
||
{
|
||
// TODO: Replace this Fermat inversion with the Montgomery ladder highlighted in
|
||
// GpuUInt128Montgomery64Benchmarks so we avoid instantiating a temporary modulus and
|
||
// reuse the pooled reduction constants.
|
||
ModPow(modulus - 2UL, new GpuUInt128(modulus));
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public void ModInv(GpuUInt128 modulus)
|
||
{
|
||
if (modulus.High == 0UL)
|
||
{
|
||
ModInv(modulus.Low);
|
||
return;
|
||
}
|
||
|
||
var exponent = new GpuUInt128(modulus);
|
||
exponent.Sub(2UL);
|
||
ModPow(exponent, modulus);
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public override readonly string ToString() =>
|
||
((UInt128)this).ToString();
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
private static void MultiplyFull(GpuUInt128 left, GpuUInt128 right, out ulong p3, out ulong p2, out ulong p1, out ulong p0)
|
||
{
|
||
// TODO: Reuse variables to reduce register pressure following the fused-limb layout
|
||
// from GpuUInt128MulModByLimbBenchmarks so the multiply helper matches the fastest
|
||
// GPU-compatible scalar routine.
|
||
var (h0, l0) = Mul64(left.Low, right.Low);
|
||
var (h1, l1) = Mul64(left.Low, right.High);
|
||
var (h2, l2) = Mul64(left.High, right.Low);
|
||
var (h3, l3) = Mul64(left.High, right.High);
|
||
|
||
p0 = l0;
|
||
ulong carry = 0UL;
|
||
ulong sum = h0;
|
||
sum += l1;
|
||
if (sum < l1)
|
||
{
|
||
carry++;
|
||
}
|
||
|
||
sum += l2;
|
||
if (sum < l2)
|
||
{
|
||
carry++;
|
||
}
|
||
|
||
p1 = sum;
|
||
sum = h1;
|
||
sum += h2;
|
||
ulong carry2 = sum < h2 ? 1UL : 0UL;
|
||
sum += l3;
|
||
if (sum < l3)
|
||
{
|
||
carry2++;
|
||
}
|
||
|
||
sum += carry;
|
||
if (sum < carry)
|
||
{
|
||
carry2++;
|
||
}
|
||
|
||
p2 = sum;
|
||
p3 = h3 + carry2;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
private static void MultiplyFull(GpuUInt128 left, ulong right, out ulong p3, out ulong p2, out ulong p1, out ulong p0)
|
||
{
|
||
var (highLow, lowLow) = Mul64(left.Low, right);
|
||
var (highHigh, lowHigh) = Mul64(left.High, right);
|
||
|
||
p0 = lowLow;
|
||
|
||
ulong mid = highLow + lowHigh;
|
||
ulong carry = mid < highLow ? 1UL : 0UL;
|
||
p1 = mid;
|
||
|
||
ulong upper = highHigh + carry;
|
||
p3 = upper < highHigh ? 1UL : 0UL;
|
||
p2 = upper;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
private static void Multiply(GpuUInt128 left, GpuUInt128 right, out ulong high, out ulong low)
|
||
{
|
||
// TODO: Reuse variables to reduce register pressure.
|
||
var (h0, l0) = Mul64(left.Low, right.Low);
|
||
low = l0;
|
||
// TODO: Since we ignore the first result element, can we create a version of the function which calculates and returns only the second element?
|
||
(_, l0) = Mul64(left.Low, right.High);
|
||
var (_, l2) = Mul64(left.High, right.Low);
|
||
|
||
// TODO: Why not just modify left instance directly instead using out parameters?
|
||
high = h0 + l0 + l2;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
private static (ulong High, ulong Low) Mul64(ulong left, ulong right)
|
||
{
|
||
// TODO: Reuse variables to reduce register pressure.
|
||
ulong a0 = (uint)left;
|
||
ulong a1 = left >> 32;
|
||
ulong b0 = (uint)right;
|
||
ulong b1 = right >> 32;
|
||
|
||
ulong lo = a0 * b0;
|
||
ulong mid1 = a1 * b0;
|
||
b0 = a0 * b1;
|
||
b1 *= a1;
|
||
|
||
a0 = (lo >> 32) + (uint)mid1 + (uint)b0;
|
||
a1 = (lo & 0xFFFFFFFFUL) | (a0 << 32);
|
||
b1 += (mid1 >> 32) + (b0 >> 32) + (a0 >> 32);
|
||
|
||
return (b1, a1);
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
private static void SquareFull(GpuUInt128 value, out ulong p3, out ulong p2, out ulong p1, out ulong p0)
|
||
{
|
||
// TODO: Reuse variables to reduce register pressure.
|
||
// Compute (H*2^64 + L)^2 = H^2*2^128 + 2*H*L*2^64 + L^2
|
||
ulong L = value.Low;
|
||
ulong H = value.High;
|
||
var (hLL, lLL) = Mul64(L, L); // L^2 -> (hLL,lLL)
|
||
var (hHH, lHH) = Mul64(H, H); // H^2 -> (hHH,lHH)
|
||
var (hLH, lLH) = Mul64(L, H); // L*H -> (hLH,lLH)
|
||
|
||
// double LH: (hLH,lLH) << 1
|
||
ulong dLH_low = lLH << 1;
|
||
ulong carry = (lLH >> 63) & 1UL;
|
||
ulong dLH_high = (hLH << 1) | carry;
|
||
|
||
// Assemble 256-bit result parts
|
||
p0 = lLL;
|
||
|
||
// p1 = hLL + dLH_low (with carry)
|
||
ulong sum1 = hLL + dLH_low;
|
||
ulong c1 = sum1 < hLL ? 1UL : 0UL;
|
||
p1 = sum1;
|
||
|
||
// p2 = lHH + dLH_high + c1
|
||
ulong sum2 = lHH + dLH_high;
|
||
ulong c2 = sum2 < lHH ? 1UL : 0UL;
|
||
sum2 += c1;
|
||
if (sum2 < c1)
|
||
{
|
||
c2++;
|
||
}
|
||
p2 = sum2;
|
||
|
||
// p3 = hHH + c2
|
||
p3 = hHH + c2;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
private static ulong MulMod64(ulong a, ulong b, ulong modulus)
|
||
{
|
||
ulong aReduced = a % modulus;
|
||
ulong bReduced = b % modulus;
|
||
if (aReduced == 0UL || bReduced == 0UL)
|
||
{
|
||
return 0UL;
|
||
}
|
||
|
||
// TODO: Can we identify where we deal with Mersenne exponent in EvenPerfectBitScanner and directly use MulModMersenne there, removing this branch?
|
||
if (TryGetMersenneExponent(modulus, out int exponent))
|
||
{
|
||
return MulModMersenne(aReduced, bReduced, modulus, exponent);
|
||
}
|
||
|
||
// Fallback for general moduli using the branch-free shift-add reducer.
|
||
ulong result = 0UL;
|
||
ulong x = aReduced;
|
||
ulong y = bReduced;
|
||
// TODO: Can we modify this loop to process multiple bits at a time? E.g. 64-bit chunks.
|
||
while (y != 0UL)
|
||
{
|
||
if ((y & 1UL) != 0UL)
|
||
{
|
||
result += x;
|
||
if (result >= modulus)
|
||
{
|
||
result -= modulus;
|
||
}
|
||
}
|
||
|
||
x <<= 1;
|
||
if (x >= modulus)
|
||
{
|
||
x -= modulus;
|
||
}
|
||
|
||
y >>= 1;
|
||
}
|
||
|
||
return result;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
private static ulong MulModMersenne(ulong a, ulong b, ulong modulus, int exponent)
|
||
{
|
||
// TODO: Reuse variables to reduce register pressure.
|
||
Mul64Parts(a, b, out ulong productHigh, out ulong productLow);
|
||
ulong mask = exponent == 64 ? ulong.MaxValue : modulus;
|
||
|
||
ulong currentHigh = productHigh;
|
||
ulong currentLow = productLow;
|
||
|
||
do
|
||
{
|
||
ulong lower = currentLow & mask;
|
||
ShiftRight128(currentHigh, currentLow, exponent, out ulong shiftedHigh, out ulong shiftedLow);
|
||
currentLow = lower + shiftedLow;
|
||
currentHigh = shiftedHigh;
|
||
if (currentLow < lower)
|
||
{
|
||
currentHigh++;
|
||
}
|
||
}
|
||
while (currentHigh != 0UL);
|
||
|
||
ulong result = currentLow;
|
||
// TODO: Can we modify this loop to process multiple bits at a time? E.g. 64-bit chunks.
|
||
while (result >= modulus)
|
||
{
|
||
result -= modulus;
|
||
}
|
||
|
||
return result;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
private static bool TryGetMersenneExponent(ulong modulus, out int exponent)
|
||
{
|
||
if (modulus <= 1UL)
|
||
{
|
||
exponent = 0;
|
||
return false;
|
||
}
|
||
|
||
modulus++;
|
||
if (modulus == 0UL)
|
||
{
|
||
exponent = 64;
|
||
return true;
|
||
}
|
||
|
||
if ((modulus & (modulus - 1UL)) != 0UL)
|
||
{
|
||
exponent = 0;
|
||
return false;
|
||
}
|
||
|
||
int bits = 0;
|
||
// TODO: Can we modify this loop to process multiple bits at a time? E.g. 64-bit chunks.
|
||
while ((modulus & 1UL) == 0UL)
|
||
{
|
||
modulus >>= 1;
|
||
bits++;
|
||
}
|
||
|
||
exponent = bits;
|
||
return true;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
private static void ShiftRight128(ulong high, ulong low, int shift, out ulong newHigh, out ulong newLow)
|
||
{
|
||
if (shift == 0)
|
||
{
|
||
newHigh = high;
|
||
newLow = low;
|
||
return;
|
||
}
|
||
|
||
if (shift < 64)
|
||
{
|
||
newLow = (low >> shift) | (high << (64 - shift));
|
||
newHigh = high >> shift;
|
||
return;
|
||
}
|
||
|
||
if (shift == 64)
|
||
{
|
||
newLow = high;
|
||
newHigh = 0UL;
|
||
return;
|
||
}
|
||
|
||
int extra = shift - 64;
|
||
newLow = high >> extra;
|
||
newHigh = 0UL;
|
||
}
|
||
|
||
// Montgomery core for 64-bit operands. Returns a*b*R^{-1} mod modulus.
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
private static ulong MontMul64(ulong aR, ulong bR, ulong modulus, ulong nPrime)
|
||
{
|
||
// TODO: Reuse variables to reduce register pressure.
|
||
// t = aR * bR (128-bit)
|
||
Mul64Parts(aR, bR, out ulong tHigh, out ulong tLow);
|
||
// m = (tLow * nPrime) mod 2^64 (low 64 bits only)
|
||
Mul64Parts(tLow, nPrime, out _, out ulong mLow);
|
||
// u = (t + m * modulus) >> 64
|
||
Mul64Parts(mLow, modulus, out ulong mmHigh, out ulong mmLow);
|
||
ulong carry = 0UL;
|
||
ulong low = tLow + mmLow;
|
||
if (low < tLow)
|
||
{
|
||
carry = 1UL;
|
||
}
|
||
|
||
ulong high = tHigh + mmHigh + carry;
|
||
ulong u = high; // (t + m*n) >> 64
|
||
if (u >= modulus)
|
||
{
|
||
u -= modulus;
|
||
}
|
||
|
||
return u;
|
||
}
|
||
|
||
// 64x64 -> 128 multiply into (high, low)
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
private static void Mul64Parts(ulong a, ulong b, out ulong high, out ulong low)
|
||
{
|
||
// TODO: Reuse variables to reduce register pressure.
|
||
ulong a0 = (uint)a;
|
||
ulong a1 = a >> 32;
|
||
ulong b0 = (uint)b;
|
||
ulong b1 = b >> 32;
|
||
|
||
ulong lo = a0 * b0;
|
||
ulong mid1 = a1 * b0;
|
||
ulong mid2 = a0 * b1;
|
||
ulong hi = a1 * b1;
|
||
|
||
ulong carry = (lo >> 32) + (uint)mid1 + (uint)mid2;
|
||
low = (lo & 0xFFFFFFFFUL) | (carry << 32);
|
||
hi += (mid1 >> 32) + (mid2 >> 32) + (carry >> 32);
|
||
high = hi;
|
||
}
|
||
|
||
private static ulong[] CreateNativeModuloBitMasks()
|
||
{
|
||
ulong[] masks = new ulong[NativeModuloBitMaskTableSize];
|
||
for (int bit = 0; bit < NativeModuloBitMaskTableSize; bit++)
|
||
{
|
||
masks[bit] = bit < 64 ? 1UL << bit : 0UL;
|
||
}
|
||
|
||
return masks;
|
||
}
|
||
}
|
||
|
||
// TODO: Check if the TODO below is still relevant.
|
||
|
||
// TODO(MOD-OPT): Montgomery/Barrett integration plan
|
||
// - Introduce caches of modulus-dependent constants:
|
||
// * Montgomery: n' (-(n^{-1}) mod 2^64 or 2^128), R2 = (R^2 mod n)
|
||
// * Barrett: mu = floor(2^k / n) for k ∈ {128, 192, 256}
|
||
// - Add fast-path for 64-bit NTT primes (modulus.High == 0UL) using pure 64-bit ops.
|
||
// - Expose helpers to retrieve/calc constants once per modulus and reuse in kernels.
|
||
// - Wire these into MulMod and SquareMod hot paths under feature toggles.
|
||
// - Ensure ILGPU compatibility (no BigInteger, no % inside kernels).
|
||
|