Merge pull request #130 from mlemanczyk/codex/optimize-number-types-for-performance

Add native-width Pollard-Brent variants

PerfectNumbers.Core/Cpu/MersenneNumberDivisorByDivisorCpuTester.cs

@@ -14,6 +14,8 @@ public sealed class MersenneNumberDivisorByDivisorCpuTester : IMersenneNumberDiv
     private ulong _lastStatusDivisor;
     private bool _isConfigured;
     private int _batchSize = 1_024;
+    private const int PollardBrentMaxRestarts = 4;
+    private const int PollardBrentIterationBudget = 262_144;
 
 
     public int BatchSize
@@ -174,6 +176,31 @@ public sealed class MersenneNumberDivisorByDivisorCpuTester : IMersenneNumberDiv
             return false;
         }
 
+        ulong pollardIterationCount = 0UL;
+        if (prime > 3UL)
+        {
+            var pollardBrent = new MersennePollardBrent(prime, _batchSize);
+            if (pollardBrent.IsSupported)
+            {
+                bool pollardFound = pollardBrent.TryFindFactor(
+                    allowedMax,
+                    PollardBrentMaxRestarts,
+                    PollardBrentIterationBudget,
+                    out ulong pollardFactor,
+                    out pollardIterationCount);
+
+                if (pollardFound)
+                {
+                    processedCount = pollardIterationCount;
+                    lastProcessed = pollardFactor;
+                    processedAll = true;
+                    return true;
+                }
+            }
+        }
+
+        processedCount = pollardIterationCount;
+
         UInt128 step = (UInt128)prime << 1;
         if (step == UInt128.Zero)
         {

PerfectNumbers.Core/Cpu/MersennePollardBrent.cs

@@ -0,0 +1,926 @@
+using System;
+using System.Buffers;
+using System.Numerics;
+using System.Security.Cryptography;
+using PerfectNumbers.Core;
+using UInt128 = System.UInt128;
+
+namespace PerfectNumbers.Core.Cpu;
+
+internal sealed class MersennePollardBrent
+{
+    private enum ArithmeticMode
+    {
+        Unsupported,
+        UInt64,
+        UInt128,
+        BigInteger,
+    }
+
+    private readonly ArithmeticMode _mode;
+    private readonly int _bitLength;
+    private readonly int _byteLength;
+    private readonly int _bitRemainder;
+    private readonly byte _maskByte;
+    private readonly BigInteger _modulus;
+    private readonly BigInteger _mask;
+    private readonly ulong _modulus64;
+    private readonly ulong _mask64;
+    private readonly UInt128 _modulus128;
+    private readonly UInt128 _mask128;
+    private readonly int _batchSize;
+    private readonly bool _isSupported;
+
+    public MersennePollardBrent(ulong exponent, int batchSize)
+    {
+        if (exponent < 3UL || exponent > int.MaxValue)
+        {
+            _mode = ArithmeticMode.Unsupported;
+            _isSupported = false;
+            _bitLength = 0;
+            _byteLength = 0;
+            _bitRemainder = 0;
+            _maskByte = 0;
+            _modulus = BigInteger.Zero;
+            _mask = BigInteger.Zero;
+            _modulus64 = 0UL;
+            _mask64 = 0UL;
+            _modulus128 = UInt128.Zero;
+            _mask128 = UInt128.Zero;
+            _batchSize = 0;
+            return;
+        }
+
+        _isSupported = true;
+        _bitLength = (int)exponent;
+        _byteLength = (_bitLength + 7) / 8;
+        _bitRemainder = _bitLength & 7;
+        _maskByte = _bitRemainder == 0 ? (byte)0xFF : (byte)((1 << _bitRemainder) - 1);
+        _batchSize = Math.Clamp(batchSize, 8, 1024);
+
+        if (_bitLength <= 64)
+        {
+            _mode = ArithmeticMode.UInt64;
+            if (_bitLength == 64)
+            {
+                _modulus64 = ulong.MaxValue;
+                _mask64 = ulong.MaxValue;
+            }
+            else
+            {
+                _modulus64 = (1UL << _bitLength) - 1UL;
+                _mask64 = _modulus64;
+            }
+
+            _modulus128 = UInt128.Zero;
+            _mask128 = UInt128.Zero;
+            _modulus = BigInteger.Zero;
+            _mask = BigInteger.Zero;
+            return;
+        }
+
+        if (_bitLength <= 128)
+        {
+            _mode = ArithmeticMode.UInt128;
+            if (_bitLength == 128)
+            {
+                _modulus128 = UInt128.MaxValue;
+                _mask128 = UInt128.MaxValue;
+            }
+            else
+            {
+                _modulus128 = (UInt128.One << _bitLength) - UInt128.One;
+                _mask128 = _modulus128;
+            }
+
+            _modulus64 = 0UL;
+            _mask64 = 0UL;
+            _modulus = BigInteger.Zero;
+            _mask = BigInteger.Zero;
+            return;
+        }
+
+        _mode = ArithmeticMode.BigInteger;
+        _modulus = (BigInteger.One << _bitLength) - BigInteger.One;
+        _mask = _modulus;
+        _modulus64 = 0UL;
+        _mask64 = 0UL;
+        _modulus128 = UInt128.Zero;
+        _mask128 = UInt128.Zero;
+    }
+
+    public bool IsSupported => _isSupported;
+
+    public bool TryFindFactor(
+        ulong allowedMax,
+        int maxRestarts,
+        int maxIterationsPerRestart,
+        out ulong factor,
+        out ulong iterationCount)
+    {
+        factor = 0UL;
+        iterationCount = 0UL;
+
+        if (!_isSupported)
+        {
+            return false;
+        }
+
+        if (allowedMax <= 1UL)
+        {
+            return false;
+        }
+
+        return _mode switch
+        {
+            ArithmeticMode.UInt64 => TryFindFactorUInt64(allowedMax, maxRestarts, maxIterationsPerRestart, out factor, out iterationCount),
+            ArithmeticMode.UInt128 => TryFindFactorUInt128(allowedMax, maxRestarts, maxIterationsPerRestart, out factor, out iterationCount),
+            ArithmeticMode.BigInteger => TryFindFactorBigInteger(allowedMax, maxRestarts, maxIterationsPerRestart, out factor, out iterationCount),
+            _ => false,
+        };
+    }
+
+    private bool TryFindFactorUInt64(
+        ulong allowedMax,
+        int maxRestarts,
+        int maxIterationsPerRestart,
+        out ulong factor,
+        out ulong iterationCount)
+    {
+        factor = 0UL;
+        iterationCount = 0UL;
+
+        int restarts = Math.Max(1, maxRestarts);
+        int iterationBudget = Math.Max(_batchSize, maxIterationsPerRestart);
+
+        for (int restart = 0; restart < restarts; restart++)
+        {
+            ulong x = NextRandomSeed64();
+            ulong y = x;
+            ulong c = NextRandomConstant64();
+            ulong q = 1UL;
+            ulong g = 1UL;
+            ulong ys = 0UL;
+            int r = 1;
+            ulong localIterations = 0UL;
+
+            while (g == 1UL && localIterations < (ulong)iterationBudget)
+            {
+                x = y;
+                for (int i = 0; i < r && localIterations < (ulong)iterationBudget; i++)
+                {
+                    y = Iterate64(y, c, ref localIterations);
+                }
+
+                if (localIterations >= (ulong)iterationBudget)
+                {
+                    break;
+                }
+
+                int k = 0;
+                while (k < r && g == 1UL && localIterations < (ulong)iterationBudget)
+                {
+                    ys = y;
+                    int t = Math.Min(_batchSize, r - k);
+                    for (int i = 0; i < t; i++)
+                    {
+                        y = Iterate64(y, c, ref localIterations);
+                        ulong diff = AbsSub64(y, x);
+                        if (diff == 0UL)
+                        {
+                            continue;
+                        }
+
+                        q = MulMod64(q, diff);
+                    }
+
+                    g = BinaryGcd64(q, _modulus64);
+                    k += t;
+                }
+
+                r <<= 1;
+            }
+
+            if (g == _modulus64)
+            {
+                do
+                {
+                    ys = Iterate64(ys, c, ref localIterations);
+                    ulong diff = AbsSub64(ys, x);
+                    if (diff == 0UL)
+                    {
+                        continue;
+                    }
+
+                    g = BinaryGcd64(diff, _modulus64);
+                }
+                while (g == 1UL);
+            }
+
+            iterationCount += localIterations;
+
+            if (g == 1UL || g == _modulus64)
+            {
+                continue;
+            }
+
+            if (g > 1UL && g <= allowedMax)
+            {
+                factor = g;
+                return true;
+            }
+        }
+
+        return false;
+    }
+
+    private bool TryFindFactorUInt128(
+        ulong allowedMax,
+        int maxRestarts,
+        int maxIterationsPerRestart,
+        out ulong factor,
+        out ulong iterationCount)
+    {
+        factor = 0UL;
+        iterationCount = 0UL;
+
+        int restarts = Math.Max(1, maxRestarts);
+        int iterationBudget = Math.Max(_batchSize, maxIterationsPerRestart);
+        UInt128 allowedMaxWide = (UInt128)allowedMax;
+
+        for (int restart = 0; restart < restarts; restart++)
+        {
+            UInt128 x = NextRandomSeed128();
+            UInt128 y = x;
+            UInt128 c = NextRandomConstant128();
+            UInt128 q = UInt128.One;
+            UInt128 g = UInt128.One;
+            UInt128 ys = UInt128.Zero;
+            int r = 1;
+            ulong localIterations = 0UL;
+
+            while (g == UInt128.One && localIterations < (ulong)iterationBudget)
+            {
+                x = y;
+                for (int i = 0; i < r && localIterations < (ulong)iterationBudget; i++)
+                {
+                    y = Iterate128(y, c, ref localIterations);
+                }
+
+                if (localIterations >= (ulong)iterationBudget)
+                {
+                    break;
+                }
+
+                int k = 0;
+                while (k < r && g == UInt128.One && localIterations < (ulong)iterationBudget)
+                {
+                    ys = y;
+                    int t = Math.Min(_batchSize, r - k);
+                    for (int i = 0; i < t; i++)
+                    {
+                        y = Iterate128(y, c, ref localIterations);
+                        UInt128 diff = AbsSub128(y, x);
+                        if (diff == UInt128.Zero)
+                        {
+                            continue;
+                        }
+
+                        q = MulMod128(q, diff);
+                    }
+
+                    g = q.BinaryGcd(_modulus128);
+                    k += t;
+                }
+
+                r <<= 1;
+            }
+
+            if (g == _modulus128)
+            {
+                do
+                {
+                    ys = Iterate128(ys, c, ref localIterations);
+                    UInt128 diff = AbsSub128(ys, x);
+                    if (diff == UInt128.Zero)
+                    {
+                        continue;
+                    }
+
+                    g = diff.BinaryGcd(_modulus128);
+                }
+                while (g == UInt128.One);
+            }
+
+            iterationCount += localIterations;
+
+            if (g == UInt128.One || g == _modulus128)
+            {
+                continue;
+            }
+
+            if (g > UInt128.One && g <= allowedMaxWide)
+            {
+                factor = (ulong)g;
+                return true;
+            }
+        }
+
+        return false;
+    }
+
+    private bool TryFindFactorBigInteger(
+        ulong allowedMax,
+        int maxRestarts,
+        int maxIterationsPerRestart,
+        out ulong factor,
+        out ulong iterationCount)
+    {
+        factor = 0UL;
+        iterationCount = 0UL;
+
+        int restarts = Math.Max(1, maxRestarts);
+        int iterationBudget = Math.Max(_batchSize, maxIterationsPerRestart);
+
+        for (int restart = 0; restart < restarts; restart++)
+        {
+            BigInteger x = NextRandomSeedBigInteger();
+            BigInteger y = x;
+            BigInteger c = NextRandomConstantBigInteger();
+            BigInteger q = BigInteger.One;
+            BigInteger g = BigInteger.One;
+            BigInteger ys = BigInteger.Zero;
+            int r = 1;
+            ulong localIterations = 0UL;
+
+            while (g.IsOne && localIterations < (ulong)iterationBudget)
+            {
+                x = y;
+                for (int i = 0; i < r && localIterations < (ulong)iterationBudget; i++)
+                {
+                    y = IterateBigInteger(y, c, ref localIterations);
+                }
+
+                if (localIterations >= (ulong)iterationBudget)
+                {
+                    break;
+                }
+
+                int k = 0;
+                while (k < r && g.IsOne && localIterations < (ulong)iterationBudget)
+                {
+                    ys = y;
+                    int t = Math.Min(_batchSize, r - k);
+                    for (int i = 0; i < t; i++)
+                    {
+                        y = IterateBigInteger(y, c, ref localIterations);
+                        BigInteger diff = AbsSubBigInteger(y, x);
+                        if (diff.IsZero)
+                        {
+                            continue;
+                        }
+
+                        q = MulModBigInteger(q, diff);
+                    }
+
+                    g = BigInteger.GreatestCommonDivisor(q, _modulus);
+                    k += t;
+                }
+
+                r <<= 1;
+            }
+
+            if (g == _modulus)
+            {
+                do
+                {
+                    ys = IterateBigInteger(ys, c, ref localIterations);
+                    BigInteger diff = AbsSubBigInteger(ys, x);
+                    if (diff.IsZero)
+                    {
+                        continue;
+                    }
+
+                    g = BigInteger.GreatestCommonDivisor(diff, _modulus);
+                }
+                while (g.IsOne);
+            }
+
+            iterationCount += localIterations;
+
+            if (g.IsOne || g == _modulus)
+            {
+                continue;
+            }
+
+            if (g <= ulong.MaxValue)
+            {
+                ulong candidate = (ulong)g;
+                if (candidate > 1UL && candidate <= allowedMax)
+                {
+                    factor = candidate;
+                    return true;
+                }
+            }
+        }
+
+        return false;
+    }
+
+    private ulong Iterate64(ulong value, ulong c, ref ulong iterationCounter)
+    {
+        iterationCounter++;
+        value = SquareMod64(value);
+        value = AddMod64(value, c);
+        return value;
+    }
+
+    private UInt128 Iterate128(UInt128 value, UInt128 c, ref ulong iterationCounter)
+    {
+        iterationCounter++;
+        value = SquareMod128(value);
+        value = AddMod128(value, c);
+        return value;
+    }
+
+    private BigInteger IterateBigInteger(BigInteger value, BigInteger c, ref ulong iterationCounter)
+    {
+        iterationCounter++;
+        value = SquareModBigInteger(value);
+        value = AddModBigInteger(value, c);
+        return value;
+    }
+
+    private ulong AddMod64(ulong a, ulong b)
+    {
+        UInt128 sum = (UInt128)a + b;
+        return Reduce64(sum);
+    }
+
+    private UInt128 AddMod128(UInt128 a, UInt128 b)
+    {
+        UInt128 sum = a + b;
+        bool carryOut = sum < a;
+        if (!carryOut && sum < _modulus128)
+        {
+            return sum;
+        }
+
+        ulong p3 = 0UL;
+        ulong p2 = carryOut ? 1UL : 0UL;
+        ulong p1 = (ulong)(sum >> 64);
+        ulong p0 = (ulong)sum;
+        return ReduceProductBitwise(p3, p2, p1, p0, _modulus128);
+    }
+
+    private BigInteger AddModBigInteger(BigInteger a, BigInteger b)
+    {
+        return ReduceBigInteger(a + b);
+    }
+
+    private ulong SquareMod64(ulong value)
+    {
+        UInt128 square = (UInt128)value * value;
+        return Reduce64(square);
+    }
+
+    private UInt128 SquareMod128(UInt128 value)
+    {
+        return MulMod128(value, value);
+    }
+
+    private BigInteger SquareModBigInteger(BigInteger value)
+    {
+        return ReduceBigInteger(value * value);
+    }
+
+    private ulong MulMod64(ulong a, ulong b)
+    {
+        UInt128 product = (UInt128)a * b;
+        return Reduce64(product);
+    }
+
+    private UInt128 MulMod128(UInt128 a, UInt128 b)
+    {
+        MultiplyFull(a, b, out ulong p3, out ulong p2, out ulong p1, out ulong p0);
+        return ReduceProductBitwise(p3, p2, p1, p0, _modulus128);
+    }
+
+    private BigInteger MulModBigInteger(BigInteger a, BigInteger b)
+    {
+        if (b == _modulus)
+        {
+            return a;
+        }
+
+        if (a == _modulus)
+        {
+            a = BigInteger.Zero;
+        }
+
+        return ReduceBigInteger(a * b);
+    }
+
+    private ulong Reduce64(UInt128 value)
+    {
+        if (_bitLength == 64)
+        {
+            UInt128 combined = (UInt128)(ulong)value + (value >> 64);
+            UInt128 modulus = _modulus64;
+            if (combined >= modulus)
+            {
+                combined -= modulus;
+                if (combined >= modulus)
+                {
+                    combined -= modulus;
+                }
+            }
+
+            return (ulong)combined;
+        }
+
+        UInt128 mask = _mask64;
+        UInt128 modulusValue = _modulus64;
+        UInt128 working = value;
+        while (true)
+        {
+            UInt128 high = working >> _bitLength;
+            if (high == UInt128.Zero)
+            {
+                break;
+            }
+
+            working = (working & mask) + high;
+        }
+
+        if (working >= modulusValue)
+        {
+            working -= modulusValue;
+            if (working >= modulusValue)
+            {
+                working %= modulusValue;
+                if (working >= modulusValue)
+                {
+                    working -= modulusValue;
+                }
+            }
+        }
+
+        return (ulong)working;
+    }
+
+    private BigInteger ReduceBigInteger(BigInteger value)
+    {
+        if (value.Sign < 0)
+        {
+            value %= _modulus;
+            if (value.Sign < 0)
+            {
+                value += _modulus;
+            }
+
+            return value;
+        }
+
+        while (true)
+        {
+            BigInteger high = value >> _bitLength;
+            if (high.IsZero)
+            {
+                break;
+            }
+
+            value &= _mask;
+            value += high;
+        }
+
+        if (value >= _modulus)
+        {
+            value -= _modulus;
+            if (value >= _modulus)
+            {
+                value %= _modulus;
+                if (value.Sign < 0)
+                {
+                    value += _modulus;
+                }
+            }
+        }
+
+        return value;
+    }
+
+    private ulong NextRandomSeed64()
+    {
+        ulong value;
+        do
+        {
+            value = NextRandomBelowModulus64();
+        }
+        while (value <= 1UL);
+
+        return value;
+    }
+
+    private UInt128 NextRandomSeed128()
+    {
+        UInt128 value;
+        do
+        {
+            value = NextRandomBelowModulus128();
+        }
+        while (value <= UInt128.One);
+
+        return value;
+    }
+
+    private BigInteger NextRandomSeedBigInteger()
+    {
+        BigInteger value;
+        do
+        {
+            value = NextRandomBelowModulusBigInteger();
+        }
+        while (value <= BigInteger.One);
+
+        return value;
+    }
+
+    private ulong NextRandomConstant64()
+    {
+        ulong value;
+        do
+        {
+            value = NextRandomBelowModulus64();
+        }
+        while (value == 0UL);
+
+        return value;
+    }
+
+    private UInt128 NextRandomConstant128()
+    {
+        UInt128 value;
+        do
+        {
+            value = NextRandomBelowModulus128();
+        }
+        while (value == UInt128.Zero);
+
+        return value;
+    }
+
+    private BigInteger NextRandomConstantBigInteger()
+    {
+        BigInteger value;
+        do
+        {
+            value = NextRandomBelowModulusBigInteger();
+        }
+        while (value.IsZero);
+
+        return value;
+    }
+
+    private ulong NextRandomBelowModulus64()
+    {
+        if (_byteLength == 0)
+        {
+            return 0UL;
+        }
+
+        Span<byte> buffer = stackalloc byte[_byteLength];
+        ulong candidate;
+        do
+        {
+            RandomNumberGenerator.Fill(buffer);
+            if (_bitRemainder != 0)
+            {
+                buffer[_byteLength - 1] &= _maskByte;
+            }
+
+            candidate = (ulong)ReadUInt128LittleEndian(buffer);
+        }
+        while (candidate >= _modulus64);
+
+        return candidate;
+    }
+
+    private UInt128 NextRandomBelowModulus128()
+    {
+        if (_byteLength == 0)
+        {
+            return UInt128.Zero;
+        }
+
+        Span<byte> buffer = stackalloc byte[_byteLength];
+        UInt128 candidate;
+        do
+        {
+            RandomNumberGenerator.Fill(buffer);
+            if (_bitRemainder != 0)
+            {
+                buffer[_byteLength - 1] &= _maskByte;
+            }
+
+            candidate = ReadUInt128LittleEndian(buffer);
+        }
+        while (candidate >= _modulus128);
+
+        return candidate;
+    }
+
+    private BigInteger NextRandomBelowModulusBigInteger()
+    {
+        if (_byteLength == 0)
+        {
+            return BigInteger.Zero;
+        }
+
+        byte[] buffer = ArrayPool<byte>.Shared.Rent(_byteLength);
+        try
+        {
+            BigInteger value;
+            do
+            {
+                RandomNumberGenerator.Fill(buffer.AsSpan(0, _byteLength));
+                if (_bitRemainder != 0)
+                {
+                    buffer[_byteLength - 1] &= _maskByte;
+                }
+
+                value = new BigInteger(buffer, isUnsigned: true, isBigEndian: false);
+            }
+            while (value >= _modulus);
+
+            return value;
+        }
+        finally
+        {
+            ArrayPool<byte>.Shared.Return(buffer, clearArray: true);
+        }
+    }
+
+    private static ulong AbsSub64(ulong a, ulong b)
+    {
+        return a >= b ? a - b : b - a;
+    }
+
+    private static UInt128 AbsSub128(UInt128 a, UInt128 b)
+    {
+        return a >= b ? a - b : b - a;
+    }
+
+    private static BigInteger AbsSubBigInteger(BigInteger a, BigInteger b)
+    {
+        BigInteger diff = a - b;
+        return diff.Sign < 0 ? -diff : diff;
+    }
+
+    private static ulong BinaryGcd64(ulong u, ulong v)
+    {
+        if (u == 0UL)
+        {
+            return v;
+        }
+
+        if (v == 0UL)
+        {
+            return u;
+        }
+
+        int shift = BitOperations.TrailingZeroCount(u | v);
+        u >>= BitOperations.TrailingZeroCount(u);
+        do
+        {
+            v >>= BitOperations.TrailingZeroCount(v);
+            if (u > v)
+            {
+                (u, v) = (v, u);
+            }
+
+            v -= u;
+        }
+        while (v != 0UL);
+
+        return u << shift;
+    }
+
+    private static void MultiplyFull(UInt128 left, UInt128 right, out ulong p3, out ulong p2, out ulong p1, out ulong p0)
+    {
+        ulong leftLow = (ulong)left;
+        ulong leftHigh = (ulong)(left >> 64);
+        ulong rightLow = (ulong)right;
+        ulong rightHigh = (ulong)(right >> 64);
+
+        (ulong High, ulong Low) partial0 = Multiply64(leftLow, rightLow);
+        (ulong High, ulong Low) partial1 = Multiply64(leftLow, rightHigh);
+        (ulong High, ulong Low) partial2 = Multiply64(leftHigh, rightLow);
+        (ulong High, ulong Low) partial3 = Multiply64(leftHigh, rightHigh);
+
+        p0 = partial0.Low;
+        ulong carry = partial0.High;
+        ulong sum = carry + partial1.Low;
+        ulong carryMid = sum < partial1.Low ? 1UL : 0UL;
+        sum += partial2.Low;
+        if (sum < partial2.Low)
+        {
+            carryMid++;
+        }
+
+        p1 = sum;
+        sum = partial1.High + partial2.High;
+        ulong carryHigh = sum < partial2.High ? 1UL : 0UL;
+        sum += partial3.Low;
+        if (sum < partial3.Low)
+        {
+            carryHigh++;
+        }
+
+        sum += carryMid;
+        if (sum < carryMid)
+        {
+            carryHigh++;
+        }
+
+        p2 = sum;
+        p3 = partial3.High + carryHigh;
+    }
+
+    private static (ulong High, ulong Low) Multiply64(ulong left, ulong right)
+    {
+        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);
+    }
+
+    private static UInt128 ReduceProductBitwise(ulong p3, ulong p2, ulong p1, ulong p0, UInt128 modulus)
+    {
+        UInt128 remainder = UInt128.Zero;
+        for (int bit = 63; bit >= 0; bit--)
+        {
+            remainder <<= 1;
+            remainder |= (UInt128)((p3 >> bit) & 1UL);
+            if (remainder >= modulus)
+            {
+                remainder -= modulus;
+            }
+        }
+
+        for (int bit = 63; bit >= 0; bit--)
+        {
+            remainder <<= 1;
+            remainder |= (UInt128)((p2 >> bit) & 1UL);
+            if (remainder >= modulus)
+            {
+                remainder -= modulus;
+            }
+        }
+
+        for (int bit = 63; bit >= 0; bit--)
+        {
+            remainder <<= 1;
+            remainder |= (UInt128)((p1 >> bit) & 1UL);
+            if (remainder >= modulus)
+            {
+                remainder -= modulus;
+            }
+        }
+
+        for (int bit = 63; bit >= 0; bit--)
+        {
+            remainder <<= 1;
+            remainder |= (UInt128)((p0 >> bit) & 1UL);
+            if (remainder >= modulus)
+            {
+                remainder -= modulus;
+            }
+        }
+
+        return remainder;
+    }
+
+    private static UInt128 ReadUInt128LittleEndian(ReadOnlySpan<byte> buffer)
+    {
+        UInt128 value = UInt128.Zero;
+        for (int i = buffer.Length - 1; i >= 0; i--)
+        {
+            value <<= 8;
+            value |= buffer[i];
+        }
+
+        return value;
+    }
+}