858 lines
24 KiB
C#
858 lines
24 KiB
C#
using System.Buffers;
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using System.Buffers.Binary;
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using System.Collections.Generic;
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using System.Diagnostics;
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using System.Numerics;
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using System.Runtime.CompilerServices;
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using System.Security.Cryptography;
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using System.Threading;
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namespace PerfectNumbers.Core;
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internal static class PrimeOrderCalculator
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{
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internal enum PrimeOrderStatus
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{
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Found,
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HeuristicUnresolved,
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}
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internal readonly struct PrimeOrderResult
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{
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public PrimeOrderResult(PrimeOrderStatus status, ulong order)
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{
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Status = status;
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Order = order;
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}
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public PrimeOrderStatus Status { get; }
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public ulong Order { get; }
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}
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internal enum PrimeOrderMode
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{
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Heuristic,
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Strict,
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}
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internal readonly struct PrimeOrderSearchConfig
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{
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public PrimeOrderSearchConfig(uint smallFactorLimit, int pollardRhoMilliseconds, int maxPowChecks, PrimeOrderMode mode)
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{
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SmallFactorLimit = smallFactorLimit;
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PollardRhoMilliseconds = pollardRhoMilliseconds;
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MaxPowChecks = maxPowChecks;
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Mode = mode;
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}
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public uint SmallFactorLimit { get; }
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public int PollardRhoMilliseconds { get; }
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public int MaxPowChecks { get; }
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public PrimeOrderMode Mode { get; }
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public static PrimeOrderSearchConfig HeuristicDefault => new(100_000, 1, 24, PrimeOrderMode.Heuristic);
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public static PrimeOrderSearchConfig StrictDefault => new(1_000_000, 0, 0, PrimeOrderMode.Strict);
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}
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public static PrimeOrderResult Calculate(ulong prime, ulong? previousOrder, PrimeOrderSearchConfig config)
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{
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if (prime <= 3UL)
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{
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return new PrimeOrderResult(PrimeOrderStatus.Found, prime == 3UL ? 2UL : 1UL);
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}
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ulong phi = prime - 1UL;
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MontgomeryDivisorData divisorData = MontgomeryDivisorDataCache.Get(prime);
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PartialFactorResult phiFactors = PartialFactor(phi, config);
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if (phiFactors.Factors is null)
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{
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return FinishStrictly(prime, config.Mode);
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}
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if (phiFactors.FullyFactored && TrySpecialMax(phi, prime, phiFactors, divisorData))
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{
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return new PrimeOrderResult(PrimeOrderStatus.Found, phi);
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}
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ulong candidateOrder = InitializeStartingOrder(prime, phi, divisorData);
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candidateOrder = ExponentLowering(candidateOrder, prime, phiFactors, divisorData);
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if (TryConfirmOrder(prime, candidateOrder, divisorData, config))
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{
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return new PrimeOrderResult(PrimeOrderStatus.Found, candidateOrder);
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}
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if (config.Mode == PrimeOrderMode.Strict)
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{
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return FinishStrictly(prime, PrimeOrderMode.Strict);
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}
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if (TryHeuristicFinish(prime, candidateOrder, previousOrder, divisorData, config, phiFactors, out ulong order))
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{
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return new PrimeOrderResult(PrimeOrderStatus.Found, order);
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}
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return new PrimeOrderResult(PrimeOrderStatus.HeuristicUnresolved, 0UL);
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}
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private static PrimeOrderResult FinishStrictly(ulong prime, PrimeOrderMode mode)
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{
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ulong strictOrder = CalculateByDoubling(prime);
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return new PrimeOrderResult(mode == PrimeOrderMode.Strict ? PrimeOrderStatus.Found : PrimeOrderStatus.HeuristicUnresolved, strictOrder);
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}
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private static bool TrySpecialMax(ulong phi, ulong prime, PartialFactorResult factors, in MontgomeryDivisorData divisorData)
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{
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ReadOnlySpan<FactorEntry> factorSpan = factors.Factors;
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int length = factors.Count;
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for (int i = 0; i < length; i++)
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{
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ulong factor = factorSpan[i].Value;
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ulong reduced = phi / factor;
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if (reduced.Pow2MontgomeryModWindowed(divisorData, keepMontgomery: false) == 1UL)
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{
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return false;
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}
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}
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return true;
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}
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private static ulong InitializeStartingOrder(ulong prime, ulong phi, in MontgomeryDivisorData divisorData)
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{
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ulong order = phi;
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if ((prime & 7UL) == 1UL || (prime & 7UL) == 7UL)
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{
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ulong half = phi >> 1;
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if (half.Pow2MontgomeryModWindowed(divisorData, keepMontgomery: false) == 1UL)
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{
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order = half;
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}
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}
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return order;
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}
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private static ulong ExponentLowering(ulong order, ulong prime, PartialFactorResult factors, in MontgomeryDivisorData divisorData)
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{
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FactorEntry[]? tempArray = null;
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try
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{
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ReadOnlySpan<FactorEntry> factorSpan = factors.Factors;
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int length = factors.Count;
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tempArray = ArrayPool<FactorEntry>.Shared.Rent(length + 1);
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Span<FactorEntry> buffer = tempArray;
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factorSpan.CopyTo(buffer);
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if (!factors.FullyFactored && factors.Cofactor > 1UL && PrimeTester.IsPrimeInternal(factors.Cofactor, CancellationToken.None))
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{
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buffer[length] = new FactorEntry(factors.Cofactor, 1);
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length++;
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}
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buffer.Slice(0, length).Sort(static (a, b) => a.Value.CompareTo(b.Value));
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for (int i = 0; i < length; i++)
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{
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ulong primeFactor = buffer[i].Value;
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int exponent = buffer[i].Exponent;
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for (int iteration = 0; iteration < exponent; iteration++)
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{
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if ((order % primeFactor) != 0UL)
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{
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break;
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}
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ulong reduced = order / primeFactor;
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if (reduced.Pow2MontgomeryModWindowed(divisorData, keepMontgomery: false) == 1UL)
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{
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order = reduced;
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continue;
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}
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break;
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}
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}
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return order;
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}
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finally
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{
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if (tempArray is not null)
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{
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ArrayPool<FactorEntry>.Shared.Return(tempArray, clearArray: false);
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}
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}
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}
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private static bool TryConfirmOrder(ulong prime, ulong order, in MontgomeryDivisorData divisorData, PrimeOrderSearchConfig config)
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{
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if (order == 0UL)
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{
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return false;
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}
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if (order.Pow2MontgomeryModWindowed(divisorData, keepMontgomery: false) != 1UL)
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{
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return false;
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}
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PartialFactorResult factorization = PartialFactor(order, config);
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if (factorization.Factors is null)
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{
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return false;
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}
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if (!factorization.FullyFactored)
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{
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if (factorization.Cofactor <= 1UL)
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{
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return false;
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}
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if (!PrimeTester.IsPrimeInternal(factorization.Cofactor, CancellationToken.None))
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{
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return false;
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}
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factorization = factorization.WithAdditionalPrime(factorization.Cofactor);
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}
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ReadOnlySpan<FactorEntry> span = factorization.Factors;
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int length = factorization.Count;
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for (int i = 0; i < length; i++)
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{
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ulong primeFactor = span[i].Value;
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ulong reduced = order;
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for (int iteration = 0; iteration < span[i].Exponent; iteration++)
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{
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if ((reduced % primeFactor) != 0UL)
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{
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break;
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}
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reduced /= primeFactor;
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if (reduced.Pow2MontgomeryModWindowed(divisorData, keepMontgomery: false) == 1UL)
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{
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return false;
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}
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}
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}
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return true;
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}
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private static bool TryHeuristicFinish(
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ulong prime,
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ulong order,
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ulong? previousOrder,
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in MontgomeryDivisorData divisorData,
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PrimeOrderSearchConfig config,
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PartialFactorResult phiFactors,
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out ulong result)
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{
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result = 0UL;
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if (order <= 1UL)
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{
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return false;
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}
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PartialFactorResult orderFactors = PartialFactor(order, config);
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if (orderFactors.Factors is null)
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{
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return false;
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}
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if (!orderFactors.FullyFactored)
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{
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if (orderFactors.Cofactor <= 1UL)
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{
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return false;
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}
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if (!PrimeTester.IsPrimeInternal(orderFactors.Cofactor, CancellationToken.None))
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{
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return false;
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}
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orderFactors = orderFactors.WithAdditionalPrime(orderFactors.Cofactor);
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}
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int capacity = config.MaxPowChecks <= 0 ? 64 : config.MaxPowChecks * 4;
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List<ulong> candidates = new(capacity);
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FactorEntry[] factorArray = orderFactors.Factors!;
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BuildCandidates(order, factorArray, orderFactors.Count, candidates, capacity);
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if (candidates.Count == 0)
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{
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return false;
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}
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SortCandidates(prime, previousOrder, candidates);
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int powBudget = config.MaxPowChecks <= 0 ? candidates.Count : config.MaxPowChecks;
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int powUsed = 0;
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int candidateCount = candidates.Count;
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for (int i = 0; i < candidateCount; i++)
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{
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if (powUsed >= powBudget)
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{
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break;
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}
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ulong candidate = candidates[i];
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powUsed++;
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if (candidate.Pow2MontgomeryModWindowed(divisorData, keepMontgomery: false) != 1UL)
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{
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continue;
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}
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if (!TryConfirmCandidate(prime, candidate, divisorData, config, ref powUsed, powBudget))
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{
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continue;
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}
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result = candidate;
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return true;
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}
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return false;
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}
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private static void SortCandidates(ulong prime, ulong? previousOrder, List<ulong> candidates)
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{
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ulong previous = previousOrder ?? 0UL;
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int previousGroup = previousOrder.HasValue ? GetGroup(previousOrder.Value, prime) : 1;
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bool hasPrevious = previousOrder.HasValue;
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candidates.Sort((x, y) =>
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{
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CandidateKey keyX = BuildKey(x, prime, previous, previousGroup, hasPrevious);
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CandidateKey keyY = BuildKey(y, prime, previous, previousGroup, hasPrevious);
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int compare = keyX.Primary.CompareTo(keyY.Primary);
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if (compare != 0)
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{
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return compare;
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}
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compare = keyX.Secondary.CompareTo(keyY.Secondary);
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if (compare != 0)
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{
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return compare;
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}
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return keyX.Tertiary.CompareTo(keyY.Tertiary);
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});
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}
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private static CandidateKey BuildKey(ulong value, ulong prime, ulong previous, int previousGroup, bool hasPrevious)
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{
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int group = GetGroup(value, prime);
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if (group == 0)
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{
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return new CandidateKey(int.MaxValue, long.MaxValue, long.MaxValue);
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}
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bool isGe = !hasPrevious || value >= previous;
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int primary = ComputePrimary(group, isGe, previousGroup);
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long secondary;
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long tertiary;
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if (group == 3)
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{
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secondary = -(long)value;
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tertiary = -(long)value;
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}
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else
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{
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ulong reference = hasPrevious ? previous : 0UL;
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ulong distance = hasPrevious ? (value > reference ? value - reference : reference - value) : value;
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secondary = (long)distance;
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tertiary = (long)value;
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}
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return new CandidateKey(primary, secondary, tertiary);
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}
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private static int ComputePrimary(int group, bool isGe, int previousGroup)
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{
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int groupOffset = group switch
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{
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1 => 0,
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2 => 2,
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3 => 4,
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_ => 6,
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};
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if (group == previousGroup)
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{
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if (group == 3)
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{
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return groupOffset + (isGe ? 0 : 3);
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}
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return groupOffset + (isGe ? 0 : 1);
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}
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return groupOffset + (isGe ? 0 : 1);
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}
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private static int GetGroup(ulong value, ulong prime)
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{
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ulong threshold1 = prime >> 3;
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if (value <= threshold1)
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{
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return 1;
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}
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ulong threshold2 = prime >> 2;
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if (value <= threshold2)
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{
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return 2;
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}
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ulong threshold3 = (prime * 3UL) >> 3;
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if (value <= threshold3)
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{
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return 3;
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}
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return 0;
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}
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private static void BuildCandidates(ulong order, FactorEntry[] factors, int count, List<ulong> candidates, int limit)
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{
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if (count == 0)
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{
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return;
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}
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Span<FactorEntry> buffer = factors.AsSpan(0, count);
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buffer.Sort(static (a, b) => a.Value.CompareTo(b.Value));
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BuildCandidatesRecursive(order, buffer, 0, 1UL, candidates, limit);
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}
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private static void BuildCandidatesRecursive(ulong order, ReadOnlySpan<FactorEntry> factors, int index, ulong divisorProduct, List<ulong> candidates, int limit)
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{
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if (candidates.Count >= limit)
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{
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return;
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}
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if (index >= factors.Length)
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{
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if (divisorProduct == 1UL || divisorProduct == order)
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{
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return;
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}
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ulong candidate = order / divisorProduct;
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if (candidate > 1UL && candidate < order)
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{
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candidates.Add(candidate);
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}
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return;
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}
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FactorEntry factor = factors[index];
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ulong primeFactor = factor.Value;
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ulong contribution = 1UL;
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for (int exponent = 0; exponent <= factor.Exponent; exponent++)
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{
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ulong nextDivisor = divisorProduct * contribution;
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if (nextDivisor > order)
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{
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break;
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}
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BuildCandidatesRecursive(order, factors, index + 1, nextDivisor, candidates, limit);
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if (candidates.Count >= limit)
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{
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return;
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}
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if (exponent == factor.Exponent)
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{
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break;
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}
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if (contribution > order / primeFactor)
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{
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break;
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}
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contribution *= primeFactor;
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}
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}
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private static bool TryConfirmCandidate(ulong prime, ulong candidate, in MontgomeryDivisorData divisorData, PrimeOrderSearchConfig config, ref int powUsed, int powBudget)
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{
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PartialFactorResult factorization = PartialFactor(candidate, config);
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if (factorization.Factors is null)
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{
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return false;
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}
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if (!factorization.FullyFactored)
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{
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if (factorization.Cofactor <= 1UL)
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{
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return false;
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}
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if (!PrimeTester.IsPrimeInternal(factorization.Cofactor, CancellationToken.None))
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{
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return false;
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}
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factorization = factorization.WithAdditionalPrime(factorization.Cofactor);
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}
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ReadOnlySpan<FactorEntry> span = factorization.Factors;
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int length = factorization.Count;
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for (int i = 0; i < length; i++)
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{
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ulong primeFactor = span[i].Value;
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ulong reduced = candidate;
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for (int iteration = 0; iteration < span[i].Exponent; iteration++)
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{
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if ((reduced % primeFactor) != 0UL)
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{
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break;
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}
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reduced /= primeFactor;
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if (powUsed >= powBudget && powBudget > 0)
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{
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return false;
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}
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powUsed++;
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if (reduced.Pow2MontgomeryModWindowed(divisorData, keepMontgomery: false) == 1UL)
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{
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return false;
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}
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}
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}
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return true;
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}
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private static PartialFactorResult PartialFactor(ulong value, PrimeOrderSearchConfig config)
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{
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if (value <= 1UL)
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{
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return PartialFactorResult.Empty;
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}
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Dictionary<ulong, int> counts = new(capacity: 8);
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ulong remaining = value;
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uint[] primes = PrimesGenerator.SmallPrimes;
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ulong[] squares = PrimesGenerator.SmallPrimesPow2;
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int primeCount = primes.Length;
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uint limit = config.SmallFactorLimit == 0 ? uint.MaxValue : config.SmallFactorLimit;
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for (int i = 0; i < primeCount; i++)
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{
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uint primeCandidate = primes[i];
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if (primeCandidate > limit)
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{
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break;
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}
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if (squares[i] > remaining)
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{
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break;
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}
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ulong primeValue = primeCandidate;
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if ((remaining % primeValue) != 0UL)
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{
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continue;
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}
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int exponent = 0;
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do
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{
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remaining /= primeValue;
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exponent++;
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}
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while ((remaining % primeValue) == 0UL);
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counts[primeValue] = exponent;
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}
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List<ulong> pending = new();
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if (remaining > 1UL)
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{
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pending.Add(remaining);
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}
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if (config.PollardRhoMilliseconds > 0 && pending.Count > 0)
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{
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Stopwatch stopwatch = Stopwatch.StartNew();
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long budgetTicks = TimeSpan.FromMilliseconds(config.PollardRhoMilliseconds).Ticks;
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Stack<ulong> stack = new();
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|
stack.Push(remaining);
|
|
pending.Clear();
|
|
|
|
while (stack.Count > 0)
|
|
{
|
|
ulong composite = stack.Pop();
|
|
if (composite == 1UL)
|
|
{
|
|
continue;
|
|
}
|
|
|
|
if (PrimeTester.IsPrimeInternal(composite, CancellationToken.None))
|
|
{
|
|
AddFactor(counts, composite, 1);
|
|
continue;
|
|
}
|
|
|
|
if (stopwatch.ElapsedTicks > budgetTicks)
|
|
{
|
|
pending.Add(composite);
|
|
continue;
|
|
}
|
|
|
|
if (!TryPollardRho(composite, stopwatch, budgetTicks, out ulong factor))
|
|
{
|
|
pending.Add(composite);
|
|
continue;
|
|
}
|
|
|
|
ulong quotient = composite / factor;
|
|
stack.Push(factor);
|
|
stack.Push(quotient);
|
|
}
|
|
}
|
|
|
|
ulong cofactor = 1UL;
|
|
int pendingCount = pending.Count;
|
|
for (int i = 0; i < pendingCount; i++)
|
|
{
|
|
ulong composite = pending[i];
|
|
if (PrimeTester.IsPrimeInternal(composite, CancellationToken.None))
|
|
{
|
|
AddFactor(counts, composite, 1);
|
|
}
|
|
else
|
|
{
|
|
cofactor = checked(cofactor * composite);
|
|
}
|
|
}
|
|
|
|
if (counts.Count == 0 && cofactor == value)
|
|
{
|
|
return new PartialFactorResult(null, value, false, 0);
|
|
}
|
|
|
|
FactorEntry[] factors = ArrayPool<FactorEntry>.Shared.Rent(counts.Count);
|
|
int index = 0;
|
|
foreach (KeyValuePair<ulong, int> entry in counts)
|
|
{
|
|
factors[index] = new FactorEntry(entry.Key, entry.Value);
|
|
index++;
|
|
}
|
|
|
|
Span<FactorEntry> span = factors.AsSpan(0, index);
|
|
span.Sort(static (a, b) => a.Value.CompareTo(b.Value));
|
|
|
|
FactorEntry[] resultArray = new FactorEntry[index];
|
|
span.CopyTo(resultArray);
|
|
ArrayPool<FactorEntry>.Shared.Return(factors, clearArray: false);
|
|
|
|
bool fullyFactored = cofactor == 1UL;
|
|
return new PartialFactorResult(resultArray, cofactor, fullyFactored, index);
|
|
}
|
|
|
|
private static bool TryPollardRho(ulong n, Stopwatch stopwatch, long budgetTicks, out ulong factor)
|
|
{
|
|
factor = 0UL;
|
|
if ((n & 1UL) == 0UL)
|
|
{
|
|
factor = 2UL;
|
|
return true;
|
|
}
|
|
|
|
Span<byte> buffer = stackalloc byte[8];
|
|
|
|
while (true)
|
|
{
|
|
if (stopwatch.ElapsedTicks > budgetTicks)
|
|
{
|
|
return false;
|
|
}
|
|
|
|
RandomNumberGenerator.Fill(buffer);
|
|
ulong c = (BinaryPrimitives.ReadUInt64LittleEndian(buffer) % (n - 1UL)) + 1UL;
|
|
RandomNumberGenerator.Fill(buffer);
|
|
ulong x = (BinaryPrimitives.ReadUInt64LittleEndian(buffer) % (n - 2UL)) + 2UL;
|
|
ulong y = x;
|
|
ulong d = 1UL;
|
|
|
|
while (d == 1UL)
|
|
{
|
|
if (stopwatch.ElapsedTicks > budgetTicks)
|
|
{
|
|
return false;
|
|
}
|
|
|
|
x = AdvancePolynomial(x, c, n);
|
|
y = AdvancePolynomial(y, c, n);
|
|
y = AdvancePolynomial(y, c, n);
|
|
ulong diff = x > y ? x - y : y - x;
|
|
d = BinaryGcd(diff, n);
|
|
}
|
|
|
|
if (d != n)
|
|
{
|
|
factor = d;
|
|
return true;
|
|
}
|
|
}
|
|
}
|
|
|
|
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
|
private static ulong AdvancePolynomial(ulong x, ulong c, ulong modulus)
|
|
{
|
|
UInt128 value = (UInt128)x * x + c;
|
|
return (ulong)(value % modulus);
|
|
}
|
|
|
|
private static ulong BinaryGcd(ulong a, ulong b)
|
|
{
|
|
if (a == 0UL)
|
|
{
|
|
return b;
|
|
}
|
|
|
|
if (b == 0UL)
|
|
{
|
|
return a;
|
|
}
|
|
|
|
int shift = BitOperations.TrailingZeroCount(a | b);
|
|
a >>= BitOperations.TrailingZeroCount(a);
|
|
|
|
while (true)
|
|
{
|
|
b >>= BitOperations.TrailingZeroCount(b);
|
|
if (a > b)
|
|
{
|
|
(a, b) = (b, a);
|
|
}
|
|
|
|
b -= a;
|
|
if (b == 0UL)
|
|
{
|
|
return a << shift;
|
|
}
|
|
}
|
|
}
|
|
|
|
private static void AddFactor(Dictionary<ulong, int> counts, ulong prime, int exponent)
|
|
{
|
|
if (counts.TryGetValue(prime, out int existing))
|
|
{
|
|
counts[prime] = existing + exponent;
|
|
}
|
|
else
|
|
{
|
|
counts[prime] = exponent;
|
|
}
|
|
}
|
|
|
|
private static ulong CalculateByDoubling(ulong prime)
|
|
{
|
|
ulong order = 1UL;
|
|
ulong pow = 2UL;
|
|
|
|
while (pow != 1UL)
|
|
{
|
|
pow <<= 1;
|
|
if (pow >= prime)
|
|
{
|
|
pow -= prime;
|
|
}
|
|
|
|
order++;
|
|
}
|
|
|
|
return order;
|
|
}
|
|
|
|
private readonly struct CandidateKey
|
|
{
|
|
public CandidateKey(int primary, long secondary, long tertiary)
|
|
{
|
|
Primary = primary;
|
|
Secondary = secondary;
|
|
Tertiary = tertiary;
|
|
}
|
|
|
|
public int Primary { get; }
|
|
|
|
public long Secondary { get; }
|
|
|
|
public long Tertiary { get; }
|
|
}
|
|
|
|
private readonly struct FactorEntry
|
|
{
|
|
public FactorEntry(ulong value, int exponent)
|
|
{
|
|
Value = value;
|
|
Exponent = exponent;
|
|
}
|
|
|
|
public ulong Value { get; }
|
|
|
|
public int Exponent { get; }
|
|
}
|
|
|
|
private readonly struct PartialFactorResult
|
|
{
|
|
public PartialFactorResult(FactorEntry[]? factors, ulong cofactor, bool fullyFactored, int count)
|
|
{
|
|
Factors = factors;
|
|
Cofactor = cofactor;
|
|
FullyFactored = fullyFactored;
|
|
Count = count;
|
|
}
|
|
|
|
public FactorEntry[]? Factors { get; }
|
|
|
|
public ulong Cofactor { get; }
|
|
|
|
public bool FullyFactored { get; }
|
|
|
|
public int Count { get; }
|
|
|
|
public static PartialFactorResult Empty => new(null, 1UL, true, 0);
|
|
|
|
public PartialFactorResult WithAdditionalPrime(ulong prime)
|
|
{
|
|
if (Factors is null)
|
|
{
|
|
FactorEntry[] local = [new FactorEntry(prime, 1)];
|
|
return new PartialFactorResult(local, 1UL, true, 1);
|
|
}
|
|
|
|
FactorEntry[] extended = new FactorEntry[Count + 1];
|
|
Array.Copy(Factors, 0, extended, 0, Count);
|
|
extended[Count] = new FactorEntry(prime, 1);
|
|
Array.Sort(extended, static (a, b) => a.Value.CompareTo(b.Value));
|
|
return new PartialFactorResult(extended, 1UL, true, Count + 1);
|
|
}
|
|
}
|
|
}
|