651 lines
19 KiB
C#
651 lines
19 KiB
C#
using System.Numerics;
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using System.Runtime.CompilerServices;
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using PerfectNumbers.Core.Gpu;
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namespace PerfectNumbers.Core;
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public static class ULongExtensions
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{
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public static ulong CalculateOrder(this ulong q)
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{
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if (q <= 2UL)
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{
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return 0UL;
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}
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ulong order = q - 1UL, prime, temp;
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uint[] smallPrimes = PrimesGenerator.SmallPrimes;
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ulong[] smallPrimesPow2 = PrimesGenerator.SmallPrimesPow2;
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int i = 0, primesLength = smallPrimes.Length;
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UInt128 q128 = q,
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cycle = MersenneDivisorCycles.GetCycle(q128);
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for (; i < primesLength; i++)
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{
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if (smallPrimesPow2[i] > order)
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{
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break;
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}
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prime = smallPrimes[i];
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while (order % prime == 0UL)
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{
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temp = order / prime;
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if (temp.PowModWithCycle(q128, cycle) == UInt128.One)
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{
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order = temp;
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}
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else
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{
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break;
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}
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}
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}
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return order;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong FastDiv64(this ulong value, ulong divisor, ulong mul)
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{
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ulong quotient = (ulong)(((UInt128)value * mul) >> 64);
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UInt128 remainder = (UInt128)value - ((UInt128)quotient * divisor);
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if (remainder >= divisor)
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{
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quotient++;
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}
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return quotient;
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}
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public const ulong WordBitMask = 0xFFFFUL;
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static bool IsPrimeCandidate(this ulong n)
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{
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int i = 0;
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uint[] smallPrimes = PrimesGenerator.SmallPrimes;
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ulong[] smallPrimesPow2 = PrimesGenerator.SmallPrimesPow2;
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int len = smallPrimes.Length;
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ulong p;
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for (; i < len; i++)
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{
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if (smallPrimesPow2[i] > n)
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{
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break;
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}
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p = smallPrimes[i];
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if ((n % p) == 0UL)
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{
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return n == p;
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}
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}
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return true;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong Mod10(this ulong value) => (value & 1UL) == 0UL
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? (value % 5UL) switch
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{
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0UL => 0UL,
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1UL => 6UL,
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2UL => 2UL,
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3UL => 8UL,
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_ => 4UL,
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}
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: (value % 5UL) switch
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{
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0UL => 5UL,
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1UL => 1UL,
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2UL => 7UL,
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3UL => 3UL,
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_ => 9UL,
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};
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong Mod128(this ulong value) => value & 127UL;
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static void Mod10_8_5_3(this ulong value, out ulong mod10, out ulong mod8, out ulong mod5, out ulong mod3)
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{
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mod8 = value & 7UL;
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byte current;
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uint byteSum = 0U;
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do
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{
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current = (byte)value;
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byteSum += current;
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value >>= 8;
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}
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while (value != 0UL);
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mod3 = byteSum % 3U;
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mod5 = byteSum % 5U;
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mod10 = (mod8 & 1UL) == 0UL
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? mod5 switch
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{
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0UL => 0UL,
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1UL => 6UL,
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2UL => 2UL,
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3UL => 8UL,
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_ => 4UL,
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}
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: mod5 switch
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{
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0UL => 5UL,
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1UL => 1UL,
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2UL => 7UL,
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3UL => 3UL,
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_ => 9UL,
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};
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static void Mod10_8_5_3Steps(this ulong value, out ulong step10, out ulong step8, out ulong step5, out ulong step3)
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{
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value.Mod10_8_5_3(out step10, out step8, out step5, out step3);
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step10 = (step10 << 1).Mod10();
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step8 = (step8 << 1) & 7UL;
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step5 = (step5 << 1) % 5UL;
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step3 = (step3 << 1) % 3UL;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static UInt128 Mul64(this ulong a, ulong b) => ((UInt128)a.MulHigh(b) << 64) | (UInt128)(a * b);
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong MulHigh(this ulong x, ulong y)
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{
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ulong xLow = (uint)x;
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ulong xHigh = x >> 32;
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ulong yLow = (uint)y;
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ulong yHigh = y >> 32;
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ulong w1 = xLow * yHigh;
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ulong w2 = xHigh * yLow;
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ulong w3 = xLow * yLow;
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// Matching the layout used in GpuUInt128.MulHigh: introducing the
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// intermediate result looks like one extra store, but it lets RyuJIT keep
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// the accumulated high word entirely in registers. Without this explicit
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// local the JIT spills the partial sum, which is where the performance
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// regression in the benchmarks came from.
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ulong result = (xHigh * yHigh) + (w1 >> 32) + (w2 >> 32);
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result += ((w3 >> 32) + (uint)w1 + (uint)w2) >> 32;
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return result;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong MulHighGpuCompatible(this ulong x, ulong y)
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{
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GpuUInt128 product = new(x);
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product.Mul64(new GpuUInt128(y));
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return product.High;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong ModPow64(this ulong value, ulong exponent, ulong modulus)
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{
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ulong result = 1UL;
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value %= modulus;
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while (exponent != 0UL)
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{
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if ((exponent & 1UL) != 0UL)
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{
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result = MulMod64(result, value, modulus);
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}
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value = MulMod64(value, value, modulus);
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exponent >>= 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 ulong MulMod64(this ulong a, ulong b, ulong modulus) => (ulong)(UInt128)(((a % modulus) * (b % modulus)) % modulus);
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong MulMod64GpuCompatible(this ulong a, ulong b, ulong modulus)
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{
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GpuUInt128 state = new(a % modulus);
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state.MulMod(b, modulus);
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return state.Low;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong MulMod64GpuCompatibleDeferred(this ulong a, ulong b, ulong modulus)
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{
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GpuUInt128 state = new(a);
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state.MulModWithNativeModulo(b, modulus);
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return state.Low;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong MontgomeryMultiply(this ulong a, ulong b, ulong modulus, ulong nPrime)
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{
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ulong tLow = unchecked(a * b);
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ulong m = unchecked(tLow * nPrime);
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ulong mTimesModulusLow = unchecked(m * modulus);
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ulong result = unchecked(a.MulHigh(b) + m.MulHigh(modulus) + (unchecked(tLow + mTimesModulusLow) < tLow ? 1UL : 0UL));
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if (result >= modulus)
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{
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result -= modulus;
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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 ulong Pow2MontgomeryMod(this ulong exponent, in MontgomeryDivisorData divisor)
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{
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ulong modulus = divisor.Modulus;
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if (modulus <= 1UL || (modulus & 1UL) == 0UL)
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{
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return 0UL;
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}
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ulong result = divisor.MontgomeryOne;
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ulong baseVal = divisor.MontgomeryTwo;
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ulong nPrime = divisor.NPrime;
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ulong remainingExponent = exponent;
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while (remainingExponent > 0UL)
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{
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if ((remainingExponent & 1UL) != 0UL)
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{
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result = result.MontgomeryMultiply(baseVal, modulus, nPrime);
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}
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remainingExponent >>= 1;
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if (remainingExponent == 0UL)
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{
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break;
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}
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baseVal = baseVal.MontgomeryMultiply(baseVal, modulus, nPrime);
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}
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return result.MontgomeryMultiply(1UL, modulus, nPrime);
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong Pow2MontgomeryModMontgomery(this ulong exponent, in MontgomeryDivisorData divisor)
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{
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ulong modulus = divisor.Modulus;
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if (modulus <= 1UL || (modulus & 1UL) == 0UL)
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{
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return 0UL;
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}
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ulong result = divisor.MontgomeryOne;
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ulong baseVal = divisor.MontgomeryTwo;
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ulong remainingExponent = exponent;
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while (remainingExponent > 0UL)
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{
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if ((remainingExponent & 1UL) != 0UL)
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{
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result = result.MontgomeryMultiply(baseVal, modulus, divisor.NPrime);
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}
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remainingExponent >>= 1;
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if (remainingExponent == 0UL)
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{
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break;
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}
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baseVal = baseVal.MontgomeryMultiply(baseVal, modulus, divisor.NPrime);
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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 ulong Pow2MontgomeryModWithCycle(this ulong exponent, ulong cycleLength, in MontgomeryDivisorData divisor)
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{
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if (cycleLength == 0UL)
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{
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return exponent.Pow2MontgomeryMod(divisor);
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}
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ulong rotationCount = exponent % cycleLength;
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return rotationCount.Pow2MontgomeryModFromCycleRemainder(divisor);
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong Pow2MontgomeryModFromCycleRemainder(this ulong reducedExponent, in MontgomeryDivisorData divisor)
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{
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if (reducedExponent == 0UL)
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{
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return 1UL;
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}
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return reducedExponent.Pow2MontgomeryMod(divisor);
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static UInt128 PowMod(this ulong exponent, UInt128 modulus)
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{
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UInt128 result = UInt128.One;
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ulong exponentLoopIndex = 0UL;
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// Return 1 because 2^0 = 1
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if (exponent == 0UL)
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return result;
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// Any number mod 1 is 0
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if (modulus == UInt128.One)
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return UInt128.Zero;
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// For small exponents, do classic method
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if (exponent < 64 || modulus < 4)
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{
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for (; exponentLoopIndex < exponent; exponentLoopIndex++)
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{
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result <<= 1;
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if (result >= modulus)
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result -= modulus;
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}
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return result;
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}
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// Special case: if modulus is a power of two, use bitmasking for efficiency and correctness
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if ((modulus & (modulus - 1)) == 0)
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{
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result = UInt128.One << (int)(exponent & 127);
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return result & (modulus - 1);
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}
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// Reusing exponentLoopIndex to iterate again over the exponent range for the general-case accumulation.
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exponentLoopIndex = 0UL;
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// Reusing result after resetting it for the general modulus accumulation phase.
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result = UInt128.One;
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for (; exponentLoopIndex < exponent; exponentLoopIndex++)
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{
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result <<= 1;
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if (result >= modulus)
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result -= modulus;
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}
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return result;
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}
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/// <summary>
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/// Computes 2^exponent mod modulus using a known cycle length.
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/// </summary>
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public static UInt128 PowModWithCycle(this ulong exponent, UInt128 modulus, ulong cycleLength)
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{
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UInt128 result = UInt128.One;
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ulong exponentLoopIndex = 0UL;
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// Return 1 because 2^0 = 1
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if (exponent == 0UL)
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return result;
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// Any number mod 1 is 0
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if (modulus == UInt128.One)
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return UInt128.Zero;
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// For small exponents, do classic method
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if (exponent < 64 || modulus < 4)
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{
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for (; exponentLoopIndex < exponent; exponentLoopIndex++)
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{
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result <<= 1;
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if (result >= modulus)
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result -= modulus;
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}
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return result;
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}
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// Special case: if modulus is a power of two, use bitmasking for efficiency and correctness
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if ((modulus & (modulus - 1)) == 0)
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{
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result = UInt128.One << (int)(exponent & 127);
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return result & (modulus - 1);
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}
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// Reusing exponentLoopIndex to iterate over the rotation count for the cycle-aware path.
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exponentLoopIndex = 0UL;
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// Reusing result after resetting it for the rotation-accumulation pass.
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result = UInt128.One;
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ulong rotationCount = exponent % cycleLength;
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for (; exponentLoopIndex < rotationCount; exponentLoopIndex++)
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{
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result <<= 1;
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if (result >= modulus)
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result -= modulus;
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}
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return result;
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}
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/// <summary>
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/// Computes 2^exponent mod modulus using a known cycle length.
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/// </summary>
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public static UInt128 PowModWithCycle(this ulong exponent, UInt128 modulus, UInt128 cycleLength)
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{
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UInt128 result = UInt128.One;
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ulong exponentLoopIndex = 0UL;
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// Return 1 because 2^0 = 1
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if (exponent == UInt128.Zero)
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return result;
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// Any number mod 1 is 0
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if (modulus == UInt128.One)
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return UInt128.Zero;
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// For small exponents, do classic method
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if (exponent < UInt128Numbers.SixtyFour || modulus < UInt128Numbers.Four)
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{
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for (; exponentLoopIndex < exponent; exponentLoopIndex++)
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{
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result <<= 1;
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if (result >= modulus)
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result -= modulus;
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}
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return result;
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}
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// Special case: if modulus is a power of two, use bitmasking for efficiency and correctness
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if ((modulus & (modulus - UInt128.One)) == UInt128.Zero)
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{
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result = UInt128.One << (int)(exponent & UInt128Numbers.OneHundredTwentySeven);
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return result & (modulus - 1);
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}
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// Reusing result after resetting it for the rotation-driven accumulation phase.
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result = UInt128.One;
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UInt128 rotationCount = exponent % cycleLength;
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UInt128 rotationIndex = UInt128.Zero;
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while (rotationIndex < rotationCount)
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{
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result <<= 1;
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if (result >= modulus)
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result -= modulus;
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rotationIndex += UInt128.One;
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}
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return result;
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}
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/// <summary>
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/// Computes 2^exponent mod modulus using a known cycle length.
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/// </summary>
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public static UInt128 PowModWithCycle(this UInt128 exponent, UInt128 modulus, UInt128 cycleLength)
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{
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UInt128 one = UInt128.One,
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result = one,
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zero = UInt128.Zero;
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ulong exponentLoopIndex = 0UL;
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// Return 1 because 2^0 = 1
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if (exponent == zero)
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return result;
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// Any number mod 1 is 0
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if (modulus == one)
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return zero;
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// For small exponents, do classic method
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if (exponent < UInt128Numbers.SixtyFour || modulus < UInt128Numbers.Four)
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{
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for (; exponentLoopIndex < exponent; exponentLoopIndex++)
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{
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result <<= 1;
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if (result >= modulus)
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result -= modulus;
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}
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return result;
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}
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// Special case: if modulus is a power of two, use bitmasking for efficiency and correctness
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if ((modulus & (modulus - one)) == zero)
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{
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result = one << (int)(exponent & UInt128Numbers.OneHundredTwentySeven);
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return result & (modulus - 1);
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}
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// Reusing result after resetting it for the rotation-driven accumulation phase.
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result = one;
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UInt128 rotationCount = exponent % cycleLength;
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// We're reusing "zero" as rotation index for just a little better performance
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while (zero < rotationCount)
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{
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result <<= 1;
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if (result >= modulus)
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result -= modulus;
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zero += one;
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}
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return result;
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}
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/// <summary>
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/// Computes 2^exponent mod modulus using iterative CRT composition from mod 10 up to modulus.
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/// Only for modulus >= 10 and reasonable size.
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/// </summary>
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public static UInt128 PowModCrt(this ulong exponent, UInt128 modulus, MersenneDivisorCycles cycles)
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{
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if (modulus < 10)
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return PowMod(exponent, modulus); // fallback to classic
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// Use cycle length 4 for mod 10
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UInt128 currentModulus = 10,
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cycle,
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modulusCandidate = 11,
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remainderForCandidate,
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result = PowModWithCycle(exponent, 10, 4),
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zero = UInt128.Zero;
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for (; modulusCandidate <= modulus; modulusCandidate++)
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{
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cycle = MersenneDivisorCycles.GetCycle(modulusCandidate);
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remainderForCandidate = cycle > zero
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? PowModWithCycle(exponent, modulusCandidate, cycle)
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: PowMod(exponent, modulusCandidate);
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// Solve x ≡ result mod currentModulus
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// x ≡ remM mod m
|
|
// Find x mod (currentModulus * m)
|
|
// Since currentModulus and m are coprime, use CRT:
|
|
// x = result + currentModulus * t, where t ≡ (remM - result) * inv(currentModulus, m) mod m
|
|
|
|
result += currentModulus * ((remainderForCandidate + modulusCandidate - (result % modulusCandidate)) * ModInverse(currentModulus, modulusCandidate) % modulusCandidate);
|
|
currentModulus *= modulusCandidate;
|
|
|
|
if (currentModulus >= modulus)
|
|
break;
|
|
}
|
|
|
|
return result % modulus;
|
|
}
|
|
|
|
// Helper: modular inverse (extended Euclidean algorithm)
|
|
private static UInt128 ModInverse(UInt128 a, UInt128 m)
|
|
{
|
|
UInt128 m0 = m,
|
|
originalA,
|
|
originalM,
|
|
temp,
|
|
x0 = 0,
|
|
x1 = 1;
|
|
|
|
if (m == 1)
|
|
{
|
|
return 0;
|
|
}
|
|
|
|
while (a > 1)
|
|
{
|
|
originalA = a;
|
|
originalM = m;
|
|
m = originalA % originalM;
|
|
a = originalM;
|
|
temp = x0;
|
|
x0 = x1 - (originalA / originalM) * x0;
|
|
x1 = temp;
|
|
}
|
|
|
|
if (x1 < 0)
|
|
{
|
|
x1 += m0;
|
|
}
|
|
|
|
return x1;
|
|
}
|
|
|
|
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
|
public static bool SharesFactorWithExponentMinusOne(this ulong exponent)
|
|
{
|
|
ulong prime, value = exponent - 1UL;
|
|
value >>= BitOperations.TrailingZeroCount(value);
|
|
uint[] smallPrimes = PrimesGenerator.SmallPrimes;
|
|
ulong[] smallPrimesPow2 = PrimesGenerator.SmallPrimesPow2;
|
|
int i = 1, smallPrimesLength = smallPrimes.Length;
|
|
|
|
for (; i < smallPrimesLength && smallPrimesPow2[i] <= value; i++)
|
|
{
|
|
prime = smallPrimes[i];
|
|
if (value % prime != 0UL)
|
|
{
|
|
continue;
|
|
}
|
|
|
|
if (exponent % prime.CalculateOrder() == 0UL)
|
|
{
|
|
return true;
|
|
}
|
|
|
|
do
|
|
{
|
|
value /= prime;
|
|
}
|
|
while (value % prime == 0UL);
|
|
}
|
|
|
|
if (value > 1UL && exponent % value.CalculateOrder() == 0UL)
|
|
{
|
|
return true;
|
|
}
|
|
|
|
return false;
|
|
}
|
|
}
|