private static void getStringRepresentation(IntegerNumber x, int base10Digits, StringBuilder result)
{
if (base10Digits <= 18)
{
string value = x.ToLong.ToString();
int diff = value.Length - base10Digits;
if (diff != 0)
{
if (diff < 0)
{
value = "".PadLeft(-diff, '0') + value;
}
else
{
value = value.Remove(0, diff);
}
}
result.Append(value);
return;
}
int secondHalfDigits = (base10Digits + 1) / 2;
IntegerNumber half = IntegerNumber.Pow(10, secondHalfDigits);
IntegerNumber reminder, quotient = IntegerNumber.DivRem(x, half, out reminder);
getStringRepresentation(quotient, base10Digits - secondHalfDigits, result);
getStringRepresentation(reminder, secondHalfDigits, result);
}
public IntegerNumber ModPow(IntegerNumber power, IntegerNumber prime, Func <IntegerNumber, IntegerNumber> modPFunction)
{
IntegerNumber a = this;
if (power.IsNegative)
{
a = a.ModInverse(prime);
power = -power;
}
int highestBitPosition = power.Digits * 64 - 1;
while (highestBitPosition >= 0 && !power.GetBitAt(highestBitPosition))
{
highestBitPosition--;
}
IntegerNumber result = highestBitPosition < 0 ? One : a;
for (int i = highestBitPosition; --i >= 0;)
{
result = modPFunction(result.Square());
if (power.GetBitAt(i))
{
result = modPFunction(result * a);
}
}
var diff = result - prime;
return(diff.IsNegative ? result : diff);
}
public static IntegerNumber DivRem(IntegerNumber number, long digit, out long remainder)
{
IntegerNumber result = new IntegerNumber(number.bits, true);
remainder = (long)AsmX64Operations.DivideDigit(result.bits, (ulong)digit, result.Digits, true);
return(result);
}
public IntegerNumber Inverse()
{
shrink(this);
IntegerNumber remainder, result = Inverse(this, out remainder);
return(result);
}
public static IntegerNumber operator /(IntegerNumber number1, long number2)
{
long remainder;
IntegerNumber quotient = IntegerNumber.DivRem(number1, number2, out remainder);
return(quotient);
}
public override string ToString()
{
IntegerNumber x = this;
bool negative = x.IsNegative;
if (negative)
{
x = -x;
}
StringBuilder result = new StringBuilder();
getStringRepresentation(x, Math.Max(1, (int)Math.Ceiling(x.BitsCount * LOG10_2)), result);
int lowzeros = 0;
while (lowzeros + 1 < result.Length && result[lowzeros] == '0')
{
lowzeros++;
}
if (lowzeros > 0)
{
result = result.Remove(0, lowzeros);
}
if (negative)
{
result = result.Insert(0, '-');
}
return(result.ToString());
}
public static IntegerNumber operator *(IntegerNumber number1, IntegerNumber number2)
{
IntegerNumber source1 = number1.ExtendTo(number2.Digits);
IntegerNumber source2 = number2.ExtendTo(number1.Digits);
ulong[] result = new ulong[source1.Digits * 2];
AsmX64Operations.FastestMultiplication(source1.bits, source2.bits, result, source1.Digits, true);
return(new IntegerNumber(result, false));
}
public static void shrink(IntegerNumber number)
{
int digits = number.RealDigits;
if (digits != number.Digits)
{
Array.Resize(ref number.bits, digits);
}
}
//f[x_] := x^2 - y
//FullSimplify[x - f[x] / D[f[x], x]] = (x + y / x) / 2 [Newton method]
//FullSimplify[x - 2*f[x]*D[f[x], x]/(2*D[f[x], x]^2 - f[x]*D[f[x], {x, 2}])] = x * (x^2 + 3 y) / (3 x^2 + y) [Halley]
//In practice Newton method is faster than Halley 3'rd order convergence for SQRT.
public IntegerNumber Sqrt()
{
if (this.IsNegative)
{
throw new ArithmeticException("NaN");
}
if (this.IsZero)
{
return(IntegerNumber.Zero);
}
shrink(this);
int n = this.Digits;
int computedDigits = (n & 1) != 0 ? 1 : 2;
if (computedDigits + 2 <= n)
{
computedDigits += 2;
}
ulong[] inputBits = this.bits;
IntegerNumber t = this.GetDigits(n - computedDigits, computedDigits, true);
double estimate = Math.Sqrt(t.ToDouble);
double highDouble = Math.Floor(estimate / Power2_64);
double lowDouble = estimate - highDouble * Power2_64;
IntegerNumber root = new IntegerNumber(new ulong[] { (ulong)lowDouble, (ulong)highDouble }, false);
if (root.IsNegative)
{
Array.Resize(ref root.bits, 3);
}
while (true)
{
IntegerNumber oldRoot = root;
root = (root + t / root) >> 1;
if (IntegerNumber.Abs(oldRoot - root) <= IntegerNumber.One)
{
break;
}
}
int additionalDigits = 1;
for (int i = computedDigits; i < n + additionalDigits;)
{
int digits = Math.Min(computedDigits, n - computedDigits + 1) >> 1;
i += computedDigits;
root <<= digits * 64;
computedDigits += digits * 2;
t = computedDigits >= n ? this : this.GetDigits(n - computedDigits, computedDigits, true);
root = (root + t / root) >> 1;
}
IntegerNumber square = root.Square();
return(this >= square ? root : root - IntegerNumber.One);
}
public static IntegerNumber operator -(IntegerNumber number1, IntegerNumber number2)
{
if (number1.IsNegative != number2.IsNegative)
{
return(number2.IsZero ? number1 : number1 + (-number2));
} //when subtracting numbers with the same sign, the absolute value of the result is less than the absoulute value of each input. (number size decreases).
IntegerNumber source1 = number1.ExtendTo(number2.Digits);
IntegerNumber source2 = number2.ExtendTo(number1.Digits);
ulong[] result = new ulong[source1.Digits];
AsmX64Operations.Subtract(source1.bits, source2.bits, result, 0, result.Length);
return(new IntegerNumber(result, false));
}
public static IntegerNumber operator -(IntegerNumber number)
{
if (number.bits[number.bits.Length - 1] == SignBit && number.isZero(0, number.bits.Length - 1))
{ //make positive the highly compact negative numbers - as a special case.
ulong[] bits = new ulong[number.bits.Length + 1];
bits[number.bits.Length - 1] = SignBit;
return(new IntegerNumber(bits, false));
}
IntegerNumber result = new IntegerNumber(number.bits, true);
AsmX64Operations.Negate(result.bits, result.Digits);
return(result);
}
public IntegerNumber ShanksSqrt(IntegerNumber p, Func <IntegerNumber, IntegerNumber> modPFunction)
{
if (this.ModPow((p - 1) >> 1, p, modPFunction) == (p - 1))
{
return(-1);
} //No Sqrt Exists
if ((p.LowestByte & 3) == 3)
{
var result = this.ModPow((p + 1) >> 2, p, modPFunction);
return(result);
}
throw new NotImplementedException("Not a special prime modulo.");
}
/// <summary>
/// return this / n modulo p.
/// </summary>
/// <param name="n"></param>
/// <param name="p"></param>
/// <returns></returns>
public IntegerNumber DivideModulo(IntegerNumber n, IntegerNumber p)
{
if (n.IsZero)
{
return(n);
}
if (p.IsZero)
{
return(p);
}
IntegerNumber a = this; //x * a + u1 * b = n
IntegerNumber c = 0; //y * c + u2 * d = p
IntegerNumber x = n;
while (p.IsEven)
{
p >>= 1;
}
IntegerNumber y = p;
while (x.IsEven)
{
x >>= 1;
a = a.IsEven ? a >> 1 : (a + p) >> 1;
}
while (!x.IsZero)
{
while (x.IsEven)
{
x >>= 1;
a = a.IsEven ? a >> 1 : (a + p) >> 1;
}
IntegerNumber diff = x - y;
if (diff.IsNegative)
{
y = x;
x = -diff;
IntegerNumber temp = a; a = c; c = temp;
}
else
{
x = diff;
}
a -= c;
if (a.IsNegative)
{
a += p;
}
}
return(c);
}
public static double Log2(IntegerNumber number)
{
if (number.IsNegative)
{
throw new InvalidOperationException("Real logarithm of a negative number does not exist.");
}
shrink(number);
int n = number.Digits;
if (n <= 1)
{
return(number.IsZero ? 0.0 : Math.Log(number.HighestDigit) * LG_E);
}
return(Math.Log(number.HighestDigit * Power2_64 + number.bits[n - 2]) * LG_E + (n - 2) * 64);
}
public static IntegerNumber operator >>(IntegerNumber x, int shift)
{
if (shift == 0)
{
return(x);
}
if (shift < 0)
{
return(x << (-shift));
}
IntegerNumber result = new IntegerNumber(x.bits, true);
AsmX64Operations.IntShr(result.bits, shift, result.Digits, true);
return(result);
}
public IntegerNumber GetBitsAt(int index, int count)
{
IntegerNumber shifted = this >> index;
int cell = count >> 6;
int pos = count & 63;
IntegerNumber result = new IntegerNumber(new ulong[cell + 1], false);
for (int i = Math.Min(cell, shifted.bits.Length); --i >= 0;)
{
result.bits[i] = shifted.bits[i];
}
if (pos != 0 && cell < shifted.bits.Length)
{
result.bits[cell] = shifted.bits[cell] & (ulong.MaxValue >> (64 - pos));
}
return(result);
}
public static IntegerNumber operator +(IntegerNumber number1, IntegerNumber number2)
{
if (number1.IsNegative != number2.IsNegative)
{
return(number2.IsZero ? number1 : number1 - (-number2));
}
IntegerNumber source1 = number1.ExtendTo(number2.Digits);
IntegerNumber source2 = number2.ExtendTo(number1.Digits);
ulong[] result = new ulong[source1.Digits];
byte carry = AsmX64Operations.Add(source1.bits, source2.bits, result, 0, result.Length);
if (carry != (byte)(result[result.Length - 1] >> 63))
{
Array.Resize(ref result, result.Length + 1);
result[result.Length - 1] = carry == 0 ? 0UL : ulong.MaxValue;
}
return(new IntegerNumber(result, false));
}
public static IntegerNumber Pow(int baseNumber, int exponent)
{
int highestBitPosition = BSR((uint)exponent);
if (highestBitPosition < 0)
{
return(IntegerNumber.One);
}
IntegerNumber number = new IntegerNumber(baseNumber);
IntegerNumber result = number;
for (int i = highestBitPosition; --i >= 0;)
{
result = result.Square();
shrink(result);
if ((exponent & (1 << i)) != 0)
{
result *= baseNumber;
}
}
return(result);
}
public static IntegerNumber Inverse(IntegerNumber number, out IntegerNumber remainder)
{
int n = number.Digits;
if (n <= IntegerDigitsBreakPoint)
{
ulong[] a = new ulong[(n + 1) * 2];
a[n * 2] = 1UL;
IntegerNumber b = number.ExtendTo(n + 1);
ulong[] q = new ulong[(n + 1) * 2];
AsmX64Operations.GetDivMod(a, b.bits, n + 1, q, true);
remainder = new IntegerNumber(a, false);
shrink(remainder);
IntegerNumber result = new IntegerNumber(q, false);
shrink(result);
return(result);
}
bool isNegative = false;
if (number.IsNegative)
{
number = -number;
isNegative = true;
n = number.Digits;
}
//Newton iteration: x <- x + x * (1 - d * x^2)
int m = (n + 5) >> 1;
IntegerNumber dx, dhi = number.GetDigits(n - m, m, false);
IntegerNumber x = Inverse(dhi, out dx);
//IntegerNumber test = x * d + dx;
//shrink(ref test);
//if (test.Digits != d.Digits * 2 + 1 || test.bits[d.Digits * 2] != 1UL || Enumerable.Range(0, d.Digits * 2).Any(idx => test.bits[idx] != 0UL))
//{
// System.Diagnostics.Debugger.Break();
//}
const int CorrectionDigits = 4;
IntegerNumber dlo = number.GetDigits(0, n - m, true);
IntegerNumber dp = (dx << ((n - m) * 64)) - dlo * x;
IntegerNumber delta = dp >> ((m - CorrectionDigits) * 64); //keep one additional correction digit.
shrink(delta);
delta *= x;
delta >>= (m + CorrectionDigits) * 64;
shrink(delta);
x <<= (n - m) * 64;
x += delta;
//FACT: number * x == (IntegerNumber.One << (n * 128)) - ((dp << ((n - m) * 64)) - delta * number);
//but : remainder = number * x;
//then: Array.Resize(ref remainder.bits, n * 2);
//then: shrink(ref remainder);
//then: remainder = -remainder;
remainder = ((dp - delta * dhi) << ((n - m) * 64)) - delta * dlo;
shrink(remainder);
int count = 0;
while (remainder.IsNegative)
{
remainder += number;
x -= One;
count++;
if (count >= 2)
{
System.Diagnostics.Debugger.Break();
}
}
while (remainder >= number)
{
remainder -= number;
x += One;
count++;
if (count >= 2)
{
System.Diagnostics.Debugger.Break();
}
}
if (isNegative)
{
x = -x;
remainder = -remainder;
}
return(x);
}
public static IntegerNumber Abs(IntegerNumber number)
{
return(number.IsNegative ? -number : number);
}
public static IntegerNumber DivRem(IntegerNumber dividend, IntegerNumber divisor, out IntegerNumber remainder)
{
shrink(dividend);
shrink(divisor);
int dividendDigits = dividend.Digits;
int divisorDigits = divisor.Digits;
IntegerNumber quotient;
int n = Math.Max(divisorDigits, (dividendDigits + 1) >> 1);
if (divisorDigits <= IntegerDigitsBreakPoint)
{
dividend = dividend.ExtendTo(n * 2);
divisor = divisor.ExtendTo(n);
quotient = new IntegerNumber(new ulong[n * 2], false);
remainder = new IntegerNumber(dividend.bits, true);
AsmX64Operations.GetDivMod(remainder.bits, divisor.bits, n, quotient.bits, true);
shrink(quotient);
shrink(remainder);
return(quotient);
}
bool sign = false;
if (dividend.IsNegative != divisor.IsNegative)
{
dividend = IntegerNumber.Abs(dividend);
divisor = IntegerNumber.Abs(divisor);
sign = true;
}
else if (dividend.IsNegative && divisor.IsNegative)
{
dividend = -dividend;
divisor = -divisor;
}
int delta = n - divisorDigits;
IntegerNumber x = (divisor << (delta * 128)).Inverse();
quotient = dividend * x >> (n * 128);
shrink(quotient);
remainder = dividend - quotient * divisor;
shrink(remainder);
int count = 0;
while (remainder.IsNegative)
{
remainder += divisor;
quotient -= One;
count++;
if (count >= 2)
{
System.Diagnostics.Debugger.Break();
}
}
while (remainder >= divisor)
{
remainder -= divisor;
quotient += One;
count++;
if (count >= 2)
{
System.Diagnostics.Debugger.Break();
}
}
if (sign)
{
quotient = -quotient;
remainder = -remainder;
}
return(quotient);
}
public IntegerNumber ModInverse(IntegerNumber p)
{
return(One.DivideModulo(this, p));
}
public static bool UnitTest()
{
// SPECIAL prime: 2^64 - 2^32 + 1
//prime: 2^64 - 2^34 + 1
//prime: 2^64 - 2^40 + 1
// SPECIAL prime: 2 ^ 128 - 2 ^ 54 + 1
//prime: 2 ^ 128 - 2 ^ 108 + 1
//prime: 2 ^ 256 - 2 ^ 168 + 1
//prime: 2 ^ 256 - 2 ^ 174 + 1
//prime: 2 ^ 512 - 2 ^ 32 + 1
//prime: 2 ^ 512 - 2 ^ 288 + 1
// SPECIAL prime: 2 ^ 1024 - 2 ^ 142 + 1
// SPECIAL prime: 2 ^ 1024 - 2 ^ 226 + 1
//prime: 2 ^ 1024 - 2 ^ 562 + 1
//prime: 2 ^ 1024 - 2 ^ 718 + 1
//prime: 2 ^ 1024 - 2 ^ 856 + 1
//prime: 2 ^ 1024 - 2 ^ 880 + 1
//prime: 2 ^ 4096 - 2 ^ 3510 + 1
//prime: 2 ^ 4096 - 2 ^ 3708 + 1
if ("not exec".Trim() == "exec")
{
StringBuilder primes = new StringBuilder();
for (int exp = 6; exp <= 12; exp++)
{
int shift = 1 << exp;
Parallel.For(32, shift, i =>
{
ulong[] baseValue = new ulong[shift / 64];
ulong[] expValue = new ulong[shift / 64];
ulong[] modulo = new ulong[shift / 64];
ulong[] subtractor = new ulong[shift / 64];
ulong[] one = new ulong[shift / 64];
ulong[] pPlusOne = new ulong[shift / 64];
subtractor[i / 64] |= 1UL << (i & 63);
modulo[0] = 1UL;
one[0] = 1UL;
AsmX64Operations.Subtract(modulo, subtractor, modulo, 0, modulo.Length);
modulo.CopyTo(expValue, 0);
modulo.CopyTo(pPlusOne, 0);
expValue[0]--;
pPlusOne[0]++;
int isPrime = 1;
for (int a = 2; a <= 128; a++)
{
one.CopyTo(baseValue, 0);
baseValue[0] = (ulong)a;
AsmX64Operations.ModularExponentiation(baseValue, expValue, modulo, modulo.Length, 5);
if (Enumerable.Range(0, modulo.Length).All(idx => baseValue[idx] == one[idx]) ||
Enumerable.Range(0, modulo.Length).All(idx => baseValue[idx] == pPlusOne[idx]))
{
continue;
}
isPrime = 0;
break;
}
if (isPrime != 0)
{
lock (primes)
{
primes.AppendLine("prime: 2^" + shift.ToString() + " - 2^" + i.ToString() + " + 1");
}
}
});
}
Clipboard.SetText(primes.ToString());
System.Windows.Forms.MessageBox.Show(primes.ToString(), "message");
}
Random random = new Random(1002);
BigInteger maxx = 0;
BigInteger minn = 0;
bool ok = true;
for (int i = 2; --i >= 0;)
{
byte[] bytes1 = new byte[112 * 8 + random.Next(32 * 8)];
byte[] bytes2 = new byte[112 * 8 + random.Next(32 * 8)];
random.NextBytes(bytes1);
random.NextBytes(bytes2);
BigInteger n1 = new BigInteger(bytes1);
BigInteger n2 = new BigInteger(bytes2);
IntegerNumber f1 = new IntegerNumber(bytes1);
IntegerNumber f2 = new IntegerNumber(bytes2);
if (n1.ToString() != f1.ToString())
{
ok = false;
}
if (n2.ToString() != f2.ToString())
{
ok = false;
}
BigInteger a1 = n1 + n2;
IntegerNumber a2 = f1 + f2;
if (a1.ToString() != a2.ToString())
{
ok = false;
}
BigInteger s1 = n1 - n2;
IntegerNumber s2 = f1 - f2;
if (s1.ToString() != s2.ToString())
{
ok = false;
}
BigInteger m1 = n1 * n2;
IntegerNumber m2 = f1 * f2;
if (m1.ToString() != m2.ToString())
{
ok = false;
}
int shrvalue = random.Next(256) + 1;
BigInteger sh1 = n1 >> shrvalue;
IntegerNumber sh2 = f1 >> shrvalue;
if (sh1.ToString() != sh2.ToString())
{
ok = false;
}
if ((-f1).ToString() != (-n1).ToString() || (-f2).ToString() != (-n2).ToString())
{
ok = false;
}
int shlvalue = random.Next(256) + 1;
BigInteger shl1 = n1 << shlvalue;
IntegerNumber shl2 = f1 << shlvalue;
if (shl1.ToString() != shl2.ToString())
{
ok = false;
}
byte[] bytesINV = new byte[(192 + 32) * 8 + random.Next(64 * 8)];
random.NextBytes(bytesINV);
BigInteger num1 = new BigInteger(bytesINV);
IntegerNumber num2 = new IntegerNumber(bytesINV);
if (num1.ToString() != num2.ToString())
{
ok = false;
}
BigInteger inv0 = (BigInteger.One << (num2.Digits * 64 * 2)) / num1;
IntegerNumber inv1 = num2.Inverse();
if (inv0.ToString() != inv1.ToString())
{
ok = false;
}
byte[] bytes4 = new byte[bytes1.Length * 4];
random.NextBytes(bytes4);
BigInteger n4 = new BigInteger(bytes4);
IntegerNumber f4 = new IntegerNumber(bytes4);
BigInteger qb4 = n4 / n1;
IntegerNumber qn4 = f4 / f1;
if (qb4.ToString() != qn4.ToString())
{
ok = false;
}
byte[] bytes3 = new byte[(bytes1.Length + bytes2.Length) & -8];
random.NextBytes(bytes3);
BigInteger square1 = BigInteger.Abs(new BigInteger(bytes3));
IntegerNumber square2 = IntegerNumber.Abs(new IntegerNumber(bytes3));
BigInteger root1 = square1.Sqrt();
IntegerNumber root2 = square2.Sqrt();
if (root1.ToString() != root2.ToString())
{
ok = false;
}
if (!ok)
{
break;
}
}
return(ok);
}
//InverseSqrt[this] = Sqrt[(this << (this.Digits * 64)).Inverse()]
//InverseSqrt[this] = Sqrt[(1 << (this.Digits * 192)) / this]
//f[x_] := (1/x)^2 - y
//FullSimplify[x - f[x] / D[f[x], x]] = x * (1/2) * (3 - x^2 y)
public IntegerNumber InverseSqrt()
{
if (this.IsNegative || this.IsZero)
{
throw new ArithmeticException("NaN");
}
shrink(this);
int n = this.Digits;
int computedDigits = (n & 1) != 0 ? 1 : 2;
if (computedDigits + 2 <= n)
{
computedDigits += 2;
}
ulong[] inputBits = this.bits;
IntegerNumber t = this.GetDigits(n - computedDigits, computedDigits, true);
double estimate = Math.Pow(Power2_64, computedDigits * 1.5) / Math.Sqrt(t.ToDouble);
List <ulong> initialDigits = new List <ulong>();
while (estimate != 0)
{
initialDigits.Add((ulong)Math.IEEERemainder(estimate, Power2_64));
estimate = Math.Floor(estimate / Power2_64);
}
IntegerNumber root = new IntegerNumber(initialDigits.ToArray(), false);
IntegerNumber three = new IntegerNumber(3L);
//repeat iteration once to reach full precision.
int iterations1 = (int)Math.Ceiling(Math.Log(initialDigits.Count * 64.0 / 52, 2.0));
for (int step = iterations1; --step >= 0;)
{
int shift = computedDigits * 64 * 3;
root = root * ((three << shift) - root.Square() * t) >> (1 + shift);
shrink(root);
}
if (computedDigits == n)
{
return(root);
}
int additionalDigits = 2;
int totalDigits = n + additionalDigits;
do
{
int digits = Math.Min(Math.Max(computedDigits - 1, 1), totalDigits - computedDigits);
computedDigits += digits;
t = this.GetDigits(n - computedDigits, computedDigits, true);
//root <<= digits * 64;
//int shift = computedDigits * 3 * 64;
//root = root * ((three << shift) - root.Square() * t) >> (1 + shift);
int shift1 = (computedDigits * 3 - digits * 2) * 64;
int shift2 = (computedDigits * 3 - digits * 3) * 64;
root = root * ((three << shift1) - root.Square() * t) >> (1 + shift2);
shrink(root);
} while (computedDigits < totalDigits);
root >>= additionalDigits * 64;
shrink(root);
return(root);
}
public static IntegerNumber operator %(IntegerNumber number1, IntegerNumber number2)
{
IntegerNumber remainder, quotient = IntegerNumber.DivRem(number1, number2, out remainder);
return(remainder);
}
