BigInteger與uint[]

用uint[]來表示非負大數，其中陣列開頭是大數的最高32位，陣列結尾是大數最低32位。其與BigInteger的轉換方法

/// <summary>
/// <see cref="uint"/>陣列轉為非負大整數
/// </summary>
private static BigInteger ValueOf(uint[] value)
{
    var result = BigInteger.Zero;
    foreach (var num in value)
    {
        result <<= 32;
        result |= (num & 0xFFFF_FFFF);
    }

    return result;
}

/// <summary>
/// 非負大整數轉為<see cref="uint"/>陣列
/// </summary>
private static uint[] ToIntArray(BigInteger value)
{
    var byteCount = value.GetByteCount();
    var len = (int)Math.Ceiling(byteCount / 4d);
    var result = new uint[len];
    for (var i = len - 1; i >= 0; --i)
    {
        result[i] = (uint)(value & 0xFFFF_FFFF);
        value >>= 32;
    }

    return result;
}
 

測試

[TestMethod]
public void ConvertTest()
{
    var bytes = new byte[32];
    Random ran = new Random();
    for (var i = 0; i < 100; ++i)
    {
        ran.NextBytes(bytes);
        var value = BigInteger.Abs(new BigInteger(bytes));
        var test = ToIntArray(value);
        Assert.AreEqual(value, ValueOf(test));
    }
            
}
 

Unsigned加法(Java BigInteger原始碼)

private static readonly long LONG_MASK = 0xFFFF_FFFFL;

/// <summary>
/// 大數加法，陣列第一個int存放最高32位，最後一個int存放最低32位。
/// </summary>
public static uint[] AddNonegative(uint[] left, uint[] right)
{
    // If left is shorter, swap the two arrays
    if (left.Length < right.Length)
    {
        var tmp = left;
        left = right;
        right = tmp;
    }

    var xIndex = left.Length;
    var yIndex = right.Length;
    var result = new uint[xIndex];
    var sum = 0L;
    if (yIndex == 1)
    {
        sum = (left[--xIndex] & LONG_MASK) + (right[0] & LONG_MASK);
        result[xIndex] = (uint)sum;
    }
    else
    {
        // Add common parts of both numbers
        while (yIndex > 0)
        {
            sum = (left[--xIndex] & LONG_MASK) +
                    (right[--yIndex] & LONG_MASK) + (long)((ulong)sum >> 32);
            result[xIndex] = (uint)sum;
        }
    }

    // Copy remainder of longer number while carry propagation is required
    var carry = ((ulong)sum >> 32) != 0;
    while (xIndex > 0 && carry)
        carry = ((result[--xIndex] = left[xIndex] + 1) == 0);

    // Copy remainder of longer number
    while (xIndex > 0)
        result[--xIndex] = left[xIndex];

    // Grow result if necessary
    if (carry)
    {
        Array.Resize(ref result, result.Length + 1);
        for (var i = result.Length - 1; i > 0; --i)
            result[i] = result[i - 1];

        result[0] = 0x01;
    }

    return result;
}
 

測試

[TestMethod]
public void AddNonegativeTest()
{
    var bytes = new byte[32];
    Random ran = new Random();
    for(var i = 0; i < 100; ++i)
    {
        ran.NextBytes(bytes);
        var left = BigInteger.Abs(new BigInteger(bytes));
        ran.NextBytes(bytes);
        var right = BigInteger.Abs(new BigInteger(bytes));
        var test = AddNonegative(ToIntArray(left), ToIntArray(right));
        var expected = left + right;
        Assert.AreEqual(expected, ValueOf(test));
    }
}

Unsigned減法(Java BigInteger原始碼)

/// <summary>
/// 大數減法，<paramref name="left"/> 大於<paramref name="right"/>。陣列第一個int存放最高32位，最後一個int存放最低32位。
/// </summary>
public static uint[] SubtractNonegative(uint[] left, uint[] right)
{
    var bigIndex = left.Length;
    var result = new uint[bigIndex];
    var littleIndex = right.Length;
    var difference = 0L;

    // Subtract common parts of both numbers
    while (littleIndex > 0)
    {
        difference = (left[--bigIndex] & LONG_MASK) -
                        (right[--littleIndex] & LONG_MASK) +
                        (difference >> 32);
        result[bigIndex] = (uint)difference;
    }

    // Subtract remainder of longer number while borrow propagates
    var borrow = (difference >> 32) != 0;
    while (bigIndex > 0 && borrow)
        borrow = ((result[--bigIndex] = left[bigIndex] - 1) == uint.MaxValue);

    // Copy remainder of longer number
    while (bigIndex > 0)
        result[--bigIndex] = left[bigIndex];

    return result;
}

測試

[TestMethod]
public void SubtractNonegativeTest()
{
    var bytes = new byte[32];
    Random ran = new Random();
    for (var i = 0; i < 100; ++i)
    {
        ran.NextBytes(bytes);
        var left = BigInteger.Abs(new BigInteger(bytes));
        ran.NextBytes(bytes);
        var right = BigInteger.Abs(new BigInteger(bytes));
        while (right > left) right >>= 1;
        var test = SubtractNonegative(ToIntArray(left), ToIntArray(right));
        var expected = left - right;
        Assert.AreEqual(expected, ValueOf(test));
    }
}

BigInteger與(uint[], bool)

上面的減法有太多限制，加法也不能計算負數，接下來推廣到通用加減法。

用(uint[], bool)來表示有符號大數，其中uint[]是大數的絕對值，bool為false時是負數。

其與BigInteger之間轉換方法

/// <summary>
/// (<see cref="uint"/>[], <see cref="bool"/>) to <see cref="BigInteger"/>
/// </summary>
private BigInteger ValueOf((uint[], bool) value)
{
    var result = BigInteger.Zero;
    foreach (var num in value.Item1)
    {
        result <<= 32;
        result |= (num & 0xFFFF_FFFF);
    }

    return value.Item2 ? result : -result;
}


/// <summary>
/// <see cref="BigInteger"/> to (<see cref="uint"/>[], <see cref="bool"/>)
/// </summary>
private (uint[], bool) ToTuple(BigInteger value)
{
    var positive = BigInteger.Abs(value);

    var byteCount = positive.GetByteCount();
    var len = (int)Math.Ceiling(byteCount / 4d);
    var result = new uint[len];
    for (var i = len - 1; i >= 0; --i)
    {
        result[i] = (uint)(positive & 0xFFFF_FFFF);
        positive >>= 32;
    }

    return (result, value >= 0);
}

測試

[TestMethod]
public void ConvertTest()
{
    var bytes = new byte[32];
    Random ran = new Random();
    for (var i = 0; i < 100; ++i)
    {
        ran.NextBytes(bytes);
        var value = new BigInteger(bytes);
        var test = ToTuple(value);
        Assert.AreEqual(value, ValueOf(test));
    }
}

Signed加法

/// <summary>
/// 大數加法，陣列第一個<see cref="uint"/>存放最高32位，最後一個<see cref="uint"/>存放最低32位。
/// </summary>
public static (uint[], bool) Add((uint[], bool) left, (uint[], bool) right)
{
    if (IsZero(left))
        return right;
    if (IsZero(right))
        return left;

    if (left.Item2 == right.Item2)//正負相同
        return (AddNonegative(left.Item1, right.Item1), left.Item2);

    var cmp = Compare(left.Item1, right.Item1);

    if (cmp < 0)//abs(left) < abs(right)
        return (SubtractNonegative(right.Item1, left.Item1), right.Item2);
    else
        return (SubtractNonegative(left.Item1, right.Item1), left.Item2);
}

private static int Compare(uint[] left, uint[] right)
{
    var len1 = left.Length;
    var len2 = right.Length;
    if (len1 < len2)
        return -1;
    if (len1 > len2)
        return 1;
    for (var i = 0; i < len1; i++)
    {
        var a = left[i];
        var b = right[i];
        if (a != b)
            return ((a & LONG_MASK) < (b & LONG_MASK)) ? -1 : 1;
    }

    return 0;
}

private static bool IsZero((uint[], bool) value)
    => value.Item1.Length == 1 && value.Item1[0] == 0;

測試

[TestMethod]
public void AddTest()
{
    var bytes = new byte[32];
    Random ran = new Random();
    for (var i = 0; i < 100; ++i)
    {
        ran.NextBytes(bytes);
        var left = new BigInteger(bytes);
        ran.NextBytes(bytes);
        var right = new BigInteger(bytes);
        var test = Add(ToTuple(left), ToTuple(right));
        var expected = left + right;
        Assert.AreEqual(expected, ValueOf(test));
    }
}

Signed減法

/// <summary>
/// 大數減法，陣列第一個<see cref="uint"/>存放最高32位，最後一個<see cref="uint"/>存放最低32位。
/// </summary>
public static (uint[], bool) Subtract((uint[], bool) left, (uint[], bool) right)
{
    if (IsZero(left))
        return (right.Item1, !right.Item2);
    if (IsZero(right))
        return left;

    if (left.Item2 != right.Item2)//正負相異
        return (AddNonegative(left.Item1, right.Item1), left.Item2);

    var cmp = Compare(left.Item1, right.Item1);
    if (cmp == 0)
        return (new uint[] { 0 }, true);
    else if (cmp < 0)//abs(left) < abs(right)
        return (SubtractNonegative(right.Item1, left.Item1), !right.Item2);
    else
        return (SubtractNonegative(left.Item1, right.Item1), left.Item2);
}

測試

[TestMethod]
public void SubtractTest()
{
    var bytes = new byte[32];
    Random ran = new Random();
    for (var i = 0; i < 100; ++i)
    {
        ran.NextBytes(bytes);
        var left = new BigInteger(bytes);
        ran.NextBytes(bytes);
        var right = new BigInteger(bytes);
        var test = Subtract(ToTuple(left), ToTuple(right));
        var expected = left - right;
        Assert.AreEqual(expected, ValueOf(test));
    }
}