【筆記】大數乘法之Karatsuba演算法 (Java BigInteger原始碼)
阿新 • • 發佈:2018-11-10
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乘法(Java BigInteger原始碼)
/// <summary> /// 非負大數Karatsuba乘法,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[] MultiplyKaratsubaNonegative(uint[] left, uint[] right) { var xlen = left.Length; var ylen = right.Length; // The number of ints in each half of the number. var half = (Math.Max(xlen, ylen) + 1) >> 1; // xl and yl are the lower halves of x and y respectively, // xh and yh are the upper halves. var xl = GetLower(left, half); var xh = GetUpper(left, half); var yl = GetLower(right, half); var yh = GetUpper(right, half); var p1 = MultiplyKaratsubaNonegative(xh, yh); // p1 = xh*yh var p2 = MultiplyKaratsubaNonegative(xl, yl); // p2 = xl*yl // p3 = (xl + xh)*(y1 + yh) - p1 - p2 = xh*yl + xl*yh var t1 = MultiplyKaratsubaNonegative(xh, yl); var t2 = MultiplyKaratsubaNonegative(xl, yh); var p3 = AddNonegative(t1, t2); // result = p1 * 2^(32*2*half) + p3 * 2^(32*half) + p2 Array.Resize(ref p1, (half << 1) + p1.Length); Array.Resize(ref p3, half + p3.Length); var result = AddNonegative(p1, p3); return AddNonegative(result, p2); } private static uint[] GetLower(uint[] value, int n) { if (value.Length <= n) { return value; } return value.Skip(value.Length - n).Take(n).ToArray(); } private static uint[] GetUpper(uint[] value, int n) { if (value.Length <= n) { return new uint[] { 0 }; } return value.Take(value.Length - n).ToArray(); }
其中大數左移(32*half)位,因為移位數是32的倍數,而uint是32位,所以只需要將陣列長度擴大half就可以了。測試
[TestMethod]
public void MultiplyKaratsubaNonegativeTest()
{
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 = MultiplyKaratsubaNonegative(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乘法
/// <summary>
/// 大數乘法,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) MultiplyKaratsuba((uint[], bool) left, (uint[], bool) right)
{
if (IsZero(left))
return left;
if (IsZero(right))
return right;
if (IsAbsOne(left))
return (right.Item1, right.Item2 == left.Item2);
if (IsAbsOne(right))
return (left.Item1, left.Item2 == right.Item2);
return (MultiplyKaratsubaNonegative(left.Item1, right.Item1), left.Item2 == right.Item2);
}
private static bool IsZero((uint[], bool) value)
=> value.Item1.Length == 1 && value.Item1[0] == 0;
private static bool IsAbsOne((uint[], bool) value)
=> value.Item1.Length == 1 && value.Item1[0] == 1;
測試
[TestMethod]
public void MultiplyKaratsubaTest()
{
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 = MultiplyKaratsuba(ToTuple(left), ToTuple(right));
var expected = left * right;
Assert.AreEqual(expected, ValueOf(test));
}
}