Class LongElementaryModMath
 java.lang.Object

 org.apfloat.internal.LongElementaryModMath

 Direct Known Subclasses:
LongModMath
public class LongElementaryModMath extends Object
Elementary modulo arithmetic functions forlong
data.Modular addition and subtraction are trivial, when the modulus is less than 2^{63} and overflow can be detected easily.
Modular multiplication is more complicated, and since it is usually the single most time consuming operation in the whole program execution, the very core of the Number Theoretic Transform (NTT), it should be carefully optimized.
The algorithm for multiplying two
long
s and taking the remainder is not entirely obvious. The basic problem is to get the full 128bit result of multiplying two 64bit integers. It would be possible to do this by splitting the arguments to high and low 32bit words and performing four multiplications. The performance of this solution would be not very good.Another approach is to use
long
s only for getting the lowest 64 bits of the result. Casting the operands todouble
and multiplying as floatingpoint numbers, we can get the highest (roughly) 52 bits of the result. However since only 116 bits can be acquired this way, it would be possible to only use 58 bits in each of the multiplication operands (not the full 64 or 63 bits). Furthermore, roundoff errors in the floatingpoint multiplications, as allowed by the IEEE specification, actually prevent getting even 52 of the top bits accurately, and actually only 57 bits can be used in the multiplication operands. This is the approach chosen in this implementation.The first observation is that since the modulus is practically constant, it should be more efficient to calculate (once) the inverse of the modulus, and then subsequently multiply by the inverse modulus instead of dividing by the modulus.
The second observation is that to get the remainder of the division, we don't necessarily need the actual result of the division (we just want the remainder). So, we should discard the topmost 50 bits of the full 114bit result whenever possible, to save a few operations.
The basic approach is to get an approximation of
a * b / modulus
(using floatingpoint operands, that isdouble
s). The approximation should be within +1 or 1 of the correct result. We first calculatea * b  approximateDivision * modulus
to get the initial remainder. This calculation can use the lowest 64 bits only and is done usinglong
s. It is enough to use adouble
to do the approximate division, as it eliminates at least 51 bits from the top of the 114bit multiplication result, leaving at most 63 bits in the remainder. The calculationresult  approximateDivision * modulus
must then be done once more to reduce the remainder since the original multiplication operands are only 57bit numbers. The second reduction reduces the results to the correct value ±modulus. It is then easy to detect the case when the approximate division was off by one (and the remainder is±modulus
off) as the final step of the algorithm. Version:
 1.0
 Author:
 Mikko Tommila


Constructor Summary
Constructors Constructor Description LongElementaryModMath()
Default constructor.

Method Summary
Modifier and Type Method Description long
getModulus()
Get the modulus.long
modAdd(long a, long b)
Modular addition.long
modMultiply(long a, long b)
Modular multiplication.long
modSubtract(long a, long b)
Modular subtraction.void
setModulus(long modulus)
Set the modulus.



Method Detail

modMultiply
public final long modMultiply(long a, long b)
Modular multiplication. Parameters:
a
 First operand.b
 Second operand. Returns:
a * b % modulus

modAdd
public final long modAdd(long a, long b)
Modular addition. Parameters:
a
 First operand.b
 Second operand. Returns:
(a + b) % modulus

modSubtract
public final long modSubtract(long a, long b)
Modular subtraction. The result is always >= 0. Parameters:
a
 First operand.b
 Second operand. Returns:
(a  b + modulus) % modulus

getModulus
public final long getModulus()
Get the modulus. Returns:
 The modulus.

setModulus
public final void setModulus(long modulus)
Set the modulus. Parameters:
modulus
 The modulus.

