Package org.apfloat.internal

Class FloatElementaryModMath

java.lang.Object

org.apfloat.internal.FloatElementaryModMath

Direct Known Subclasses:

FloatModMath

public class FloatElementaryModMath
extends Object

Elementary modulo arithmetic functions for float data. Note that although a floating-point data type is used, the data will always be integers.

Since the moduli are close to 2²⁴ some attention must be paid to avoiding overflow in modular addition and subtraction. This can be done easily e.g. by casting the operands to double. Note that an IEEE float has a mantissa with a precision of 24 bits (1 + 23).

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.

Some obvious (but not very efficient) algorithms for multiplying two floats and taking the remainder would be to call Math.IEEEremainder(), or cast the operands to long, e.g.

(float) ((long) a * (long) b % (long) modulus)

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 algorithm used in this implementation casts the operands to double, performs the multiplication, multiplies by the inverse modulus, then takes the integer part. Getting the integer part is typically a lot faster by casting to int compared to e.g. calling Math.floor(). An int, holding 32 bits, can easily contain the result of the cast, which will have a maximum of 24 bits.

Overflow is not a problem, since a double can hold 53 bits precisely in the mantissa – more than doubly what a float can. Note that multiplying by the inverse modulus is also trivial, when the inverse modulus has more than twice accurate bits than what are in each of the multiplicands. Since the modulus is assumed to be prime, there can be no situations where multiplication by the inverse modulus would have a near-integer result that would be rounded incorrectly, e.g. as in 0.333... * 3 = 0.999....

Version:

1.0

Author:

Mikko Tommila

Constructor Summary

Constructors

Constructor	Description
FloatElementaryModMath()	Default constructor.

Method Summary

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

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Details

FloatElementaryModMath

public FloatElementaryModMath()

Default constructor.

Method Details

modMultiply

public final float modMultiply(float a,
 float b)

Modular multiplication.

Parameters:

a - First operand.

b - Second operand.

Returns:

a * b % modulus

modAdd

public final float modAdd(float a,
 float b)

Modular addition.

Parameters:

a - First operand.

b - Second operand.

Returns:

(a + b) % modulus

modSubtract

public final float modSubtract(float a,
 float b)

Modular subtraction. The result is always >= 0.

Parameters:

a - First operand.

b - Second operand.

Returns:

(a - b + modulus) % modulus

getModulus

public final float getModulus()

Get the modulus.

Returns:

The modulus.

setModulus

public final void setModulus(float modulus)

Set the modulus.

Parameters:

modulus - The modulus.