Field elements
Last updated
Last updated
Field element is the most fundamental datatype in Keelung.
They can be constructed with numeric literals like 3 , 42, or -7.
Negative numbers don't really exist in finite fields, they will be normalized to their positive counterparts after compilation. For example, -7 would become:
in GF181 where there are only 1552511030102430251236801561344621993261920897571225601 numbers.
Field elements are constructed with the following constructor under the hood:
A numeric literal represents the application of the function fromInteger to the appropriate value of type Integer.
A floating literal stands for an application of fromRational to a value of type Rational.
Example usage: fromRational 3.5
BtoF converts to Booleans to field elements:
Example usage: 3 + 4
Example usage: 3 - 4
Example usage: 3 * 4
Example usage: 3 / 4
Example usage: recip 4
Example usage: negate 4
Since there are no negative numbers in Field, the abs function has the same effect as the id function.
signum would always return 1 on any field element.
pow takes a Field as the base and an Integer as the power, and computes the exponentiation.
If you are multiplying the same stuff for more than 3 times, for example:
You should consider replacing it with:
To allow the exponentiation by squaring optimization to kick in.
Example usage: false `eq` false or eq false true.
You can also use EqF , which is the underlying implementation of eq on field elements.
neq is implemented as such under the hood:
Since Field is an instance of and , all methods of Num and Fractional works on Numbers.
Returns when two field elements are the same:
Returns when two numbers are the same: