Montgomery-form Wider words, on the Montgomery domain defined by the secp256k1 field prime.

>>> let one = 1 :: Montgomery
>>> one
1
>>> putStrLn (render one)
(4294968273, 0, 0, 0)

Render a Montgomery value as a String, showing its individual Limbs.

>>> putStrLn (render 1)
(4294968273, 0, 0, 0)

Convert a Wider word to the Montgomery domain.

Retrieve a Montgomery word from the Montgomery domain.

This function is a synonym for retr.

Zero (the additive unit) in the Montgomery domain.

One (the multiplicative unit) in the Montgomery domain.

Constant-time equality comparison.

Variable-time equality comparison.

Montgomery reduction.

The first argument represents the low words, and the second the high words, of an extra-large eight-limb word in Montgomery form.

Montgomery reduction.

Retrieve a Montgomery value from the Montgomery domain, producing a Wider word.

Return a if c is truthy, otherwise return b.

>>> import qualified Data.Choice as C
>>> select 0 1 (C.true# ())
1

Addition in the Montgomery domain.

Note that Montgomery is an instance of Num, so you can use + to apply this function.

>>> 1 + 1 :: Montgomery
2

Subtraction in the Montgomery domain.

Note that Montgomery is an instance of Num, so you can use - to apply this function.

>>> 1 - 1 :: Montgomery
0

Multiplication in the Montgomery domain.

Note that Montgomery is an instance of Num, so you can use * to apply this function.

>>> 1 * 1 :: Montgomery
1

Squaring in the Montgomery domain.

>>> sqr 1
1
>>> sqr 2
4
>>> sqr (negate 2)
4

Additive inverse in the Montgomery domain.

Note that Montgomery is an instance of Num, so you can use negate to apply this function.

>>> negate 1 :: Montgomery
115792089237316195423570985008687907853269984665640564039457584007908834671662
>>> (negate 1 :: Montgomery) + 1
0

Multiplicative inverse in the Montgomery domain.

> inv 2

57896044618658097711785492504343953926634992332820282019728792003954417335832 >> inv 2 * 2 1

Square root (Tonelli-Shanks) in the Montgomery domain.

Returns Nothing if the square root doesn't exist.

Note that the square root calculation itself is performed in constant time; we branch only when casting to Maybe at the end.

>>> sqrt_vartime 4
Just 2
>>> sqrt_vartime 15
Just 69211104694897500952317515077652022726490027694212560352756646854116994689233
>>> (*) <$> sqrt_vartime 15 <*> sqrt_vartime 15
Just 15

Exponentiation in the Montgomery domain.

>>> exp 2 3
8
>>> exp 2 10
1024

Check if a Montgomery value is odd.

Note that the comparison is performed in constant time, but we branch when converting to Bool.

>>> odd 1
True
>>> odd 2
False
>>> Data.Word.Wider.odd (retr 3) -- parity is preserved
True