scalarff

A minimal, opinionated, library for working with scalar finite fields.

Usage

This library exports a FieldElement trait and concrete implementations for the following curves:

• OxfoiFieldElement - 2^64 - 2^32 + 1 field element
• Curve25519FieldElement - curve25519 field element powered by
• Bn128FieldElement - alt_bn128 field element powered by

use scalarff::FieldElement; // Bring the trait in scope
// Import 1 or more concrete instances
use scalarff::Bn128FieldElement;
use scalarff::Curve25519FieldElement;
use scalarff::OxfoiFieldElement;

// later in a function
{
    let x = Curve25519FieldElement::from(100_u64);
    let y = Curve25519FieldElement::from(200_u64);
    let z = x * y; // multiplication with modular reduction
}

Math

This library is designed to provide easy access to field elements and their corresponding real number representations as BigUint instances. This combination is used to implement higher level functions.

See the 1000 residues example to get started.

finding the next 10 residues in field alt_bn128: starting at 360
    -361_alt_bn128 = 19 * 279774667609210862_L60
    -362_alt_bn128 = 234352660719301385_L60 * 45422006889909496_L60
    -363_alt_bn128 = 1031407624700784573_L60 * 401288547515273284_L60
    -373_alt_bn128 = 739289271740562462_L60 * 693406900475495395_L60
    -374_alt_bn128 = 118418780304769778_L60 * 161355887304441103_L60
    -377_alt_bn128 = 612775390531146123_L60 * 819920781684911734_L60
    -380_alt_bn128 = 282368138647270618_L60 * 1150328033568787239_L60
    -382_alt_bn128 = 992674259711445781_L60 * 440021912504612076_L60
    -383_alt_bn128 = 867565972486375475_L60 * 565130199729682382_L60
    -384_alt_bn128 = 109351148931867722_L60 * 170423518677343159_L60
^^^^^^^^^^ function executed in 22 ms ^^^^^^^^^^
||||||||||||||||||||||||||||||||||||||||
finding the next 10 residues in field curve25519: starting at 360
    -361_curve25519 = 19 * 581636266764129242_L60
    -362_curve25519 = 181526123365884781_L60 * 400110143398244480_L60
    -363_curve25519 = 59257890370718404_L60 * 522378376393410857_L60
    -364_curve25519 = 948291384789065767_L60 * 786266386581910470_L60
    -365_curve25519 = 436801762965341433_L60 * 144834503798787828_L60
    -369_curve25519 = 557032144741475498_L60 * 24604122022653763_L60
    -371_curve25519 = 111337551809662451_L60 * 470298714954466810_L60
    -373_curve25519 = 847841932944108498_L60 * 886715838426867739_L60
    -375_curve25519 = 777357655433650296_L60 * 957200115937325941_L60
    -377_curve25519 = 1097662937271356306_L60 * 636894834099619931_L60
^^^^^^^^^^ function executed in 3 ms ^^^^^^^^^^
||||||||||||||||||||||||||||||||||||||||
finding the next 10 residues in field oxfoi: starting at 360
    -360_oxfoi = 4886810760654287587 * 13559933308760296734
    -361_oxfoi = 19 * 18446744069414584302
    -363_oxfoi = 3096224742375424 * 18443647844672208897
    -364_oxfoi = 640366319723949669 * 17806377749690634652
    -368_oxfoi = 8139125605395827597 * 10307618464018756724
    -369_oxfoi = 3284662639461963411 * 15162081429952620910
    -371_oxfoi = 2993791755975720565 * 15452952313438863756
    -373_oxfoi = 8308875621651992349 * 10137868447762591972
    -375_oxfoi = 8637146607354536426 * 9809597462060047895
    -384_oxfoi = 1152912708379604992 * 17293831361034979329
^^^^^^^^^^ function executed in 3 ms ^^^^^^^^^^
||||||||||||||||||||||||||||||||||||||||
10 quadratic residues in alt_bn128 executed in 22 ms
10 quadratic residues in curve25519 executed in 3 ms
10 quadratic residues in oxfoi executed in 3 ms