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## modular-arithmetic

### Author

### Requirements

### Description

This library allows you to perform modular arithmetic on finite fields with arbitrarily large integer moduli.

### API

*[procedure]*

`(xgcd a b)`

Computes the "extended greatest common divisor" of a and b, ie. it returns the two values x and y that constitute the solution of the equation

x*a + y*b = gcd(a, b)

*[procedure]*

`((mod+ modulus) n ...)`

Computes the sum of all `n` parameters just like the standard procedure `+` does, but operating on the finite field with the given `modulus`.

*[procedure]*

`((mod- modulus) a n ...)`

Computes the difference of `a` and all `n` parameters or the additive inverse of `a` just like the standard procedure `-` does, but operating on the finite field with the given `modulus`.

*[procedure]*

`((mod* modulus) n ...)`

Computes the product of all `n` parameters just like the standard procedure `*` does, but operating on the finite field with the given `modulus`.

*[procedure]*

`((mod/ modulus) a n ...)`

Computes the quotient of `a` and all `n` parameters or the multiplicative inverse of `a` just like the standard procedure `/` does, but operating on the finite field with the given `modulus`.

Note that a unique multiplicative inverse of an element in a finite field only exists if the element and the modulus are coprime. This procedure only works in that case.

*[procedure]*

`((modexpt modulus) a b)`

Computes a raised to the power of b modulo the given modulus, but does so far more efficiently than using

(modulo (* a b) modulus).

If `b` is negative, the multiplicative inverse of `a` is raised to the power of `(abs b)`.

*[syntax]*

`(with-modulus modulus body ...)`

Overloads the symbols `+`, `add1`, `-`, `sub1`, `*`, `/` and `expt` inside `body` with the modular versions operating on the finite field with the given `modulus` instead of the standard arithmetic procedures.

### License

Copyright (c) 2010, Thomas Chust All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: * Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. * Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. * Neither the name of the author nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDERS OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

### Version history

- 1.0.0
- Initial release