Modular arithmetic

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Modular arithmetic is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Modular arithmetic is a form of arithmetic (a calculation technique involving the concepts of addition and multiplication) which is done on numbers with a defined equivalence relation called congruence.

For any positive integer called the congruence modulus, two numbers and are said to be congruent modulo p whenever there exists an integer such that:

The corresponding set of equivalence classes forms a ring denoted . Addition and multiplication on this ring have the same algebraic structure as in usual arithmetics, so that a function such as a polynomial expression could receive a ring element as argument and give a consistent result.

The purpose of this task is to show, if your programming language allows it, how to redefine operators so that they can be used transparently on modular integers. You can do it either by using a dedicated library, or by implementing your own class.

You will use the following function for demonstration:

You will use as the congruence modulus and you will compute .

It is important that the function is agnostic about whether or not its argument is modular; it should behave the same way with normal and modular integers. In other words, the function is an algebraic expression that could be used with any ring, not just integers.

ALGOL 68[edit]

Works with: ALGOL 68G version Any - tested with release 2.8.win32
# allow for large integers in Algol 68G #
PR precision 200 PR
# modular integer type #
# modular integer + and * operators #
# where both operands are modular, they must have the same modulus #
OP + = ( MODULARINT a, b )MODULARINT: ( ( v OF a + v OF b ) MOD modulus OF a, modulus OF a );
OP + = ( MODULARINT a, INT b )MODULARINT: ( ( v OF a + b ) MOD modulus OF a, modulus OF a );
OP * = ( MODULARINT a, b )MODULARINT: ( ( v OF a * v OF b ) MOD modulus OF a, modulus OF a );
OP ** = ( MODULARINT a, INT b )MODULARINT: ( ( v OF a ** b ) MOD modulus OF a, modulus OF a );
# f(x) function - can be applied to either LONG LONG INT or MODULARINT values #
# the result is always a LONG LONG INT #
IN ( LONG LONG INT ix ): ( ix**100 + ix + 1 )
, ( MODULARINT mx ): v OF ( mx**100 + mx + 1 )
print( ( whole( f( MODULARINT( 10, 13 ) ), 0 ), newline ) )


Go does not allow redefinition of operators. That element of the task cannot be done in Go. The element of defining f so that it can be used with any ring however can be done, just not with the syntactic sugar of operator redefinition.

package main
import "fmt"
// Define enough of a ring to meet the needs of the task. Addition and
// multiplication are mentioned in the task; multiplicative identity is not
// mentioned but is useful for the power function.
type ring interface {
add(ringElement, ringElement) ringElement
mul(ringElement, ringElement) ringElement
mulIdent() ringElement
type ringElement interface{}
// Define a power function that works for any ring.
func ringPow(r ring, a ringElement, p uint) (pow ringElement) {
for pow = r.mulIdent(); p > 0; p-- {
pow = r.mul(pow, a)
// The task function f has that constant 1 in it.
// Define a special kind of ring that has this element.
type oneRing interface {
one() ringElement // return ring element corresponding to '1'
// Now define the required function f.
// It works for any ring (that has a "one.")
func f(r oneRing, x ringElement) ringElement {
return r.add(r.add(ringPow(r, x, 100), x),
// With rings and the function f defined in a general way, now define
// the specific ring of integers modulo n.
type modRing uint // value is congruence modulus n
func (m modRing) add(a, b ringElement) ringElement {
return (a.(uint) + b.(uint)) % uint(m)
func (m modRing) mul(a, b ringElement) ringElement {
return (a.(uint) * b.(uint)) % uint(m)
func (modRing) mulIdent() ringElement { return uint(1) }
func (modRing) one() ringElement { return uint(1) }
// Demonstrate the general function f on the specific ring with the
// specific values.
func main() {
fmt.Println(f(modRing(13), uint(10)))


-- We use a couple of GHC extensions to make the program cooler.  They let us
-- use / as an operator and 13 as a literal at the type level. (The library
-- also provides the fancy Zahlen (ℤ) symbol as a synonym for Integer.)
{-# Language DataKinds #-}
{-# Language TypeOperators #-}
import Data.Modular
f ::/13 ->/13
f x = x^100 + x + 1
main :: IO ()
main = print (f 10)


This feature exists natively in GP:



There is a CPAN module called Math::ModInt which does the job.

use Math::ModInt qw(mod);
sub f { my $x = shift; $x**100 + $x + 1 };
print f mod(10, 13);
mod(1, 13)

Perl 6[edit]

There is a Panda module called Modular which works basically as Perl 5's Math::ModInt.

use Modular;
sub f(\x) { x**100 + x + 1};
say f( 10 Mod 13 )
1 「mod 13」


Works with SWI-Prolog versin 6.4.1 and module lambda (found there : ).

:- use_module(library(lambda)).
congruence(Congruence, In, Fun, Out) :-
maplist(Congruence +\X^Y^(Y is X mod Congruence), In, In1),
call(Fun, In1, Out1),
maplist(Congruence +\X^Y^(Y is X mod Congruence), Out1, Out).
fun_1([X], [Y]) :-
Y is X^100 + X + 1.
fun_2(L, [R]) :-
sum_list(L, R).
 ?- congruence(13, [10], fun_1, R).
R = [1].

 ?- congruence(13, [10, 15, 13, 9, 22], fun_2, R).
R = [4].

 ?- congruence(13, [10, 15, 13, 9, 22], maplist(\X^Y^(Y is X * 13)), R).
R = [0,0,0,0,0].


Works with: Python version 3.x

We need to implement a Modulo type first, then give one of its instances to the "f" function.
Thanks to duck typing, the function doesn't need to care about the actual type it's given. We also use the dynamic nature of Python to dynamically build the operator overload methods and avoid repeating very similar code.

import operator
import functools
class Mod:
__slots__ = ['val','mod']
def __init__(self, val, mod):
if not isinstance(val, int):
raise ValueError('Value must be integer')
if not isinstance(mod, int) or mod<=0:
raise ValueError('Modulo must be positive integer')
self.val = val % mod
self.mod = mod
def __repr__(self):
return 'Mod({}, {})'.format(self.val, self.mod)
def __int__(self):
return self.val
def __eq__(self, other):
if isinstance(other, Mod):
if self.mod == other.mod:
return self.val==other.val
return NotImplemented
elif isinstance(other, int):
return self.val == other
return NotImplemented
def __lt__(self, other):
if isinstance(other, Mod):
if self.mod == other.mod:
return self.val<other.val
return NotImplemented
elif isinstance(other, int):
return self.val < other
return NotImplemented
def _check_operand(self, other):
if not isinstance(other, (int, Mod)):
raise TypeError('Only integer and Mod operands are supported')
if isinstance(other, Mod) and self.mod != other.mod:
raise ValueError('Inconsistent modulus: {} vs. {}'.format(self.mod, other.mod))
def __pow__(self, other):
# We use the built-in modular exponentiation function, this way we can avoid working with huge numbers.
return Mod(pow(self.val, int(other), self.mod), self.mod)
def __neg__(self):
return Mod(self.mod - self.val, self.mod)
def __pos__(self):
return self # The unary plus operator does nothing.
def __abs__(self):
return self # The value is always kept non-negative, so the abs function should do nothing.
# Helper functions to build common operands based on a template.
# They need to be implemented as functions for the closures to work properly.
def _make_op(opname):
op_fun = getattr(operator, opname) # Fetch the operator by name from the operator module
def op(self, other):
return Mod(op_fun(self.val, int(other)) % self.mod, self.mod)
return op
def _make_reflected_op(opname):
op_fun = getattr(operator, opname)
def op(self, other):
return Mod(op_fun(int(other), self.val) % self.mod, self.mod)
return op
# Build the actual operator overload methods based on the template.
for opname, reflected_opname in [('__add__', '__radd__'), ('__sub__', '__rsub__'), ('__mul__', '__rmul__')]:
setattr(Mod, opname, _make_op(opname))
setattr(Mod, reflected_opname, _make_reflected_op(opname))
def f(x):
return x**100+x+1
# Output: Mod(1, 13)


#lang racket
(require racket/require
 ;; grab all "mod*" names, but get them without the "mod", so
 ;; `+' and `expt' is actually `mod+' and `modexpt'
(filtered-in (λ(n) (and (regexp-match? #rx"^mod" n)
(regexp-replace #rx"^mod" n "")))
(only-in math with-modulus))
(define (f x) (+ (expt x 100) x 1))
(with-modulus 13 (f 10))
;; => 1


# stripped version of Andrea Fazzi's submission to Ruby Quiz #179
class Modulo
include Comparable
def initialize(n = 0, m = 13)
@n, @m = n % m, m
def to_i
def <=>(other_n)
@n <=> other_n.to_i
[:+, :-, :*, :**].each do |meth|
define_method(meth) { |other_n|, other_n.to_i), @m) }
def coerce(numeric)
[numeric, @n]
# Demo
x, y =,
p x > y # true
p x == y # false
p [x,y].sort #[#<Modulo:0x000000012ae0f8 @n=7, @m=13>, #<Modulo:0x000000012ae148 @n=10, @m=13>]
p x + y ##<Modulo:0x0000000117e110 @n=4, @m=13>
p 2 + y # 9
p y + 2 ##<Modulo:0x00000000ad1d30 @n=9, @m=13>
p x**100 + x +1 ##<Modulo:0x00000000ad1998 @n=1, @m=13>


Translation of: Ruby
class Modulo(n=0, m=13) {
method init {
(n, m) = (n % m, m)
method to_n { n }
< + - * ** >.each { |meth|
Modulo.def_method(meth, method(n2) { Modulo(n.(meth)(n2.to_n), m) })
method to_s { "#{n} 「mod #{m}」" }
func f(x) { x**100 + x + 1 }
say f(Modulo(10, 13))
1 「mod 13」


Tcl does not permit overriding of operators, but does not force an expression to be evaluated as a standard expression.
Creating a parser and custom evaluation engine is relatively straight-forward, as is shown here.

Library: Tcllib (Package: pt::pgen)
package require Tcl 8.6
package require pt::pgen
### A simple expression parser for a subset of Tcl's expression language
# Define the grammar of expressions that we want to handle
set grammar {
PEG Calculator (Expression)
Expression <- Term (' '* AddOp ' '* Term)* ;
Term <- Factor (' '* MulOp ' '* Factor)* ;
Fragment <- '(' ' '* Expression ' '* ')' / Number / Var ;
Factor <- Fragment (' '* PowOp ' '* Fragment)* ;
Number <- Sign? Digit+ ;
Var <- '$' ( 'x'/'y'/'z' ) ;
Digit <- '0'/'1'/'2'/'3'/'4'/'5'/'6'/'7'/'8'/'9' ;
Sign <- '-' / '+' ;
MulOp <- '*' / '/' ;
AddOp <- '+' / '-' ;
PowOp <- '**' ;
# Instantiate the parser class
catch [pt::pgen peg $grammar snit -class Calculator -name Grammar]
# An engine that compiles an expression into Tcl code
oo::class create CompileAST {
variable sourcecode opns
constructor {semantics} {
set opns $semantics
method compile {script} {
# Instantiate the parser
set c [Calculator]
set sourcecode $script
try {
return [my {*}[$c parset $script]]
} finally {
$c destroy
method Expression-Empty args {}
method Expression-Compound {from to args} {
foreach {o p} [list Expression-Empty {*}$args] {
set o [my {*}$o]; set p [my {*}$p]
set v [expr {$o ne "" ? "$o \[$v\] \[$p\]" : $p}]
return $v
forward Expression my Expression-Compound
forward Term my Expression-Compound
forward Factor my Expression-Compound
forward Fragment my Expression-Compound
method Expression-Operator {from to args} {
list ${opns} [string range $sourcecode $from $to]
forward AddOp my Expression-Operator
forward MulOp my Expression-Operator
forward PowOp my Expression-Operator
method Number {from to args} {
list ${opns} value [string range $sourcecode $from $to]
method Var {from to args} {
list ${opns} variable [string range $sourcecode [expr {$from+1}] $to]

None of the code above knows about modular arithmetic at all, or indeed about actual expression evaluation. Now we define the semantics that we want to actually use.

# The semantic evaluation engine; this is the part that knows mod arithmetic
oo::class create ModEval {
variable mod
constructor {modulo} {set mod $modulo}
method value {literal} {return [expr {$literal}]}
method variable {name} {return [expr {[set ::$name]}]}
method + {a b} {return [expr {($a + $b) % $mod}]}
method - {a b} {return [expr {($a - $b) % $mod}]}
method * {a b} {return [expr {($a * $b) % $mod}]}
method / {a b} {return [expr {($a / $b) % $mod}]}
method ** {a b} {
# Tcl supports bignums natively, so we use the naive version
return [expr {($a ** $b) % $mod}]
export + - * / **
# Put all the pieces together
set comp [CompileAST new [ModEval create mod13 13]]

Finally, demonstrating…

set compiled [$comp compile {$x**100 + $x + 1}]
set x 10
puts "[eval $compiled] = $compiled"
1 = ::mod13 + [::mod13 + [::mod13 ** [::mod13 variable x] [::mod13 value 100]] [::mod13 variable x]] [::mod13 value 1]


Doing just enough to perform the task:

class MC{
fcn init(n,mod){ var N=n,M=mod; }
fcn toString { String(N.divr(M)[1],"M",M) }
fcn pow(p) { self( N.pow(p).divr(M)[1], M ) }
fcn __opAdd(mc){
if(mc.isType(Int)) z:=N+mc; else z:=N*M + mc.N*mc.M;

Using GNU GMP lib to do the big math (to avoid writing a powmod function):

var BN=Import("zklBigNum");
fcn f(n){ n.pow(100) + n + 1 }
f(1).println(" <-- 1^100 + 1 + 1");
(n+3).println(" <-- 10M13 + 3");
f(n).println(" <-- 10M13^100 + 10M13 + 1");
3 <-- 1^100 + 1 + 1
0M13 <-- 10M13 + 3
1M13 <-- 10M13^100 + 10M13 + 1