Modular arithmetic

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 $p$ called the congruence modulus, two numbers $a$ and $b$ are said to be congruent modulo p whenever there exists an integer $k$ such that:

$a=b+k\,p$

The corresponding set of equivalence classes forms a ring denoted ${\frac {\mathbb {Z} }{p\mathbb {Z} }}$. 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:

$f(x)=x^{100}+x+1$

You will use $13$ as the congruence modulus and you will compute $f(10)$.

It is important that the function $f$ 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.

Haskell

<lang haskell>-- 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 :: ℤ/13 -> ℤ/13 f x = x^100 + x + 1

main :: IO () main = print (f 10)</lang>

Output:

./modarith 
1

PARI/GP

This feature exists natively in GP: <lang parigp>Mod(3,7)+Mod(4,7)</lang>

Perl

There is a CPAN module called Math::ModInt which does the job. <lang Perl>use Math::ModInt qw(mod); sub f { my $x = shift; $x**100 + $x + 1 }; print f mod(10, 13);</lang>

Output:

mod(1, 13)

Perl 6

There is a Panda module called Modular which works basically as Perl 5's Math::ModInt. <lang Perl 6>use Modular; sub f(\x) { x**100 + x + 1}; say f( 10 Mod 13 )</lang>

Output:

1 「mod 13」

Prolog

Works with SWI-Prolog versin 6.4.1 and module lambda (found there : http://www.complang.tuwien.ac.at/ulrich/Prolog-inedit/lambda.pl ). <lang Prolog>:- 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). </lang>

Output:

 ?- 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].

Python

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.

<lang Python>import operator import functools

@functools.total_ordering 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
           else:
               return NotImplemented
       elif isinstance(other, int):
           return self.val == other
       else:
           return NotImplemented

   def __lt__(self, other):
       if isinstance(other, Mod):
           if self.mod == other.mod:
               return self.val<other.val
           else:
               return NotImplemented
       elif isinstance(other, int):
           return self.val < other
       else:
           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):
       self._check_operand(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.

1. Helper functions to build common operands based on a template.
2. 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):
       self._check_operand(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):
       self._check_operand(other)
       return Mod(op_fun(int(other), self.val) % self.mod, self.mod)
   return op

1. 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

print(f(Mod(10,13)))

1. Output: Mod(1, 13)</lang>

Racket

<lang racket>#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 "")))
                     math)
        (only-in math with-modulus))

(define (f x) (+ (expt x 100) x 1)) (with-modulus 13 (f 10))

=> 1</lang>

Ruby

<lang ruby># 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
 end

 def to_i
   @n
 end

 def <=>(other_n)
   @n <=> other_n.to_i
 end

 [:+, :-, :*, :**].each do |meth|
   define_method(meth) { |other_n| Modulo.new(@n.send(meth, other_n.to_i), @m) }
 end

 def coerce(numeric)
   [numeric, @n]
 end

end

1. Demo

x, y = Modulo.new(10), Modulo.new(20)

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> </lang>

Sidef

Translation of: Ruby

<lang 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}, #{m})" }

}

func f(x) { x**100 + x + 1 } say f(Modulo(10, 13))</lang>

Output:

(1, 13)

Tcl

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)

<lang tcl>package require Tcl 8.6 package require pt::pgen

1. 1. 2. A simple expression parser for a subset of Tcl's expression language

1. 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	<- '**'						;

END; }

1. Instantiate the parser class

catch [pt::pgen peg $grammar snit -class Calculator -name Grammar]

1. 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]

   }

}</lang> 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. <lang tcl># 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 + - * / **

}

1. Put all the pieces together

set comp [CompileAST new [ModEval create mod13 13]]</lang> Finally, demonstrating… <lang tcl>set compiled [$comp compile {$x**100 + $x + 1}] set x 10 puts "[eval $compiled] = $compiled"</lang>

Output:

1 = ::mod13 + [::mod13 + [::mod13 ** [::mod13 variable x] [::mod13 value 100]] [::mod13 variable x]] [::mod13 value 1]

zkl

Doing just enough to perform the task: <lang zkl>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;
     self(z.divr(M)[1],M) 
  }

}</lang> Using GNU GMP lib to do the big math (to avoid writing a powmod function): <lang zkl>var BN=Import("zklBigNum"); fcn f(n){ n.pow(100) + n + 1 } f(1).println(" <-- 1^100 + 1 + 1"); n:=MC(BN(10),13); (n+3).println(" <-- 10M13 + 3"); f(n).println(" <-- 10M13^100 + 10M13 + 1");</lang>

Output:

3 <-- 1^100 + 1 + 1
0M13 <-- 10M13 + 3
1M13 <-- 10M13^100 + 10M13 + 1