Modular arithmetic
(Redirected from Modular Arithmetic)
You are encouraged to solve this task according to the task description, using any language you may know.
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 . When p is a prime number, this ring becomes a field denoted , but you won't have to implement the multiplicative inverse for this task.
Addition and multiplication on this ring have the same algebraic structure as in usual arithmetic, 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.
- Related tasks
Ada
Ada has modular types.
with Ada.Text_IO;
procedure Modular_Demo is
type Modul_13 is mod 13;
function F (X : Modul_13) return Modul_13 is
begin
return X**100 + X + 1;
end F;
package Modul_13_IO is
new Ada.Text_IO.Modular_IO (Modul_13);
use Ada.Text_IO;
use Modul_13_IO;
X_Integer : constant Integer := 10;
X_Modul_13 : constant Modul_13 := Modul_13'Mod (X_Integer);
F_10 : constant Modul_13 := F (X_Modul_13);
begin
Put ("f("); Put (X_Modul_13); Put (" mod "); Put (Modul_13'Modulus'Image); Put (") = ");
Put (F_10); Put (" mod "); Put (Modul_13'Modulus'Image);
New_Line;
end Modular_Demo;
- Output:
f( 10 mod 13) = 1 mod 13
ALGOL 68
# allow for large integers in Algol 68G #
PR precision 200 PR
# modular integer type #
MODE MODULARINT = STRUCT( LONG LONG INT v, INT modulus );
# 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 #
PROC f = ( UNION( LONG LONG INT, MODULARINT ) x )LONG LONG INT:
CASE x
IN ( LONG LONG INT ix ): ( ix**100 + ix + 1 )
, ( MODULARINT mx ): v OF ( mx**100 + mx + 1 )
ESAC;
print( ( whole( f( MODULARINT( 10, 13 ) ), 0 ), newline ) );
# additional test to avoid a warning that the * operator isn't used... #
print( ( whole( f( MODULARINT( 10, 13 ) * MODULARINT( 10, 13 ) ), 0 ), newline ) )- Output:
1 6
ATS
The following program uses the unsigned __int128 type of GNU C. I use the compiler extension to implement modular multiplication that works for all moduli possible with the ordinary integer types on AMD64.
And I let "modulus 0" mean to do ordinary unsigned arithmetic, so I might be tempted to say the code is transparently supporting both modular and non-modular integers. However, 10**100 would overflow the register, so the result would actually be modulo 2**(wordsize). There is, in fact, with C semantics for unsigned integers, no way to do non-modular arithmetic! You automatically get 2**(wordsize) as a modulus. And you cannot safely use signed integers at all.
(There is support for multiple precision exact rationals, with overloaded operators, in the ats2-xprelude package. If the denominators are one, then such numbers are integers. The macro g(x) defined near the end of the program should work with those numbers.)
(* The program is compiled to C and the integer types have C
semantics. This means ordinary unsigned arithmetic is already
modular! However, the modulus is fixed at 2**n, where n is the
number of bits in the unsigned integer type.
Below, I let a "modulus" of zero mean to use 2**n as the
modulus. *)
(*------------------------------------------------------------------*)
#include "share/atspre_staload.hats"
staload UN = "prelude/SATS/unsafe.sats"
(*------------------------------------------------------------------*)
(* An abstract type, the size of @(g0uint tk, g1uint (tk, modulus)) *)
abst@ype modular_g0uint (tk : tkind, modulus : int) =
@(g0uint tk, g1uint (tk, modulus))
(* Because the type is abstract, we need a constructor: *)
extern fn {tk : tkind}
modular_g0uint_make
{m : int} (* "For any integer m" *)
(a : g0uint tk,
m : g1uint (tk, m))
:<> modular_g0uint (tk, m)
(* A deconstructor: *)
extern fn {tk : tkind}
modular_g0uint_unmake
{m : int}
(a : modular_g0uint (tk, m))
:<> @(g0uint tk, g1uint (tk, m))
extern fn {tk : tkind}
modular_g0uint_succ (* "Successor" *)
{m : int}
(a : modular_g0uint (tk, m))
:<> modular_g0uint (tk, m)
extern fn {tk : tkind} (* This won't be used, but let us write it. *)
modular_g0uint_pred (* "Predecessor" *)
{m : int}
(a : modular_g0uint (tk, m))
:<> modular_g0uint (tk, m)
extern fn {tk : tkind} (* This won't be used, but let us write it.*)
modular_g0uint_neg
{m : int}
(a : modular_g0uint (tk, m))
:<> modular_g0uint (tk, m)
extern fn {tk : tkind}
modular_g0uint_add
{m : int}
(a : modular_g0uint (tk, m),
b : modular_g0uint (tk, m))
:<> modular_g0uint (tk, m)
extern fn {tk : tkind} (* This won't be used, but let us write it.*)
modular_g0uint_sub
{m : int}
(a : modular_g0uint (tk, m),
b : modular_g0uint (tk, m))
:<> modular_g0uint (tk, m)
extern fn {tk : tkind}
modular_g0uint_mul
{m : int}
(a : modular_g0uint (tk, m),
b : modular_g0uint (tk, m))
:<> modular_g0uint (tk, m)
extern fn {tk : tkind}
modular_g0uint_npow
{m : int}
(a : modular_g0uint (tk, m),
i : intGte 0)
:<> modular_g0uint (tk, m)
overload succ with modular_g0uint_succ
overload pred with modular_g0uint_pred
overload ~ with modular_g0uint_neg
overload + with modular_g0uint_add
overload - with modular_g0uint_sub
overload * with modular_g0uint_mul
overload ** with modular_g0uint_npow
(*------------------------------------------------------------------*)
local
(* We make the type be @(g0uint tk, g1uint (tk, modulus)).
The first element is the least residue, the second is the
modulus. A modulus of 0 indicates that the modulus is 2**n, where
n is the number of bits in the typekind. *)
typedef _modular_g0uint (tk : tkind, modulus : int) =
@(g0uint tk, g1uint (tk, modulus))
in (* local *)
assume modular_g0uint (tk, modulus) = _modular_g0uint (tk, modulus)
implement {tk}
modular_g0uint_make (a, m) =
if m = g1i2u 0 then
@(a, m)
else
@(a mod m, m)
implement {tk}
modular_g0uint_unmake a =
a
implement {tk}
modular_g0uint_succ a =
let
val @(a, m) = a
in
if (m = g1i2u 0) || (succ a <> m) then
@(succ a, m)
else
@(g1i2u 0, m)
end
implement {tk}
modular_g0uint_pred a =
let
val @(a, m) = a
prval () = lemma_g1uint_param m
in
(* An exercise for the advanced reader: how come in
modular_g0uint_succ I could use "||", but here I have to use
"+" instead? *)
if (m = g1i2u 0) + (a <> g1i2u 0) then
@(pred a, m)
else
@(pred m, m)
end
implement {tk}
modular_g0uint_neg a =
let
val @(a, m) = a
in
if m = g1i2u 0 then
@(succ (lnot a), m) (* Two's complement. *)
else if a = g0i2u 0 then
@(a, m)
else
@(m - a, m)
end
implement {tk}
modular_g0uint_add (a, b) =
let
(* The modulus of b WILL be same as that of a. The type system
guarantees this at compile time. *)
val @(a, m) = a
and @(b, _) = b
in
if m = g1i2u 0 then
@(a + b, m)
else
@((a + b) mod m, m)
end
implement {tk}
modular_g0uint_mul (a, b) =
(* For multiplication there is a complication, which is that the
product might overflow the register and so end up reduced
modulo the 2**(wordsize). Approaches to that problem are
discussed here:
https://en.wikipedia.org/w/index.php?title=Modular_arithmetic&oldid=1145603919#Example_implementations
However, what I will do is inline some C, and use a GNU C
extension for an integer type that (on AMD64, at least) is
twice as large as uintmax_t.
In so doing, perhaps I help demonstrate how suitable ATS is for
low-level systems programming. Inlining the C is very easy to
do. *)
let
val @(a, m) = a
and @(b, _) = b
in
if m = g1i2u 0 then
@(a * b, m)
else
let
typedef big = $extype"unsigned __int128"
(* A call to _modular_g0uint_mul will actually be a call to
a C function or macro, which happens also to be named
_modular_g0uint_mul. *)
extern fn
_modular_g0uint_mul
(a : big,
b : big,
m : big)
:<> big = "mac#_modular_g0uint_mul"
in
(* The following will work only as long as the C compiler
itself knows how to cast the integer types. There are
safer methods of casting, but, for this task, let us
ignore that. *)
@($UN.cast (_modular_g0uint_mul ($UN.cast a,
$UN.cast b,
$UN.cast m)),
m)
end
end
(* The following puts a static inline function _modular_g0uint_mul
near the top of the C source file. *)
%{^
ATSinline() unsigned __int128
_modular_g0uint_mul (unsigned __int128 a,
unsigned __int128 b,
unsigned __int128 m)
{
return ((a * b) % m);
}
%}
end (* local *)
implement {tk}
modular_g0uint_sub (a, b) =
a + (~b)
implement {tk}
modular_g0uint_npow {m} (a, i) =
(* To compute a power, the multiplication implementation devised
above can be used. The algorithm here is simply the squaring
method:
https://en.wikipedia.org/w/index.php?title=Exponentiation_by_squaring&oldid=1144956501 *)
let
fun
repeat {i : nat} (* <-- This number consistently shrinks. *)
.<i>. (* <-- Proof the recursion will terminate. *)
(accum : modular_g0uint (tk, m), (* "Accumulator" *)
base : modular_g0uint (tk, m),
i : int i)
:<> modular_g0uint (tk, m) =
if i = 0 then
accum
else
let
val i_halved = half i (* Integer division. *)
and base_squared = base * base
in
if i_halved + i_halved = i then
repeat (accum, base_squared, i_halved)
else
repeat (base * accum, base_squared, i_halved)
end
val @(_, m) = modular_g0uint_unmake<tk> a
in
repeat (modular_g0uint_make<tk> (g0i2u 1, m), a, i)
end
(*------------------------------------------------------------------*)
extern fn {tk : tkind}
f : {m : int} modular_g0uint (tk, m) -<> modular_g0uint (tk, m)
(* Using the "successor" function below means that, to add 1, we do
not need to know the modulus. That is why I added "succ". *)
implement {tk}
f(x) = succ (x**100 + x)
(* Using a macro, and thanks to operator overloading, we can use the
same code for modular integers, floating point, etc. *)
macdef g(x) =
let
val x_ = ,(x) (* Evaluate the argument just once. *)
in
succ (x_**100 + x_)
end
implement
main0 () =
let
val x = modular_g0uint_make (10U, 13U)
in
println! ((modular_g0uint_unmake (f(x))).0);
println! ((modular_g0uint_unmake (g(x))).0);
println! (g(10.0))
end
(*------------------------------------------------------------------*)- Output:
$ patscc -std=gnu2x -g -O2 modular_arithmetic_task.dats && ./a.out 1 1 10000000000000002101697803323328251387822715387464188032188166609887360023982790799717755191065313280.000000
C
#include <stdio.h>
struct ModularArithmetic {
int value;
int modulus;
};
struct ModularArithmetic make(const int value, const int modulus) {
struct ModularArithmetic r = { value % modulus, modulus };
return r;
}
struct ModularArithmetic add(const struct ModularArithmetic a, const struct ModularArithmetic b) {
return make(a.value + b.value, a.modulus);
}
struct ModularArithmetic addi(const struct ModularArithmetic a, const int v) {
return make(a.value + v, a.modulus);
}
struct ModularArithmetic mul(const struct ModularArithmetic a, const struct ModularArithmetic b) {
return make(a.value * b.value, a.modulus);
}
struct ModularArithmetic pow(const struct ModularArithmetic b, int pow) {
struct ModularArithmetic r = make(1, b.modulus);
while (pow-- > 0) {
r = mul(r, b);
}
return r;
}
void print(const struct ModularArithmetic v) {
printf("ModularArithmetic(%d, %d)", v.value, v.modulus);
}
struct ModularArithmetic f(const struct ModularArithmetic x) {
return addi(add(pow(x, 100), x), 1);
}
int main() {
struct ModularArithmetic input = make(10, 13);
struct ModularArithmetic output = f(input);
printf("f(");
print(input);
printf(") = ");
print(output);
printf("\n");
return 0;
}
- Output:
f(ModularInteger(10, 13)) = ModularInteger(1, 13)
C#
using System;
namespace ModularArithmetic {
interface IAddition<T> {
T Add(T rhs);
}
interface IMultiplication<T> {
T Multiply(T rhs);
}
interface IPower<T> {
T Power(int pow);
}
interface IOne<T> {
T One();
}
class ModInt : IAddition<ModInt>, IMultiplication<ModInt>, IPower<ModInt>, IOne<ModInt> {
private int modulo;
public ModInt(int value, int modulo) {
Value = value;
this.modulo = modulo;
}
public int Value { get; }
public ModInt One() {
return new ModInt(1, modulo);
}
public ModInt Add(ModInt rhs) {
return this + rhs;
}
public ModInt Multiply(ModInt rhs) {
return this * rhs;
}
public ModInt Power(int pow) {
return Pow(this, pow);
}
public override string ToString() {
return string.Format("ModInt({0}, {1})", Value, modulo);
}
public static ModInt operator +(ModInt lhs, ModInt rhs) {
if (lhs.modulo != rhs.modulo) {
throw new ArgumentException("Cannot add rings with different modulus");
}
return new ModInt((lhs.Value + rhs.Value) % lhs.modulo, lhs.modulo);
}
public static ModInt operator *(ModInt lhs, ModInt rhs) {
if (lhs.modulo != rhs.modulo) {
throw new ArgumentException("Cannot add rings with different modulus");
}
return new ModInt((lhs.Value * rhs.Value) % lhs.modulo, lhs.modulo);
}
public static ModInt Pow(ModInt self, int p) {
if (p < 0) {
throw new ArgumentException("p must be zero or greater");
}
int pp = p;
ModInt pwr = self.One();
while (pp-- > 0) {
pwr *= self;
}
return pwr;
}
}
class Program {
static T F<T>(T x) where T : IAddition<T>, IMultiplication<T>, IPower<T>, IOne<T> {
return x.Power(100).Add(x).Add(x.One());
}
static void Main(string[] args) {
ModInt x = new ModInt(10, 13);
ModInt y = F(x);
Console.WriteLine("x ^ 100 + x + 1 for x = {0} is {1}", x, y);
}
}
}
- Output:
x ^ 100 + x + 1 for x = ModInt(10, 13) is ModInt(1, 13)
C++
#include <iostream>
#include <ostream>
template<typename T>
T f(const T& x) {
return (T) pow(x, 100) + x + 1;
}
class ModularInteger {
private:
int value;
int modulus;
void validateOp(const ModularInteger& rhs) const {
if (modulus != rhs.modulus) {
throw std::runtime_error("Left-hand modulus does not match right-hand modulus.");
}
}
public:
ModularInteger(int v, int m) {
modulus = m;
value = v % m;
}
int getValue() const {
return value;
}
int getModulus() const {
return modulus;
}
ModularInteger operator+(const ModularInteger& rhs) const {
validateOp(rhs);
return ModularInteger(value + rhs.value, modulus);
}
ModularInteger operator+(int rhs) const {
return ModularInteger(value + rhs, modulus);
}
ModularInteger operator*(const ModularInteger& rhs) const {
validateOp(rhs);
return ModularInteger(value * rhs.value, modulus);
}
friend std::ostream& operator<<(std::ostream&, const ModularInteger&);
};
std::ostream& operator<<(std::ostream& os, const ModularInteger& self) {
return os << "ModularInteger(" << self.value << ", " << self.modulus << ")";
}
ModularInteger pow(const ModularInteger& lhs, int pow) {
if (pow < 0) {
throw std::runtime_error("Power must not be negative.");
}
ModularInteger base(1, lhs.getModulus());
while (pow-- > 0) {
base = base * lhs;
}
return base;
}
int main() {
using namespace std;
ModularInteger input(10, 13);
auto output = f(input);
cout << "f(" << input << ") = " << output << endl;
return 0;
}
- Output:
f(ModularInteger(10, 13)) = ModularInteger(1, 13)
Common Lisp
In Scheme all procedures are anonymous, and names such as + and expt are really the names of variables. Thus one can trivially redefine "functions" locally, by storing procedures in variables having the same names as the global ones. Common Lisp has functions with actual names, and which are not the contents of variables. There is no attempt below to copy the Scheme example's trickery with names. (Some less trivial method would be necessary.)
(defvar *modulus* nil)
(defmacro define-enhanced-op (enhanced-op op)
`(defun ,enhanced-op (&rest args)
(if *modulus*
(mod (apply ,op args) *modulus*)
(apply ,op args))))
(define-enhanced-op enhanced+ #'+)
(define-enhanced-op enhanced-expt #'expt)
(defun f (x)
(enhanced+ (enhanced-expt x 100) x 1))
;; Use f on regular integers.
(princ "No modulus: ")
(princ (f 10))
(terpri)
;; Use f on modular integers.
(let ((*modulus* 13))
(princ "modulus 13: ")
(princ (f 10))
(terpri))
- Output:
$ sbcl --script modular_arithmetic_task.lisp No modulus: 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011 modulus 13: 1
D
import std.stdio;
version(unittest) {
void assertEquals(T)(T actual, T expected) {
import core.exception;
import std.conv;
if (actual != expected) {
throw new AssertError("Actual [" ~ to!string(actual) ~ "]; Expected [" ~ to!string(expected) ~ "]");
}
}
}
void main() {
auto input = ModularInteger(10,13);
auto output = f(input);
writeln("f(", input, ") = ", output);
}
V f(V)(const V x) {
return x^^100 + x + 1;
}
/// Integer tests on f
unittest {
assertEquals(f(1), 3);
assertEquals(f(0), 1);
}
/// Floating tests on f
unittest {
assertEquals(f(1.0), 3.0);
assertEquals(f(0.0), 1.0);
}
struct ModularInteger {
private:
int value;
int modulus;
public:
this(int value, int modulus) {
this.modulus = modulus;
this.value = value % modulus;
}
ModularInteger opBinary(string op : "+")(ModularInteger rhs) const in {
assert(this.modulus == rhs.modulus);
} body {
return ModularInteger((this.value + rhs.value) % this.modulus, this.modulus);
}
ModularInteger opBinary(string op : "+")(int rhs) const {
return ModularInteger((this.value + rhs) % this.modulus, this.modulus);
}
ModularInteger opBinary(string op : "*")(ModularInteger rhs) const in {
assert(this.modulus == rhs.modulus);
assert(this.value < this.modulus);
assert(rhs.value < this.modulus);
} body {
return ModularInteger((this.value * rhs.value) % this.modulus, this.modulus);
}
ModularInteger opBinary(string op : "^^")(int pow) const in {
assert(pow >= 0);
} body {
auto base = ModularInteger(1, this.modulus);
while (pow-- > 0) {
base = base * this;
}
return base;
}
string toString() {
import std.format;
return format("ModularInteger(%s, %s)", value, modulus);
}
}
/// Addition with same type of int
unittest {
auto a = ModularInteger(2,5);
auto b = ModularInteger(3,5);
assertEquals(a+b, ModularInteger(0,5));
}
/// Addition with differnt int types
unittest {
auto a = ModularInteger(2,5);
assertEquals(a+0, a);
assertEquals(a+1, ModularInteger(3,5));
}
/// Muliplication
unittest {
auto a = ModularInteger(2,5);
auto b = ModularInteger(3,5);
assertEquals(a*b, ModularInteger(1,5));
}
/// Power
unittest {
const a = ModularInteger(3,13);
assertEquals(a^^2, ModularInteger(9,13));
assertEquals(a^^3, ModularInteger(1,13));
const b = ModularInteger(10,13);
assertEquals(b^^1, ModularInteger(10,13));
assertEquals(b^^2, ModularInteger(9,13));
assertEquals(b^^3, ModularInteger(12,13));
assertEquals(b^^4, ModularInteger(3,13));
assertEquals(b^^5, ModularInteger(4,13));
assertEquals(b^^6, ModularInteger(1,13));
assertEquals(b^^7, ModularInteger(10,13));
assertEquals(b^^8, ModularInteger(9,13));
assertEquals(b^^10, ModularInteger(3,13));
assertEquals(b^^20, ModularInteger(9,13));
assertEquals(b^^30, ModularInteger(1,13));
assertEquals(b^^50, ModularInteger(9,13));
assertEquals(b^^75, ModularInteger(12,13));
assertEquals(b^^90, ModularInteger(1,13));
assertEquals(b^^95, ModularInteger(4,13));
assertEquals(b^^97, ModularInteger(10,13));
assertEquals(b^^98, ModularInteger(9,13));
assertEquals(b^^99, ModularInteger(12,13));
assertEquals(b^^100, ModularInteger(3,13));
}
- Output:
f(ModularInteger(10, 13)) = ModularInteger(1, 13)
Factor
While it's probably not the best idea to define methods in arithmetic words that specialize on custom classes, it can be done. There are a few pitfalls to doing so, which is why custom types typically implement their own arithmetic words. Examples are words like v+ from the math.vectors vocabulary and q+ from the math.quaternions vocabulary.
The pitfalls are as follows:
First, arithmetic words are declared using MATH:, which means they use the math method combination. These methods will dispatch on both their arguments, and promote lower-priority numeric types to higher-priority types when both types are different. The math method combination also means that methods added to MATH: words cannot specialize on any classes except for fixnum,
bignum, ratio, float, complex, object, or unions of them.
This is a bit of a problem, because we must specialize on object and then do a bunch of manual type checking and stack shuffling to make sure we are performing the correct operations on the correct objects.
Second, if any other vocabularies add methods that specialize on arithmetic words, they will conflict with our modular arithmetic vocabulary due to the aforementioned inability to specialize on specific classes.
For these reasons, I would normally opt to define my own arithmetic words, with the added bonus of being able to use non-MATH: multiple dispatch (from the multi-methods vocabulary) to cleanly implement mixed-type dispatch.
Also note that since ^ is not a generic word, we employ the strategy of renaming it to ** inside our vocabulary and defining a new word named ^ that can also handle modular integers. This is an acceptable way to handle it because Factor has pretty good word-disambiguation faculties. I just wouldn't want to have to employ them for more frequently-used arithmetic.
USING: accessors generalizations io kernel math math.functions
parser prettyprint prettyprint.custom sequences ;
IN: rosetta-code.modular-arithmetic
RENAME: ^ math.functions => **
! Define a modular integer class.
TUPLE: mod-int
{ n integer read-only } { mod integer read-only } ;
! Define a constructor for mod-int.
C: <mod-int> mod-int
ERROR: non-equal-modulus m1 m2 ;
! Define a literal syntax for mod-int.
<< SYNTAX: MI{ \ } [ first2 <mod-int> ] parse-literal ; >>
! Implement prettyprinting for mod-int custom syntax.
M: mod-int pprint-delims drop \ MI{ \ } ;
M: mod-int >pprint-sequence [ n>> ] [ mod>> ] bi { } 2sequence ;
M: mod-int pprint* pprint-object ;
<PRIVATE
! Helper words for displaying the results of an arithmetic
! operation.
: show ( quot -- )
[ unparse 2 tail but-last "= " append write ] [ call . ] bi
; inline
: 2show ( quots -- )
[ 2curry show ] map-compose [ call( -- ) ] each ; inline
! Check whether two mod-ints have the same modulus and throw an
! error if not.
: check-mod ( m1 m2 -- )
2dup [ mod>> ] bi@ = [ 2drop ] [ non-equal-modulus ] if ;
! Apply quot to the integer parts of two mod-ints and create a
! new mod-int from the result.
: mod-int-op ( m1 m2 quot -- m3 )
[ [ n>> ] bi@ ] prepose [ 2dup check-mod ] dip over
mod>> [ call( x x -- x ) ] dip [ mod ] keep <mod-int>
; inline
! Promote an integer to a mod-int and call mod-int-op.
: integer-op ( obj1 obj2 quot -- mod-int )
[
dup integer?
[ over mod>> <mod-int> ]
[ dup [ mod>> <mod-int> ] dip ] if
] dip mod-int-op ; inline
! Apply quot, a binary function, to any combination of integers
! and mod-ints.
: binary-op ( obj1 obj2 quot -- mod-int )
2over [ mod-int? ] both? [ mod-int-op ] [ integer-op ] if
; inline
PRIVATE>
! This is where the arithmetic words are 'redefined' by adding a
! method to them that specializes on the object class.
M: object + [ + ] binary-op ;
M: object - [ - ] binary-op ;
M: object * [ * ] binary-op ;
M: object /i [ /i ] binary-op ;
! ^ is a special case because it is not generic.
: ^ ( obj1 obj2 -- obj3 )
2dup [ mod-int? ] either? [ [ ** ] binary-op ] [ ** ] if ;
: fn ( obj -- obj' ) dup 100 ^ + 1 + ;
: modular-arithmetic-demo ( -- )
[ MI{ 10 13 } fn ]
[ 2 fn ] [ show ] bi@
{
[ MI{ 10 13 } MI{ 5 13 } [ + ] ]
[ MI{ 10 13 } 5 [ + ] ]
[ 5 MI{ 10 13 } [ + ] ]
[ MI{ 10 13 } 2 [ /i ] ]
[ 5 10 [ * ] ]
[ MI{ 3 7 } MI{ 4 7 } [ * ] ]
[ MI{ 3 7 } 50 [ ^ ] ]
} 2show ;
MAIN: modular-arithmetic-demo
- Output:
MI{ 10 13 } fn = MI{ 1 13 }
2 fn = 1267650600228229401496703205379
MI{ 10 13 } MI{ 5 13 } + = MI{ 2 13 }
MI{ 10 13 } 5 + = MI{ 2 13 }
5 MI{ 10 13 } + = MI{ 2 13 }
MI{ 10 13 } 2 /i = MI{ 5 13 }
5 10 * = 50
MI{ 3 7 } MI{ 4 7 } * = MI{ 5 7 }
MI{ 3 7 } 50 ^ = MI{ 2 7 }
Forth
Contrary to other contributions, this present a modular package that is complete, i.e. they contain a full set of operators, notably division. It relies not on a library supplied with Forth, but presents the implementation, defined using only Forth kernel definitions.
\ We would normally define operators that have a suffix `m' in order
\ not to be confused: +m -m *m /m **m
\ Also useful is %:m reduce a number modulo.
\ Words that may be not be present in the kernel.
\ This example loads them in ciforth.
WANT ALIAS VOCABULARY
VARIABLE _m ( Modulo number)
\ Set the modulus to m .
: set-modulus _m ! ;
\ For A B return C GCD where C*A+x*B=GCD
: XGCD 1 0 2SWAP BEGIN OVER /MOD OVER WHILE >R SWAP
2SWAP OVER R> * - SWAP 2SWAP REPEAT 2DROP NIP ;
\ Suffix n : normalized number.
: _norm_-m DUP 0< _m @ AND + ; ( x -- xn ) \ -m<xn<+m
: +m + _m @ - _norm_-m ; ( an bn -- sumn )
: -m - _norm_-m ; ( an bn -- diffn)
: *m M* _m @ SM/REM DROP ; ( an bn -- prodn)
: /m _m @ XGCD DROP _norm_-m *m ; ( a b -- quotn)
: %:m S>D _m @ SM/REM DROP _norm_-m ; ( a -- an)
\ Both steps: For A B and C: return A B en C. Invariant A*B^C.
: _reduce_1- 1- >R >R R@ *m R> R> ;
: _reduce_2/ 2/ >R DUP *m R> ;
( a b -- apowbn )
: **m 1 ROT ROT BEGIN DUP 1 AND IF _reduce_1- THEN
_reduce_2/ DUP 0= UNTIL 2DROP ;
\ The solution is
13 set-modulus
10 DUP 100 **m +m 1 +m . CR
\ In order to comply with the problem we can generate a separate namespace
\ and import the above definitions.
VOCABULARY MODULO
ALSO MODULO DEFINITIONS
' set-modulus ALIAS set-modulus
' +m ALIAS +
' -m ALIAS -
' *m ALIAS *
' /m ALIAS /
' **m ALIAS **
' %:m ALIAS %:m
\ now the calculation becomes
13 set-modulus
10 DUP 100 ** + 1 + . CR
Fortran
Program 1
This program requires the C preprocessor (or, if your Fortran compiler has it, the "fortranized" preprocessor fpp). For gfortran, one gets the C preprocessor simply by capitalizing the source file extension: .F90
module modular_arithmetic
implicit none
type :: modular
integer :: val
integer :: modulus
end type modular
interface operator(+)
module procedure modular_modular_add
module procedure modular_integer_add
end interface operator(+)
interface operator(**)
module procedure modular_integer_pow
end interface operator(**)
contains
function modular_modular_add (a, b) result (c)
type(modular), intent(in) :: a
type(modular), intent(in) :: b
type(modular) :: c
if (a%modulus /= b%modulus) error stop
c%val = modulo (a%val + b%val, a%modulus)
c%modulus = a%modulus
end function modular_modular_add
function modular_integer_add (a, i) result (c)
type(modular), intent(in) :: a
integer, intent(in) :: i
type(modular) :: c
c%val = modulo (a%val + i, a%modulus)
c%modulus = a%modulus
end function modular_integer_add
function modular_integer_pow (a, i) result (c)
type(modular), intent(in) :: a
integer, intent(in) :: i
type(modular) :: c
! One cannot simply use the integer ** operator and then compute
! the least residue, because the integers will overflow. Let us
! instead use the right-to-left binary method:
! https://en.wikipedia.org/w/index.php?title=Modular_exponentiation&oldid=1136216610#Right-to-left_binary_method
integer :: modulus
integer :: base
integer :: exponent
modulus = a%modulus
exponent = i
if (modulus < 1) error stop
if (exponent < 0) error stop
c%modulus = modulus
if (modulus == 1) then
c%val = 0
else
c%val = 1
base = modulo (a%val, modulus)
do while (exponent > 0)
if (modulo (exponent, 2) /= 0) then
c%val = modulo (c%val * base, modulus)
end if
exponent = exponent / 2
base = modulo (base * base, modulus)
end do
end if
end function modular_integer_pow
end module modular_arithmetic
! If one uses the extension .F90 instead of .f90, then gfortran will
! pass the program through the C preprocessor. Thus one can write f(x)
! without considering the type of the argument
#define f(x) ((x)**100 + (x) + 1)
program modular_arithmetic_task
use, intrinsic :: iso_fortran_env
use, non_intrinsic :: modular_arithmetic
implicit none
type(modular) :: x, y
x = modular(10, 13)
y = f(x)
write (*, '(" modulus 13: ", I0)') y%val
write (*, '("floating point: ", E55.50)') f(10.0_real64)
end program modular_arithmetic_task
- Output:
$ gfortran -O2 -fbounds-check -Wall -Wextra -g modular_arithmetic_task.F90 && ./a.out
modulus 13: 1
floating point: .10000000000000000159028911097599180468360808563945+101
Program 2
This program uses unlimited runtime polymorphism.
module modular_arithmetic
implicit none
type :: modular_integer
integer :: val
integer :: modulus
end type modular_integer
interface operator(+)
module procedure add
end interface operator(+)
interface operator(*)
module procedure mul
end interface operator(*)
interface operator(**)
module procedure npow
end interface operator(**)
contains
function modular (val, modulus) result (modint)
integer, intent(in) :: val, modulus
type(modular_integer) :: modint
modint%val = modulo (val, modulus)
modint%modulus = modulus
end function modular
subroutine write_number (x)
class(*), intent(in) :: x
select type (x)
class is (modular_integer)
write (*, '(I0)', advance = 'no') x%val
type is (integer)
write (*, '(I0)', advance = 'no') x
class default
error stop
end select
end subroutine write_number
function add (a, b) result (c)
class(*), intent(in) :: a, b
class(*), allocatable :: c
select type (a)
class is (modular_integer)
select type (b)
class is (modular_integer)
if (a%modulus /= b%modulus) error stop
allocate (c, source = modular (a%val + b%val, a%modulus))
type is (integer)
allocate (c, source = modular (a%val + b, a%modulus))
class default
error stop
end select
type is (integer)
select type (b)
class is (modular_integer)
allocate (c, source = modular (a + b%val, b%modulus))
type is (integer)
allocate (c, source = a + b)
class default
error stop
end select
class default
error stop
end select
end function add
function mul (a, b) result (c)
class(*), intent(in) :: a, b
class(*), allocatable :: c
select type (a)
class is (modular_integer)
select type (b)
class is (modular_integer)
if (a%modulus /= b%modulus) error stop
allocate (c, source = modular (a%val * b%val, a%modulus))
type is (integer)
allocate (c, source = modular (a%val * b, a%modulus))
class default
error stop
end select
type is (integer)
select type (b)
class is (modular_integer)
allocate (c, source = modular (a * b%val, b%modulus))
type is (integer)
allocate (c, source = a * b)
class default
error stop
end select
class default
error stop
end select
end function mul
function npow (a, i) result (c)
class(*), intent(in) :: a
integer, intent(in) :: i
class(*), allocatable :: c
class(*), allocatable :: base
integer :: exponent, exponent_halved
if (i < 0) error stop
select type (a)
class is (modular_integer)
allocate (c, source = modular (1, a%modulus))
class default
c = 1
end select
allocate (base, source = a)
exponent = i
do while (exponent /= 0)
exponent_halved = exponent / 2
if (2 * exponent_halved /= exponent) c = base * c
base = base * base
exponent = exponent_halved
end do
end function npow
end module modular_arithmetic
program modular_arithmetic_task
use, non_intrinsic :: modular_arithmetic
implicit none
write (*, '("f(10) ≅ ")', advance = 'no')
call write_number (f (modular (10, 13)))
write (*, '(" (mod 13)")')
write (*, '()')
write (*, '("f applied to a regular integer would overflow, so, in what")')
write (*, '("follows, instead we use g(x) = x**2 + x + 1")')
write (*, '()')
write (*, '("g(10) = ")', advance = 'no')
call write_number (g (10))
write (*, '()')
write (*, '("g(10) ≅ ")', advance = 'no')
call write_number (g (modular (10, 13)))
write (*, '(" (mod 13)")')
contains
function f(x) result (y)
class(*), intent(in) :: x
class(*), allocatable :: y
y = x**100 + x + 1
end function f
function g(x) result (y)
class(*), intent(in) :: x
class(*), allocatable :: y
y = x**2 + x + 1
end function g
end program modular_arithmetic_task
- Output:
$ gfortran -O2 -fbounds-check -Wall -Wextra -g modular_arithmetic_task-2.f90 && ./a.out f(10) ≅ 1 (mod 13) f applied to a regular integer would overflow, so, in what follows, instead we use g(x) = x**2 + x + 1 g(10) = 111 g(10) ≅ 7 (mod 13)
FreeBASIC
Type ModInt
As Ulongint Value
As Ulongint Modulo
End Type
Function Add_(lhs As ModInt, rhs As ModInt) As ModInt
If lhs.Modulo <> rhs.Modulo Then Print "Cannot add rings with different modulus": End
Dim res As ModInt
res.Value = (lhs.Value + rhs.Value) Mod lhs.Modulo
res.Modulo = lhs.Modulo
Return res
End Function
Function Multiply(lhs As ModInt, rhs As ModInt) As ModInt
If lhs.Modulo <> rhs.Modulo Then Print "Cannot multiply rings with different modulus": End
Dim res As ModInt
res.Value = (lhs.Value * rhs.Value) Mod lhs.Modulo
res.Modulo = lhs.Modulo
Return res
End Function
Function One(self As ModInt) As ModInt
Dim res As ModInt
res.Value = 1
res.Modulo = self.Modulo
Return res
End Function
Function Power(self As ModInt, p As Ulongint) As ModInt
If p < 0 Then Print "p must be zero or greater": End
Dim pp As Ulongint = p
Dim pwr As ModInt = One(self)
While pp > 0
pp -= 1
pwr = Multiply(pwr, self)
Wend
Return pwr
End Function
Function F(x As ModInt) As ModInt
Return Add_(Power(x, 100), Add_(x, One(x)))
End Function
Dim x As ModInt
x.Value = 10
x.Modulo = 13
Dim y As ModInt = F(x)
Print Using "x ^ 100 + x + 1 for x = ModInt(&, &) is ModInt(&, &)"; x.Value; x.Modulo; y.Value; y.Modulo
Sleep
- Output:
x ^ 100 + x + 1 for x = ModInt(10, 13) is ModInt(1, 13)
Go
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)
}
return
}
// The task function f has that constant 1 in it.
// Define a special kind of ring that has this element.
type oneRing interface {
ring
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), r.one())
}
// 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)))
}
- Output:
1
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)
- Output:
./modarith 1
J
J does not allow "operator redefinition", but J's operators are capable of operating consistently on values which represent modular integers:
f=: (+./1 1,:_101{.1x)&p.
Task example:
13|f 10
1
Java
public class ModularArithmetic {
private interface Ring<T> {
Ring<T> plus(Ring<T> rhs);
Ring<T> times(Ring<T> rhs);
int value();
Ring<T> one();
default Ring<T> pow(int p) {
if (p < 0) {
throw new IllegalArgumentException("p must be zero or greater");
}
int pp = p;
Ring<T> pwr = this.one();
while (pp-- > 0) {
pwr = pwr.times(this);
}
return pwr;
}
}
private static class ModInt implements Ring<ModInt> {
private int value;
private int modulo;
private ModInt(int value, int modulo) {
this.value = value;
this.modulo = modulo;
}
@Override
public Ring<ModInt> plus(Ring<ModInt> other) {
if (!(other instanceof ModInt)) {
throw new IllegalArgumentException("Cannot add an unknown ring.");
}
ModInt rhs = (ModInt) other;
if (modulo != rhs.modulo) {
throw new IllegalArgumentException("Cannot add rings with different modulus");
}
return new ModInt((value + rhs.value) % modulo, modulo);
}
@Override
public Ring<ModInt> times(Ring<ModInt> other) {
if (!(other instanceof ModInt)) {
throw new IllegalArgumentException("Cannot multiple an unknown ring.");
}
ModInt rhs = (ModInt) other;
if (modulo != rhs.modulo) {
throw new IllegalArgumentException("Cannot multiply rings with different modulus");
}
return new ModInt((value * rhs.value) % modulo, modulo);
}
@Override
public int value() {
return value;
}
@Override
public Ring<ModInt> one() {
return new ModInt(1, modulo);
}
@Override
public String toString() {
return String.format("ModInt(%d, %d)", value, modulo);
}
}
private static <T> Ring<T> f(Ring<T> x) {
return x.pow(100).plus(x).plus(x.one());
}
public static void main(String[] args) {
ModInt x = new ModInt(10, 13);
Ring<ModInt> y = f(x);
System.out.print("x ^ 100 + x + 1 for x = ModInt(10, 13) is ");
System.out.println(y);
System.out.flush();
}
}
- Output:
x ^ 100 + x + 1 for x = ModInt(10, 13) is ModInt(1, 13)
jq
Works with gojq, the Go implementation of jq
This entry focuses on the requirement that the function, f, "should behave the same way with normal and modular integers."
To illustrate that the function `ring::f` defined here does satisfy this requirement, we will evaluate it at both the integer 1 and the element «10 % 13» of ℤ/13ℤ.
To keep the distinctions between functions defined on different types clear, this entry uses jq's support for name spaces. Specifically, we will use the prefix "modint::" for the modular arithmetic functions, and "ring::" for the generic ring functions.
Since jq supports neither redefining any of the symbolic operators (such as "+") nor dynamic dispatch, ring functions (such as `ring::add`) must be written with the specific rings that are to be supported in mind.
Preliminaries
def assert($e; $msg): if $e then . else "assertion violation @ \($msg)" | error end;
def is_integer: type=="number" and floor == .;
# To take advantage of gojq's arbitrary-precision integer arithmetic:
def power($b): . as $in | reduce range(0;$b) as $i (1; . * $in);Modular Arithmetic
# "ModularArithmetic" objects are represented by JSON objects of the form: {value, mod}.
# The function modint::assert/0 checks the input is of this form with integer values.
def is_modint: type=="object" and has("value") and has("mod");
def modint::assert:
assert(type=="object"; "object expected")
| assert(has("value"); "object should have a value")
| assert(has("mod"); "object should have a mod")
| assert(.value | is_integer; "value should be an integer")
| assert(.mod | is_integer; "mod should be an integer");
def modint::make($value; $mod):
assert($value|is_integer; "value should be an integer")
| assert($mod|is_integer; "mod should be an integer")
| { value: ($value % $mod), mod: $mod};
def modint::add($A; $B):
if ($B|type) == "object"
then assert($A.mod == $B.mod ; "modint::add")
| modint::make( $A.value + $B.value; $A.mod )
else modint::make( $A.value + $B; $A.mod )
end;
def modint::mul($A; $B):
if ($B|type) == "object"
then assert($A.mod == $B.mod ; "mul")
| modint::make( $A.value * $B.value; $A.mod )
else modint::make( $A.value * $B; $A.mod )
end;
def modint::pow($A; $pow):
assert($pow | is_integer; "pow")
| reduce range(0; $pow) as $i ( modint::make(1; $A.mod);
modint::mul( .; $A) );
# pretty print
def modint::pp: "«\(.value) % \(.mod)»";Ring Functions
def ring::add($A; $B):
if $A|is_modint then modint::add($A; $B)
elif $A|is_integer then $A + $B
else "ring::add" | error
end;
def ring::mul($A; $B):
if $A|is_modint then modint::mul($A; $B)
elif $A|is_integer then $A * $B
else "ring::mul" | error
end;
def ring::pow($A; $B):
if $A|is_modint then modint::pow($A; $B)
elif $A|is_integer then $A|power($B)
else "ring::pow" | error
end;
def ring::pp:
if is_modint then modint::pp
elif is_integer then .
else "ring::pp" | error
end;
def ring::f($x):
ring::add( ring::add( ring::pow($x; 100); $x); 1);Evaluating ring::f
def main:
(ring::f(1) | "f(\(1)) => \(.)"),
(modint::make(10;13)
| ring::f(.) as $out
| "f(\(ring::pp)) => \($out|ring::pp)");- Output:
f(1) => 3 f(«10 % 13») => «1 % 13»
Julia
Implements the Modulo struct and basic operations.
struct Modulo{T<:Integer} <: Integer
val::T
mod::T
Modulo(n::T, m::T) where T = new{T}(mod(n, m), m)
end
modulo(n::Integer, m::Integer) = Modulo(promote(n, m)...)
Base.show(io::IO, md::Modulo) = print(io, md.val, " (mod $(md.mod))")
Base.convert(::Type{T}, md::Modulo) where T<:Integer = convert(T, md.val)
Base.copy(md::Modulo{T}) where T = Modulo{T}(md.val, md.mod)
Base.:+(md::Modulo) = copy(md)
Base.:-(md::Modulo) = Modulo(md.mod - md.val, md.mod)
for op in (:+, :-, :*, :÷, :^)
@eval function Base.$op(a::Modulo, b::Integer)
val = $op(a.val, b)
return Modulo(mod(val, a.mod), a.mod)
end
@eval Base.$op(a::Integer, b::Modulo) = $op(b, a)
@eval function Base.$op(a::Modulo, b::Modulo)
if a.mod != b.mod throw(InexactError()) end
val = $op(a.val, b.val)
return Modulo(mod(val, a.mod), a.mod)
end
end
f(x) = x ^ 100 + x + 1
@show f(modulo(10, 13))
- Output:
f(modulo(10, 13)) = 11 (mod 13)
Kotlin
// version 1.1.3
interface Ring<T> {
operator fun plus(other: Ring<T>): Ring<T>
operator fun times(other: Ring<T>): Ring<T>
val value: Int
val one: Ring<T>
}
fun <T> Ring<T>.pow(p: Int): Ring<T> {
require(p >= 0)
var pp = p
var pwr = this.one
while (pp-- > 0) pwr *= this
return pwr
}
class ModInt(override val value: Int, val modulo: Int): Ring<ModInt> {
override operator fun plus(other: Ring<ModInt>): ModInt {
require(other is ModInt && modulo == other.modulo)
return ModInt((value + other.value) % modulo, modulo)
}
override operator fun times(other: Ring<ModInt>): ModInt {
require(other is ModInt && modulo == other.modulo)
return ModInt((value * other.value) % modulo, modulo)
}
override val one get() = ModInt(1, modulo)
override fun toString() = "ModInt($value, $modulo)"
}
fun <T> f(x: Ring<T>): Ring<T> = x.pow(100) + x + x.one
fun main(args: Array<String>) {
val x = ModInt(10, 13)
val y = f(x)
println("x ^ 100 + x + 1 for x == ModInt(10, 13) is $y")
}
- Output:
x ^ 100 + x + 1 for x == ModInt(10, 13) is ModInt(1, 13)
Lua
function make(value, modulo)
local v = value % modulo
local tbl = {value=v, modulo=modulo}
local mt = {
__add = function(lhs, rhs)
if type(lhs) == "table" then
if type(rhs) == "table" then
if lhs.modulo ~= rhs.modulo then
error("Cannot add rings with different modulus")
end
return make(lhs.value + rhs.value, lhs.modulo)
else
return make(lhs.value + rhs, lhs.modulo)
end
else
error("lhs is not a table in +")
end
end,
__mul = function(lhs, rhs)
if lhs.modulo ~= rhs.modulo then
error("Cannot multiply rings with different modulus")
end
return make(lhs.value * rhs.value, lhs.modulo)
end,
__pow = function(b,p)
if p<0 then
error("p must be zero or greater")
end
local pp = p
local pwr = make(1, b.modulo)
while pp > 0 do
pp = pp - 1
pwr = pwr * b
end
return pwr
end,
__concat = function(lhs, rhs)
if type(lhs) == "table" and type(rhs) == "string" then
return "ModInt("..lhs.value..", "..lhs.modulo..")"..rhs
elseif type(lhs) == "string" and type(rhs) == "table" then
return lhs.."ModInt("..rhs.value..", "..rhs.modulo..")"
else
return "todo"
end
end
}
setmetatable(tbl, mt)
return tbl
end
function func(x)
return x ^ 100 + x + 1
end
-- main
local x = make(10, 13)
local y = func(x)
print("x ^ 100 + x + 1 for "..x.." is "..y)
- Output:
x ^ 100 + x + 1 for ModInt(10, 13) is ModInt(1, 13)
Mathematica /Wolfram Language
For versions prior to 13.3, the best way to do it is probably to use the finite fields package.
<< FiniteFields`
x^100 + x + 1 /. x -> GF[13]@{10}
- Output:
{1}13
Version 13.3 has a "complete, consistent coverage of all finite fields":
OutputForm[
x^100 + x + 1 /. x -> FiniteField[13][10]
]
We have to show the `OutputForm` though, because the `StandardForm` is not easy to render here.
- Output:
FiniteFieldElement[<1,13,1,+>]
Mercury
Numbers have to be converted to ordinary(Number) or modular(Number, Modulus) before they are plugged into f. The two kinds can be mixed: if any operation involves a "modular" number, then the result will be "modular", but otherwise the result will be "ordinary".
(There are limitations in Mercury's current system of overloads that make it more difficult than I care to deal with, to do this so that f could be invoked directly on the integer type.)
%% -*- mode: mercury; prolog-indent-width: 2; -*-
:- module modular_arithmetic_task.
:- interface.
:- import_module io.
:- pred main(io::di, io::uo) is det.
:- implementation.
:- import_module exception.
:- import_module integer.
:- type modular_integer
---> modular(integer, integer)
; ordinary(integer).
:- func operate((func(integer, integer) = integer),
modular_integer, modular_integer) = modular_integer.
operate(OP, modular(A, M1), modular(B, M2)) = C :-
if (M1 = M2)
then (C = modular(mod(OP(A, B), M1), M1))
else throw("mismatched moduli").
operate(OP, modular(A, M), ordinary(B)) = C :-
C = modular(mod(OP(A, B), M), M).
operate(OP, ordinary(A), modular(B, M)) = C :-
C = modular(mod(OP(A, B), M), M).
operate(OP, ordinary(A), ordinary(B)) = C :-
C = ordinary(OP(A, B)).
:- func '+'(modular_integer, modular_integer) = modular_integer.
(A : modular_integer) + (B : modular_integer) = operate(+, A, B).
:- func pow(modular_integer, modular_integer) = modular_integer.
pow(A : modular_integer, B : modular_integer) = operate(pow, A, B).
:- pred display(modular_integer::in, io::di, io::uo) is det.
display(X, !IO) :-
if (X = modular(A, _)) then print(A, !IO)
else if (X = ordinary(A)) then print(A, !IO)
else true.
:- func f(modular_integer) = modular_integer.
f(X) = Y :-
Y = pow(X, ordinary(integer(100))) + X
+ ordinary(integer(1)).
main(!IO) :-
X1 = ordinary(integer(10)),
X2 = modular(integer(10), integer(13)),
print("No modulus: ", !IO),
display(f(X1), !IO),
nl(!IO),
print("modulus 13: ", !IO),
display(f(X2), !IO),
nl(!IO).
:- end_module modular_arithmetic_task.- Output:
$ mmc --use-subdirs --make modular_arithmetic_task && ./modular_arithmetic_task Making Mercury/int3s/modular_arithmetic_task.int3 Making Mercury/ints/modular_arithmetic_task.int Making Mercury/cs/modular_arithmetic_task.c Making Mercury/os/modular_arithmetic_task.o Making modular_arithmetic_task No modulus: 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011 modulus 13: 1
Nim
Modular integers are represented as distinct integers with a modulus N managed by the compiler.
import macros, sequtils, strformat, strutils
const Subscripts: array['0'..'9', string] = ["₀", "₁", "₂", "₃", "₄", "₅", "₆", "₇", "₈", "₉"]
# Modular integer with modulus N.
type ModInt[N: static int] = distinct int
#---------------------------------------------------------------------------------------------------
# Creation.
func initModInt[N](n: int): ModInt[N] =
## Create a modular integer from an integer.
static:
when N < 2: error "Modulus must be greater than 1."
if n >= N: raise newException(ValueError, &"value must be in 0..{N - 1}.")
result = ModInt[N](n)
#---------------------------------------------------------------------------------------------------
# Arithmetic operations: ModInt op ModInt, ModInt op int and int op ModInt.
func `+`*[N](a, b: ModInt[N]): ModInt[N] =
ModInt[N]((a.int + b.int) mod N)
func `+`*[N](a: ModInt[N]; b: int): ModInt[N] =
a + initModInt[N](b)
func `+`*[N](a: int; b: ModInt[N]): ModInt[N] =
initModInt[N](a) + b
func `*`*[N](a, b: ModInt[N]): ModInt[N] =
ModInt[N]((a.int * b.int) mod N)
func `*`*[N](a: ModInt[N]; b: int): ModInt[N] =
a * initModInt[N](b)
func `*`*[N](a: int; b: ModInt[N]): ModInt[N] =
initModInt[N](a) * b
func `^`*[N](a: ModInt[N]; n: Natural): ModInt[N] =
var a = a
var n = n
result = initModInt[N](1)
while n > 0:
if (n and 1) != 0:
result = result * a
n = n shr 1
a = a * a
#---------------------------------------------------------------------------------------------------
# Representation of a modular integer as a string.
template subscript(n: Natural): string =
mapIt($n, Subscripts[it]).join()
func `$`(a: ModInt): string =
&"{a.int}{subscript(a.N)})"
#---------------------------------------------------------------------------------------------------
# The function "f" defined for any modular integer, the same way it would be defined for an
# integer argument (except that such a function would be of no use as it would overflow for
# any argument different of 0 and 1).
func f(x: ModInt): ModInt = x^100 + x + 1
#———————————————————————————————————————————————————————————————————————————————————————————————————
when isMainModule:
var x = initModInt[13](10)
echo &"f({x}) = {x}^100 + {x} + 1 = {f(x)}."
- Output:
f(10₁₃) = 10₁₃^100 + 10₁₃ + 1 = 1₁₃.
ObjectIcon
# -*- ObjectIcon -*-
#
# Object Icon has a "Number" class (with subclasses) that has "add"
# and "mul" methods. These methods can be implemented in a modular
# numbers class, even though we cannot redefine the symbolic operators
# "+" and "*". Neither can we inherit from Number, but that turns out
# not to get in our way.
#
import io
import ipl.types
import numbers (Rat)
import util (need_integer)
procedure main ()
local x
x := Rat (10) # The number 10 as a rational with denominator 1.
write ("no modulus: ", f(x).n)
x := Modular (10, 13)
write ("modulus 13: ", f(x).least_residue)
end
procedure f(x)
return npow(x, 100).add(x).add(1)
end
procedure npow (x, i)
# Raise a number to a non-negative power, using the methods of its
# class. The algorithm is the squaring method.
local accum, i_halved
if i < 0 then runerr ("Non-negative number expected", i)
accum := typeof(x) (1)
# Perhaps the following hack can be eliminated?
if is (x, Modular) then accum := Modular (1, x.modulus)
while i ~= 0 do
{
i_halved := i / 2
if i_halved + i_halved ~= i then accum := x.mul(accum)
x := x.mul(x)
i := i_halved
}
return accum
end
class Modular ()
public const least_residue
public const modulus
public new (num, m)
if /m & is (num, Modular) then
{
self.least_residue := num.least_residue
self.modulus := num.modulus
}
else
{
/m := 0
m := need_integer (m)
if m < 0 then runerr ("Non-negative number expected", m)
self.modulus := m
num := need_integer (num)
if m = 0 then
self.least_residue := num # A regular integer.
else
self.least_residue := residue (num, modulus)
}
return
end
public add (x)
if is (x, Modular) then x := x.least_residue
return Modular (least_residue + x, need_modulus (self, x))
end
public mul (x)
if is (x, Modular) then x := x.least_residue
return Modular (least_residue * x, need_modulus (self, x))
end
end
procedure need_modulus (x, y)
local mx, my
mx := if is (x, Modular) then x.modulus else 0
my := if is (y, Modular) then y.modulus else 0
if mx = 0 then
{
if my = 0 then runerr ("Cannot determine the modulus", [x, y])
mx := my
}
else if my = 0 then
my := mx
if mx ~= my then runerr ("Mismatched moduli", [x, y])
return mx
end
procedure residue(i, m, j)
# Residue for j-based integers, taken from the Arizona Icon IPL
# (which is in the public domain). With the default value j=0, this
# is what we want for reducing numbers to their least residues.
/j := 0
i %:= m
if i < j then i +:= m
return i
end- Output:
$ oiscript modular_arithmetic_task_OI.icn no modulus: 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011 modulus 13: 1
Owl Lisp
;; Owl Lisp, though a dialect of Scheme, has no true variables: it has
;; only value-bindings. We cannot use "make-parameter" to specify an
;; optional modulus. Instead let us introduce a new type for modular
;; integers.
(define (modular? x)
;; The new type is simply a pair of integers.
(and (pair? x) (integer? (car x)) (integer? (cdr x))))
(define (enhanced-op op)
(lambda (x y)
(if (modular? x)
(if (modular? y)
(begin
(unless (= (cdr x) (cdr y))
(error "mismatched moduli"))
(cons (floor-remainder (op (car x) (car y)) (cdr x))
(cdr x)))
(cons (floor-remainder (op (car x) y) (cdr x))
(cdr x)))
(if (modular? y)
(cons (floor-remainder (op x (car y)) (cdr y))
(cdr y))
(op x y)))))
(define enhanced+ (enhanced-op +))
(define enhanced-expt (enhanced-op expt))
(define (f x)
;; Temporarily redefine + and expt so they can handle either regular
;; numbers or modular integers.
(let ((+ enhanced+)
(expt enhanced-expt))
;; Here is a definition of f(x), in the notation of Owl Lisp:
(+ (+ (expt x 100) x) 1)))
;; Use f on regular integers.
(display "No modulus: ")
(display (f 10))
(newline)
(display "modulus 13: ")
(display (car (f (cons 10 13))))
(newline)
- Output:
$ ol modular_arithmetic_task_Owl.scm No modulus: 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011 modulus 13: 1
PARI/GP
This feature exists natively in GP:
Mod(3,7)+Mod(4,7)Perl
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);
- Output:
mod(1, 13)
Phix
Phix does not allow operator overloading, but an f() which is agnostic about whether its parameter is a modular or normal int, we can do.
type mi(object m) return sequence(m) and length(m)=2 and integer(m[1]) and integer(m[2]) end type type mii(object m) return mi(m) or atom(m) end type function mi_one(mii a) if atom(a) then a=1 else a = {1,a[2]} end if return a end function function mi_add(mii a, mii b) if atom(a) then if not atom(b) then throw("error") end if return a+b end if if a[2]!=b[2] then throw("error") end if a[1] = mod(a[1]+b[1],a[2]) return a end function function mi_mul(mii a, mii b) if atom(a) then if not atom(b) then throw("error") end if return a*b end if if a[2]!=b[2] then throw("error") end if a[1] = mod(a[1]*b[1],a[2]) return a end function function mi_power(mii x, integer p) mii res = mi_one(x) for i=1 to p do res = mi_mul(res,x) end for return res end function function mi_print(mii m) return sprintf(iff(atom(m)?"%g":"modint(%d,%d)"),m) end function function f(mii x) return mi_add(mi_power(x,100),mi_add(x,mi_one(x))) end function procedure test(mii x) printf(1,"x^100 + x + 1 for x == %s is %s\n",{mi_print(x),mi_print(f(x))}) end procedure test(10) test({10,13})
- Output:
x^100 + x + 1 for x == 10 is 1e+100 x^100 + x + 1 for x == modint(10,13) is modint(1,13)
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 ).
:- 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).
- 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
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
@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.
# 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):
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
# 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)))
# Output: Mod(1, 13)
Quackery
Quackery is an extensible assembler for the Quackery Virtual Processor, which is implemented in Python3 (but could be implemented in any language). The QVP recognises three static types; Number (Python Int), Nest (Python List) and Operator (Python function). Adding more static types would require adding functionality to the QVP by modifying the source code for Quackery.
However it is possible to extend the assembler to include dynamic typing without modifying the QVP. The first part of the code presented here adds just sufficient dynamic typing to Quackery to fulfil the requirements of this task. It could be considered a first sketch towards adding more comprehensive dynamic typing to Quackery.
The second part of the code uses this to overload the Quackery words + and **.
The third part fulfils the requirements of this task.
[ stack ] is modulus ( --> s )
[ this ] is modular ( --> [ )
[ modulus share mod
modular nested join ] is modularise ( n --> N )
[ dup nest? iff
[ -1 peek modular oats ]
else [ drop false ] ] is modular? ( N --> b )
[ modular? swap
modular? or ] is 2modular? ( N N --> b )
[ dup modular? if [ 0 peek ] ] is demodularise ( N --> n )
[ demodularise swap
demodularise swap ] is 2demodularise ( N N --> n )
[ dup $ '' = if
[ $ '"modularify(2-->1)" '
$ "needs a name after it."
join message put bail ]
nextword
$ "[ 2dup 2modular? iff
[ 2demodularise " over join
$ " modularise ]
else " join over join
$ " ] is " join swap join
space join
swap join ] builds modularify(2-->1) ( --> )
( --------------------------------------------------------------- )
modularify(2-->1) + ( N N --> N )
modularify(2-->1) ** ( N N --> N )
( --------------------------------------------------------------- )
[ dup 100 ** + 1 + ] is f ( N --> N )
13 modulus put
10 f echo cr
10 modularise f echo
modulus release crOutput:
10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011 [ 1 modular ]
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
Raku
(formerly Perl 6)
We'll use the FiniteFields repo.
use FiniteField;
$*modulus = 13;
sub f(\x) { x**100 + x + 1};
say f(10);
- Output:
1
Red
This implementation of +,-,*,/ uses a loose test (object?) to check operands type. As soon as one is a modular integer, the other one is treated as a modular integer too.
Red ["Modular arithmetic"]
; defining the modular integer class, and a constructor
modulus: 13
m: function [n] [
either object? n [make n []] [context [val: n % modulus]]
]
; redefining operators +, -, *, / to include modular integers
foreach [op fun][+ add - subtract * multiply / divide][
set op make op! function [a b] compose/deep [
either any [object? a object? b][
a: m a
b: m b
m (fun) a/val b/val
][(fun) a b]
]
]
; redefining power - ** ; second operand must be an integer
**: make op! function [a n] [
either object? a [
tmp: 1
loop n [tmp: tmp * a/val % modulus]
m tmp
][power a n]
]
; testing
f: function [x] [x ** 100 + x + 1]
print ["f definition is:" mold :f]
print ["f((integer) 10) is:" f 10]
print ["f((modular) 10) is: (modular)" f m 10]
- Output:
f definition is: func [x][x ** 100 + x + 1] f((integer) 10) is: 1.0e100 f((modular) 10) is: (modular) val: 1
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
# 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>
Scala
- Output:
Best seen running in your browser either by ScalaFiddle (ES aka JavaScript, non JVM) or Scastie (remote JVM).
object ModularArithmetic extends App {
private val x = new ModInt(10, 13)
private val y = f(x)
private def f[T](x: Ring[T]) = (x ^ 100) + x + x.one
private trait Ring[T] {
def +(rhs: Ring[T]): Ring[T]
def *(rhs: Ring[T]): Ring[T]
def one: Ring[T]
def ^(p: Int): Ring[T] = {
require(p >= 0, "p must be zero or greater")
var pp = p
var pwr = this.one
while ( {
pp -= 1;
pp
} >= 0) pwr = pwr * this
pwr
}
}
private class ModInt(var value: Int, var modulo: Int) extends Ring[ModInt] {
def +(other: Ring[ModInt]): Ring[ModInt] = {
require(other.isInstanceOf[ModInt], "Cannot add an unknown ring.")
val rhs = other.asInstanceOf[ModInt]
require(modulo == rhs.modulo, "Cannot add rings with different modulus")
new ModInt((value + rhs.value) % modulo, modulo)
}
def *(other: Ring[ModInt]): Ring[ModInt] = {
require(other.isInstanceOf[ModInt], "Cannot multiple an unknown ring.")
val rhs = other.asInstanceOf[ModInt]
require(modulo == rhs.modulo,
"Cannot multiply rings with different modulus")
new ModInt((value * rhs.value) % modulo, modulo)
}
override def one = new ModInt(1, modulo)
override def toString: String = f"ModInt($value%d, $modulo%d)"
}
println("x ^ 100 + x + 1 for x = ModInt(10, 13) is " + y)
}
Scheme
The program is for R7RS Scheme.
"Modular integers" are not introduced here as a type distinct from "integers". Instead, a modulus may be imposed on "enhanced" versions of arithmeti operations.
Note the use of floor-remainder instead of truncate-remainder. The latter would function incorrectly if there were negative numbers.
(cond-expand
(r7rs)
(chicken (import r7rs)))
(import (scheme base))
(import (scheme write))
(define *modulus*
(make-parameter
#f
(lambda (mod)
(if (or (not mod)
(and (exact-integer? mod)
(positive? mod)))
mod
(error "not a valid modulus")))))
(define-syntax enhanced-op
(syntax-rules ()
((_ op)
(lambda args
(let ((mod (*modulus*))
(tentative-result (apply op args)))
(if mod
(floor-remainder tentative-result mod)
tentative-result))))))
(define enhanced+ (enhanced-op +))
(define enhanced-expt (enhanced-op expt))
(define (f x)
;; Temporarily redefine + and expt so they can handle either regular
;; numbers or modular integers.
(let ((+ enhanced+)
(expt enhanced-expt))
;; Here is a definition of f(x), in the notation of Scheme:
(+ (expt x 100) x 1)))
;; Use f on regular integers.
(display "No modulus: ")
(display (f 10))
(newline)
;; Use f on modular integers.
(parameterize ((*modulus* 13))
(display "modulus 13: ")
(display (f 10))
(newline))
- Output:
$ gosh modular_arithmetic_task.scm No modulus: 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011 modulus 13: 1
Sidef
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))
- Output:
1 「mod 13」
Swift
precedencegroup ExponentiationGroup {
higherThan: MultiplicationPrecedence
}
infix operator ** : ExponentiationGroup
protocol Ring {
associatedtype RingType: Numeric
var one: Self { get }
static func +(_ lhs: Self, _ rhs: Self) -> Self
static func *(_ lhs: Self, _ rhs: Self) -> Self
static func **(_ lhs: Self, _ rhs: Int) -> Self
}
extension Ring {
static func **(_ lhs: Self, _ rhs: Int) -> Self {
var ret = lhs.one
for _ in stride(from: rhs, to: 0, by: -1) {
ret = ret * lhs
}
return ret
}
}
struct ModInt: Ring {
typealias RingType = Int
var value: Int
var modulo: Int
var one: ModInt { ModInt(1, modulo: modulo) }
init(_ value: Int, modulo: Int) {
self.value = value
self.modulo = modulo
}
static func +(lhs: ModInt, rhs: ModInt) -> ModInt {
precondition(lhs.modulo == rhs.modulo)
return ModInt((lhs.value + rhs.value) % lhs.modulo, modulo: lhs.modulo)
}
static func *(lhs: ModInt, rhs: ModInt) -> ModInt {
precondition(lhs.modulo == rhs.modulo)
return ModInt((lhs.value * rhs.value) % lhs.modulo, modulo: lhs.modulo)
}
}
func f<T: Ring>(_ x: T) -> T { (x ** 100) + x + x.one }
let x = ModInt(10, modulo: 13)
let y = f(x)
print("x ^ 100 + x + 1 for x = ModInt(10, 13) is \(y)")
- Output:
x ^ 100 + x + 1 for x = ModInt(10, 13) is ModInt(value: 1, modulo: 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.
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 <- '**' ;
END;
}
# 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"
- Output:
1 = ::mod13 + [::mod13 + [::mod13 ** [::mod13 variable x] [::mod13 value 100]] [::mod13 variable x]] [::mod13 value 1]
VBA
Option Base 1
Private Function mi_one(ByVal a As Variant) As Variant
If IsArray(a) Then
a(1) = 1
Else
a = 1
End If
mi_one = a
End Function
Private Function mi_add(ByVal a As Variant, b As Variant) As Variant
If IsArray(a) Then
If IsArray(b) Then
If a(2) <> b(2) Then
mi_add = CVErr(2019)
Else
a(1) = (a(1) + b(1)) Mod a(2)
mi_add = a
End If
Else
mi_add = CVErr(2018)
End If
Else
If IsArray(b) Then
mi_add = CVErr(2018)
Else
a = a + b
mi_add = a
End If
End If
End Function
Private Function mi_mul(ByVal a As Variant, b As Variant) As Variant
If IsArray(a) Then
If IsArray(b) Then
If a(2) <> b(2) Then
mi_mul = CVErr(2019)
Else
a(1) = (a(1) * b(1)) Mod a(2)
mi_mul = a
End If
Else
mi_mul = CVErr(2018)
End If
Else
If IsArray(b) Then
mi_mul = CVErr(2018)
Else
a = a * b
mi_mul = a
End If
End If
End Function
Private Function mi_power(x As Variant, p As Integer) As Variant
res = mi_one(x)
For i = 1 To p
res = mi_mul(res, x)
Next i
mi_power = res
End Function
Private Function mi_print(m As Variant) As Variant
If IsArray(m) Then
s = "modint(" & m(1) & "," & m(2) & ")"
Else
s = CStr(m)
End If
mi_print = s
End Function
Private Function f(x As Variant) As Variant
f = mi_add(mi_power(x, 100), mi_add(x, mi_one(x)))
End Function
Private Sub test(x As Variant)
Debug.Print "x^100 + x + 1 for x == " & mi_print(x) & " is " & mi_print(f(x))
End Sub
Public Sub main()
test 10
test [{10,13}]
End Sub
- Output:
x^100 + x + 1 for x == 10 is 1E+100 x^100 + x + 1 for x == modint(10,13) is modint(1,13)
Visual Basic .NET
Module Module1
Interface IAddition(Of T)
Function Add(rhs As T) As T
End Interface
Interface IMultiplication(Of T)
Function Multiply(rhs As T) As T
End Interface
Interface IPower(Of T)
Function Power(pow As Integer) As T
End Interface
Interface IOne(Of T)
Function One() As T
End Interface
Class ModInt
Implements IAddition(Of ModInt), IMultiplication(Of ModInt), IPower(Of ModInt), IOne(Of ModInt)
Sub New(value As Integer, modulo As Integer)
Me.Value = value
Me.Modulo = modulo
End Sub
ReadOnly Property Value As Integer
ReadOnly Property Modulo As Integer
Public Function Add(rhs As ModInt) As ModInt Implements IAddition(Of ModInt).Add
Return Me + rhs
End Function
Public Function Multiply(rhs As ModInt) As ModInt Implements IMultiplication(Of ModInt).Multiply
Return Me * rhs
End Function
Public Function Power(pow_ As Integer) As ModInt Implements IPower(Of ModInt).Power
Return Pow(Me, pow_)
End Function
Public Function One() As ModInt Implements IOne(Of ModInt).One
Return New ModInt(1, Modulo)
End Function
Public Overrides Function ToString() As String
Return String.Format("ModInt({0}, {1})", Value, Modulo)
End Function
Public Shared Operator +(lhs As ModInt, rhs As ModInt) As ModInt
If lhs.Modulo <> rhs.Modulo Then
Throw New ArgumentException("Cannot add rings with different modulus")
End If
Return New ModInt((lhs.Value + rhs.Value) Mod lhs.Modulo, lhs.Modulo)
End Operator
Public Shared Operator *(lhs As ModInt, rhs As ModInt) As ModInt
If lhs.Modulo <> rhs.Modulo Then
Throw New ArgumentException("Cannot multiply rings with different modulus")
End If
Return New ModInt((lhs.Value * rhs.Value) Mod lhs.Modulo, lhs.Modulo)
End Operator
Public Shared Function Pow(self As ModInt, p As Integer) As ModInt
If p < 0 Then
Throw New ArgumentException("p must be zero or greater")
End If
Dim pp = p
Dim pwr = self.One()
While pp > 0
pp -= 1
pwr *= self
End While
Return pwr
End Function
End Class
Function F(Of T As {IAddition(Of T), IMultiplication(Of T), IPower(Of T), IOne(Of T)})(x As T) As T
Return x.Power(100).Add(x).Add(x.One)
End Function
Sub Main()
Dim x As New ModInt(10, 13)
Dim y = F(x)
Console.WriteLine("x ^ 100 + x + 1 for x = {0} is {1}", x, y)
End Sub
End Module
- Output:
x ^ 100 + x + 1 for x = ModInt(10, 13) is ModInt(1, 13)
Wren
// Semi-abstract though we can define a 'pow' method in terms of the other operations.
class Ring {
+(other) {}
*(other) {}
one {}
pow(p) {
if (p.type != Num || !p.isInteger || p < 0) {
Fiber.abort("Argument must be non-negative integer.")
}
var pwr = one
while (p > 0) {
pwr = pwr * this
p = p - 1
}
return pwr
}
}
class ModInt is Ring {
construct new(value, modulo) {
_value = value
_modulo = modulo
}
value { _value }
modulo { _modulo }
+(other) {
if (other.type != ModInt || _modulo != other.modulo) {
Fiber.abort("Argument must be a ModInt with the same modulus.")
}
return ModInt.new((_value + other.value) % _modulo, _modulo)
}
*(other) {
if (other.type != ModInt || _modulo != other.modulo) {
Fiber.abort("Argument must be a ModInt with the same modulus.")
}
return ModInt.new((_value * other.value) % _modulo, _modulo)
}
one { ModInt.new(1, _modulo) }
toString { "Modint(%(_value), %(_modulo))" }
}
var f = Fn.new { |x|
if (!(x is Ring)) Fiber.abort("Argument must be a Ring.")
return x.pow(100) + x + x.one
}
var x = ModInt.new(10, 13)
System.print("x^100 + x + 1 for x = %(x) is %(f.call(x))")
- Output:
x^100 + x + 1 for x = Modint(10, 13) is Modint(1, 13)
zkl
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;
self(z.divr(M)[1],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:=MC(BN(10),13);
(n+3).println(" <-- 10M13 + 3");
f(n).println(" <-- 10M13^100 + 10M13 + 1");- Output:
3 <-- 1^100 + 1 + 1 0M13 <-- 10M13 + 3 1M13 <-- 10M13^100 + 10M13 + 1
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