Task: Montgomery reduction





Java: import java.math.BigInteger;

public class MontgomeryReduction {
    private static final BigInteger ZERO = BigInteger.ZERO;
    private static final BigInteger ONE = BigInteger.ONE;
    private static final BigInteger TWO = BigInteger.valueOf(2);

    public static class Montgomery {
        public static final int BASE = 2;

        BigInteger m;
        BigInteger rrm;
        int n;

        public Montgomery(BigInteger m) {
            if (m.compareTo(BigInteger.ZERO) <= 0 || !m.testBit(0)) {
                throw new IllegalArgumentException();
            }
            this.m = m;
            this.n = m.bitLength();
            this.rrm = ONE.shiftLeft(n * 2).mod(m);
        }

        public BigInteger reduce(BigInteger t) {
            BigInteger a = t;
            for (int i = 0; i < n; i++) {
                if (a.testBit(0)) a = a.add(this.m);
                a = a.shiftRight(1);
            }
            if (a.compareTo(m) >= 0) a = a.subtract(this.m);
            return a;
        }
    }

    public static void main(String[] args) {
        BigInteger m  = new BigInteger("750791094644726559640638407699");
        BigInteger x1 = new BigInteger("540019781128412936473322405310");
        BigInteger x2 = new BigInteger("515692107665463680305819378593");

        Montgomery mont = new Montgomery(m);
        BigInteger t1 = x1.multiply(mont.rrm);
        BigInteger t2 = x2.multiply(mont.rrm);

        BigInteger r1 = mont.reduce(t1);
        BigInteger r2 = mont.reduce(t2);
        BigInteger r = ONE.shiftLeft(mont.n);

        System.out.printf("b :  %s\n", Montgomery.BASE);
        System.out.printf("n :  %s\n", mont.n);
        System.out.printf("r :  %s\n", r);
        System.out.printf("m :  %s\n", mont.m);
        System.out.printf("t1:  %s\n", t1);
        System.out.printf("t2:  %s\n", t2);
        System.out.printf("r1:  %s\n", r1);
        System.out.printf("r2:  %s\n", r2);
        System.out.println();
        System.out.printf("Original x1       :  %s\n", x1);
        System.out.printf("Recovered from r1 :  %s\n", mont.reduce(r1));
        System.out.printf("Original x2       :  %s\n", x2);
        System.out.printf("Recovered from r2 :  %s\n", mont.reduce(r2));

        System.out.println();
        System.out.println("Montgomery computation of x1 ^ x2 mod m :");
        BigInteger prod = mont.reduce(mont.rrm);
        BigInteger base = mont.reduce(x1.multiply(mont.rrm));
        BigInteger exp = x2;
        while (exp.bitLength()>0) {
            if (exp.testBit(0)) prod=mont.reduce(prod.multiply(base));
            exp = exp.shiftRight(1);
            base = mont.reduce(base.multiply(base));
        }
        System.out.println(mont.reduce(prod));

        System.out.println();
        System.out.println("Library-based computation of x1 ^ x2 mod m :");
        System.out.println(x1.modPow(x2, m));
    }
}




VB: Imports System.Numerics
Imports System.Runtime.CompilerServices

Module Module1

    <Extension()>
    Function BitLength(v As BigInteger) As Integer
        If v < 0 Then
            v *= -1
        End If

        Dim result = 0
        While v > 0
            v >>= 1
            result += 1
        End While
        Return result
    End Function

    Structure Montgomery
        Shared ReadOnly BASE = 2
        Dim m As BigInteger
        Dim rrm As BigInteger
        Dim n As Integer

        Sub New(m As BigInteger)
            If m < 0 OrElse m.IsEven Then
                Throw New ArgumentException()
            End If

            Me.m = m
            n = m.BitLength
            rrm = (BigInteger.One << (n * 2)) Mod m
        End Sub

        Function Reduce(t As BigInteger) As BigInteger
            Dim a = t
            For i = 1 To n
                If Not a.IsEven Then
                    a += m
                End If
                a >>= 1
            Next
            If a >= m Then
                a -= m
            End If
            Return a
        End Function
    End Structure

    Sub Main()
        Dim m = BigInteger.Parse("750791094644726559640638407699")
        Dim x1 = BigInteger.Parse("540019781128412936473322405310")
        Dim x2 = BigInteger.Parse("515692107665463680305819378593")

        Dim mont As New Montgomery(m)
        Dim t1 = x1 * mont.rrm
        Dim t2 = x2 * mont.rrm

        Dim r1 = mont.Reduce(t1)
        Dim r2 = mont.Reduce(t2)
        Dim r = BigInteger.One << mont.n

        Console.WriteLine("b :  {0}", Montgomery.BASE)
        Console.WriteLine("n :  {0}", mont.n)
        Console.WriteLine("r :  {0}", r)
        Console.WriteLine("m :  {0}", mont.m)
        Console.WriteLine("t1:  {0}", t1)
        Console.WriteLine("t2:  {0}", t2)
        Console.WriteLine("r1:  {0}", r1)
        Console.WriteLine("r2:  {0}", r2)
        Console.WriteLine()
        Console.WriteLine("Original x1       : {0}", x1)
        Console.WriteLine("Recovered from r1 : {0}", mont.Reduce(r1))
        Console.WriteLine("Original x2       : {0}", x2)
        Console.WriteLine("Recovered from r2 : {0}", mont.Reduce(r2))

        Console.WriteLine()
        Console.WriteLine("Montgomery computation of x1 ^ x2 mod m :")
        Dim prod = mont.Reduce(mont.rrm)
        Dim base = mont.Reduce(x1 * mont.rrm)
        Dim exp = x2
        While exp.BitLength > 0
            If Not exp.IsEven Then
                prod = mont.Reduce(prod * base)
            End If
            exp >>= 1
            base = mont.Reduce(base * base)
        End While
        Console.WriteLine(mont.Reduce(prod))
        Console.WriteLine()
        Console.WriteLine("Alternate computation of x1 ^ x2 mod m :")
        Console.WriteLine(BigInteger.ModPow(x1, x2, m))
    End Sub

End Module