Stinson Exercise 5.6 Galois Field

/*
Author:  Pate Williams (c) 1997

Exercise
"5.6 The field GF(2 ^ 5) can be constructed as
Z_2[x]/(x ^ 5 + x ^ 2 + 1). Perform the following
computations in this field.
(a) Compute (x ^ 4 + x ^ 2) * (x ^ 3 + x + 1).
(b) Using the Extended Euclidean algorithm
compute (x ^ 3 + x ^ 2) ^ - 1.
(c) Using the square and multiply algorithm,
compute x ^ 25." -Douglas R. Stinson-
See "Cryptography: Theory and Practice" by
Douglas R. Stinson page 201.
*/

#include <math.h>
#include <stdio.h>

#define SIZE 32l

void poly_mul(long m, long n, long *a, long *b, long *c, long *p)
{
	long ai, bj, i, j, k, sum;

	*p = m + n;
	for (k = 0; k <= *p; k++) {
		sum = 0;
		for (i = 0; i <= k; i++) {
			j = k - i;
			if (i > m) ai = 0; else ai = a[i];
			if (j > n) bj = 0; else bj = b[j];
			sum += ai * bj;
		}
		c[k] = sum;
	}
}

void poly_div(long m, long n, long *u, long *v,
	long *q, long *r, long *p, long *s)
{
	long j, jk, k, nk, vn = v[n];

	for (j = 0; j <= m; j++) r[j] = u[j];
	if (m < n) {
		*p = 0, *s = m;
		q[0] = 0;
	}
	else {
		*p = m - n, *s = n - 1;
		for (k = *p; k >= 0; k--) {
			nk = n + k;
			q[k] = r[nk] * pow(vn, k);
			for (j = nk - 1; j >= 0; j--) {
				jk = j - k;
				if (jk >= 0)
					r[j] = vn * r[j] - r[nk] * v[j - k];
				else
					r[j] = vn * r[j];
			}
		}
		while (*p > 0 && q[*p] == 0) *p = *p - 1;
		while (*s > 0 && r[*s] == 0) *s = *s - 1;
	}
}

void poly_exp_mod(long degreeA, long degreem, long n,
	long modulus, long *A, long *m, long *s,
	long *ds)
{
	int zero;
	long dp, dq, dx = degreeA, i;
	long p[SIZE], q[SIZE], x[SIZE];

	*ds = 0, s[0] = 1;
	for (i = 0; i <= dx; i++) x[i] = A[i];
	while (n > 0) {
		if ((n & 1) == 1) {
			/* s = (s * x) % m; */
			poly_mul(*ds, dx, s, x, p, &dp);
			poly_div(dp, degreem, p, m, q, s, &dq, ds);
			for (i = 0; i <= *ds; i++) s[i] %= modulus;
			zero = s[*ds] == 0, i = *ds;
			while (i > 0 && zero) {
				if (zero) *ds = *ds - 1;
				zero = s[--i] == 0;
			}
		}
		n >>= 1;
		/* x = (x * x) % m; */
		poly_mul(dx, dx, x, x, p, &dp);
		poly_div(dp, degreem, p, m, q, x, &dq, &dx);
		for (i = 0; i <= dx; i++) x[i] %= modulus;
		zero = x[dx] == 0, i = dx;
		while (i > 0 && zero) {
			if (zero) dx--;
			zero = x[--i] == 0;
		}
	}
}

void poly_copy(long db, long *a, long *b, long *da)
/* a = b */
{
	long i;

	*da = db;
	for (i = 0; i <= db; i++) a[i] = b[i];
}

int poly_Extended_Euclidean(long db, long dn,
	long *b, long *n,
	long *t, long *dt)
{
	int nonzero;
	long db0, dn0, dq, dr, dt0 = 0, dtemp, du, i;
	long b0[SIZE], n0[SIZE], q[SIZE];
	long r[SIZE], t0[SIZE], temp[SIZE], u[SIZE];

	*dt = 0;
	poly_copy(dn, n0, n, &dn0);
	poly_copy(db, b0, b, &db0);
	t0[0] = 0;
	t[0] = 1;
	poly_div(dn0, db0, n0, b0, q, r, &dq, &dr);
	nonzero = r[0] != 0;
	for (i = 1; !nonzero && i <= dr; i++)
		nonzero = r[i] != 0;
	while (nonzero) {
		poly_mul(dq, *dt, q, t, u, &du);
		if (dt0 < du)
			for (i = dt0 + 1; i <= du; i++) t0[i] = 0;
		for (i = 0; i <= du; i++)
			temp[i] = t0[i] - u[i];
		dtemp = du;
		poly_copy(*dt, t0, t, &dt0);
		poly_copy(dtemp, t, temp, dt);
		poly_copy(db0, n0, b0, &dn0);
		poly_copy(dr, b0, r, &db0);
		poly_div(dn0, db0, n0, b0, q, r, &dq, &dr);
		nonzero = r[0] != 0;
		for (i = 1; !nonzero && i <= dr; i++)
			nonzero = r[i] != 0;
	}
	if (db0 != 0 && b0[0] != 1) return 0;
	return 1;
}

void poly_mod(long da, long p, long *a, long *new_da)
{
	int zero;
	long i;

	for (i = 0; i <= da; i++) {
		a[i] %= p;
		if (a[i] < 0) a[i] += p;
	}
	zero = a[da] == 0;
	for (i = da - 1; zero && i >= 0; i--) {
		da--;
		zero = a[i] == 0;
	}
	*new_da = da;
}

void poly_write(char *label, long da, long *a)
{
	long i;

	printf("%s", label);
	for (i = da; i >= 0; i--)
		printf("%ld ", a[i]);
	printf("\n");
}

int main(void)
{
	long da = 4, db = 3, dc = 3, dd = 5;
	long de, dq, dr, ds, p = 2;
	long a[5] = { 0, 0, 1, 0, 1 };
	long b[4] = { 1, 1, 0, 1 };
	long c[4] = { 0, 0, 1, 1 };
	long d[6] = { 1, 0, 1, 0, 0, 1 };
	long e[SIZE], q[SIZE], r[SIZE], s[SIZE];

	poly_write("A = ", da, a);
	poly_write("B = ", db, b);
	poly_write("C = ", dc, c);
	poly_write("D = ", dd, d);
	poly_mul(da, db, a, b, e, &de);
	poly_mod(de, p, e, &de);
	poly_div(de, dd, e, d, q, r, &dq, &dr);
	poly_mod(dq, p, q, &dq);
	poly_mod(dr, p, r, &dr);
	poly_write("A * B mod 2 = ", de, e);
	poly_write("A * B / D mod 2 = ", dq, q);
	poly_write("A * B mod D, 2  = ", dr, r);
	if (!poly_Extended_Euclidean(dc, dd, c, d, e, &de))
		printf("*error*\in poly_Extended_Euclidean\n");
	poly_mod(de, p, e, &de);
	poly_write("C ^ - 1 mod D = ", de, e);
	poly_mul(dc, de, c, e, s, &ds);
	poly_mod(ds, p, s, &ds);
	poly_div(ds, dd, s, d, q, r, &dq, &dr);
	poly_mod(dr, p, r, &dr);
	poly_write("C * C ^ - 1 mod D, 2 = ", dr, r);
	dq = 1;
	q[0] = 0;
	q[1] = 1;
	poly_exp_mod(dq, dd, 7, p, q, d, s, &ds);
	poly_mod(ds, p, s, &ds);
	poly_write("x ^  7 mod D, 2 = ", ds, s);
	poly_exp_mod(dq, dd, 9, p, q, d, s, &ds);
	poly_mod(ds, p, s, &ds);
	poly_write("x ^  9 mod D, 2 = ", ds, s);
	poly_exp_mod(dq, dd, 10, p, q, d, s, &ds);
	poly_mod(ds, p, s, &ds);
	poly_write("x ^ 10 mod D, 2 = ", ds, s);
	poly_exp_mod(dq, dd, 25, p, q, d, s, &ds);
	poly_mod(ds, p, s, &ds);
	poly_write("x ^ 25 mod D, 2 = ", ds, s);
	return 0;
}

More Elliptic Curve Point Counting Results Using the Shanks-Mestre Algorithm by James Pate Williams, Jr.

As can be seen the C++ application appears to be much faster than the C# application. Also, in my humble opinion, C++ with header files and C++ source code files is much more elegant than C# with only class source code. I leave it to someone else to add C# interfaces to my class source code files.

Elliptic Curve Point Counting Using the Shanks-Mestre Algorithm in C# and C++ by James Pate Williams, Jr.

A few years ago, I implemented the Shanks-Mestre elliptic curve point counting algorithm in C# using the BigInteger data structure and the algorithm found in Henri Cohen’s textbook A Course in Computational Algebraic Number Theory. I translated the C# application to C++ in August 21-22, 2023. The C++ code uses the long long 64-bit signed integer data type. Below are some results. I also implemented Schoof’s point counting algorithm in C#.

Blast from the Past 1997 Image of a Matrix by James Pate Williams, Jr.

/*
  Author:  Pate Williams (c) 1997

  "Algorithm 2.3.2 (Image of a Matrix). Given an
  m by n matrix M = (m[i][i]) with 1 <= i <= m and
  1 <= j <= n having coefficients in the field K,
  this algorithm outputs a basis of the image of
  M, i. e. vector space spanned by the columns of
  M." -Henri Cohen- See "A Course in Computational
  Algebraic Number Theory" by Henri Cohen pages
  58-59. We specialize the code to the field Zp.
*/

#include <stdio.h>
#include <stdlib.h>

long** create_matrix(long m, long n)
{
    long i, ** matrix = (long**)calloc(m, sizeof(long*));

    if (!matrix) {
        fprintf(stderr, "fatal error\ninsufficient memory\n");
        fprintf(stderr, "from create_matrix\n");
        exit(1);
    }
    for (i = 0; i < m; i++) {
        matrix[i] = (long*)calloc(n, sizeof(long));
        if (!matrix[i]) {
            fprintf(stderr, "fatal error\ninsufficient memory\n");
            fprintf(stderr, "from create_matrix\n");
            exit(1);
        }
    }
    return matrix;
}

void delete_matrix(long m, long** matrix)
{
    long i;

    for (i = 0; i < m; i++) free(matrix[i]);
    free(matrix);
}

void Euclid_extended(long a, long b, long* u,
    long* v, long* d)
{
    long q, t1, t3, v1, v3;

    *u = 1, * d = a;
    if (b == 0) {
        *v = 0;
        return;
    }
    v1 = 0, v3 = b;
#ifdef DEBUG
    printf("----------------------------------\n");
    printf(" q    t3   *u   *d   t1   v1   v3\n");
    printf("----------------------------------\n");
#endif
    while (v3 != 0) {
        q = *d / v3;
        t3 = *d - q * v3;
        t1 = *u - q * v1;
        *u = v1, * d = v3;
#ifdef DEBUG
        printf("%4ld %4ld %4ld ", q, t3, *u);
        printf("%4ld %4ld %4ld %4ld\n", *d, t1, v1, v3);
#endif
        v1 = t1, v3 = t3;
    }
    *v = (*d - a * *u) / b;
#ifdef DEBUG
    printf("----------------------------------\n");
#endif
}

long inv(long number, long modulus)
{
    long d, u, v;

    Euclid_extended(number, modulus, &u, &v, &d);
    if (d == 1) return u;
    return 0;
}

void image(long m, long n, long p,
    long** M, long** X, long* r)
{
    int found;
    long D, i, j, k, s;
    long* c = (long*)calloc(m, sizeof(long));
    long* d = (long*)calloc(n, sizeof(long));
    long** N = create_matrix(m, n);

    if (!c || !d) {
        fprintf(stderr, "fatal error\ninsufficient memory\n");
        fprintf(stderr, "from kernel\n");
        exit(1);
    }
    for (i = 0; i < m; i++) {
        c[i] = -1;
        for (j = 0; j < n; j++) N[i][j] = M[i][j];
    }
    *r = 0;
    for (k = 0; k < n; k++) {
        found = 0, j = 0;
        while (!found && j < m) {
            found = M[j][k] != 0 && c[j] == -1;
            if (!found) j++;
        }
        if (found) {
            D = p - inv(M[j][k], p);
            M[j][k] = p - 1;
            for (s = k + 1; s < n; s++)
                M[j][s] = (D * M[j][s]) % p;
            for (i = 0; i < m; i++) {
                if (i != j) {
                    D = M[i][k];
                    M[i][k] = 0;
                    for (s = k + 1; s < n; s++) {
                        M[i][s] = (M[i][s] + D * M[j][s]) % p;
                        if (M[i][s] < 0) M[i][s] += p;
                    }
                }
            }
            c[j] = k;
            d[k] = j;
        }
        else {
            *r = *r + 1;
            d[k] = -1;
        }
    }
    for (j = 0; j < m; j++) {
        if (c[j] != -1) {
            for (i = 0; i < n; i++) {
                if (i < m) X[i][j] = N[i][c[j]];
                else X[i][j] = 0;
            }
        }
    }
    delete_matrix(m, N);
    free(c);
    free(d);
}

void print_matrix(long m, long n, long** a)
{
    long i, j;

    for (i = 0; i < m; i++) {
        for (j = 0; j < n; j++)
            printf("%2ld ", a[i][j]);
        printf("\n");
    }
}

int main(void)
{
    long i, j, m = 8, n = 8, p = 13, r;
    long a[8][8] = { {0,  0,  0,  0,  0,  0,  0,  0},
                    {2,  0,  7, 11, 10, 12,  5, 11},
                    {3,  6,  3,  3,  0,  4,  7,  2},
                    {4,  3,  6,  4,  1,  6,  2,  3},
                    {2, 11,  8,  8,  2,  1,  3, 11},
                    {6, 11,  8,  6,  2,  6, 10,  9},
                    {5, 11,  7, 10,  0, 11,  6, 12},
                    {3,  3, 12,  5,  0, 11,  9, 11} };
    long** M = create_matrix(m, n);
    long** X = create_matrix(n, n);

    for (i = 0; i < m; i++)
        for (j = 0; j < n; j++)
            M[i][j] = a[j][i];
    printf("the original matrix is as follows:\n");
    print_matrix(m, n, M);
    image(m, n, p, M, X, &r);
    printf("the image of the matrix is as follows:\n");
    print_matrix(n, n - r, X);
    printf("the rank of the matrix is: %ld\n", n - r);
    delete_matrix(m, M);
    delete_matrix(n, X);
    return 0;
}

First Blast from the Past (1997) Computing the Inverse Image of a Matrix by James Pate Williams, Jr.

/*
  Author:  Pate Williams (c) 1997

  "Algorithm 2.3.5 (Inverse Image Matrix). Let M be
  an m by n matrix and V be an m by r matrix, where
  n <= m. This algorithm either outputs a message
  saying that some column vector of V is not in the
  image of M, or outputs an n by r matrix X such
  that V = MX." -Henri Cohen- See "A Course in Com-
  putational Algebraic Number Theory" by Henri
  Cohen pages 60-61. We specialize to the field Q.
*/

#include <stdio.h>
#include <stdlib.h>

double** create_matrix(long m, long n)
{
    double** matrix = (double**)calloc(m, sizeof(double*));
    long i;

    if (!matrix) {
        fprintf(stderr, "fatal error\ninsufficient memory\n");
        fprintf(stderr, "from create_matrix\n");
        exit(1);
    }
    for (i = 0; i < m; i++) {
        matrix[i] = (double*)calloc(n, sizeof(double));
        if (!matrix[i]) {
            fprintf(stderr, "fatal error\ninsufficient memory\n");
            fprintf(stderr, "from create_matrix\n");
            exit(1);
        }
    }
    return matrix;
}

void delete_matrix(long m, double** matrix)
{
    long i;

    for (i = 0; i < m; i++) free(matrix[i]);
    free(matrix);
}

void inverse_image_matrix(long m, long n, long r,
    double** M, double** V,
    double** X)
{
    double ck, d, sum, t;
    double** B = create_matrix(m, r);
    int found;
    long i, j, k, l;

    for (i = 0; i < m; i++)
        for (j = 0; j < r; j++)
            B[i][j] = V[i][j];
    for (j = 0; j < n; j++) {
        found = 0, i = j;
        while (!found && i < m) {
            found = M[i][j] != 0;
            if (!found) i++;
        }
        if (!found) {
            fprintf(stderr, "fatal error\nnot linearly independent\n");
            fprintf(stderr, "from inverse_image_matrix\n");
            exit(1);
        }
        if (i > j) {
            for (l = 0; l < n; l++)
                t = M[i][l], M[i][l] = M[j][l], M[j][l] = t;
            for (l = 0; l < r; l++)
                t = B[i][l], B[i][l] = B[j][l], B[j][l] = t;
        }
        d = 1.0 / M[j][j];
        for (k = j + 1; k < m; k++) {
            ck = d * M[k][j];
            for (l = j + 1; l < n; l++)
                M[k][l] -= ck * M[j][l];
            for (l = 0; l < r; l++)
                B[k][l] -= ck * B[j][l];
        }
    }
    for (i = n - 1; i >= 0; i--) {
        for (k = 0; k < r; k++) {
            sum = 0;
            for (j = i + 1; j < n; j++)
                sum += M[i][j] * X[j][k];
            X[i][k] = (B[i][k] - sum) / M[i][i];
        }
    }
    for (k = n + 1; k < m; k++) {
        for (j = 0; j < r; j++) {
            sum = 0;
            for (i = 0; i < n; i++)
                sum += M[k][i] * X[i][j];
            if (sum != B[k][j]) {
                fprintf(stderr, "fatal error\ncolumn not in image\n");
                fprintf(stderr, "from inverse_image_matrix\n");
                exit(1);
            }
        }
    }
    delete_matrix(m, B);
}

void matrix_multiply(long m, long n, long r,
    double** a, double** b,
    double** c)
    /* c = a * b */
{
    double sum;
    long i, j, k;

    for (i = 0; i < m; i++) {
        for (j = 0; j < r; j++) {
            sum = 0.0;
            for (k = 0; k < n; k++)
                sum += a[i][k] * b[k][j];
            c[i][j] = sum;
        }
    }
}

void print_matrix(long m, long n, double** a)
{
    long i, j;

    for (i = 0; i < m; i++) {
        for (j = 0; j < n; j++)
            printf("%+10.6lf ", a[i][j]);
        printf("\n");
    }
}

int main(void)
{
    long i, j, m = 4, n = 4, r = 4;
    double** c = create_matrix(m, n);
    double** M = create_matrix(m, n);
    double** V = create_matrix(m, r);
    double** X = create_matrix(n, r);

    for (i = 0; i < m; i++) {
        c[i][i] = M[i][i] = 2.0;
        if (i > 0)
            c[i][i - 1] = M[i][i - 1] = -1.0;
        if (i < m - 1)
            c[i][i + 1] = M[i][i + 1] = -1.0;
    }
    for (i = 0; i < m; i++)
        for (j = 0; j < r; j++)
            V[i][j] = i + j + 1;
    printf("M is as follows:\n");
    print_matrix(m, n, M);
    printf("V is as follows:\n");
    print_matrix(m, r, V);
    inverse_image_matrix(m, n, r, M, V, X);
    printf("X is as follows:\n");
    print_matrix(n, r, X);
    matrix_multiply(m, n, r, c, X, M);
    printf("MX is as follows:\n");
    print_matrix(m, r, M);
    delete_matrix(m, c);
    delete_matrix(m, M);
    delete_matrix(m, V);
    delete_matrix(n, X);
    return 0;
}

Solving the Low-Density Subset Sum Problem with the LLL Lattice Reduction Algorithm C# Implementation by James Pate Williams, Jr.

A few years ago, I implemented the LLL lattice reduction algorithm found in two references: “Handbook of Applied Cryptography” Edited by Alfred J. Menezes et al and “A Course in Computational Algebraic Number Theory” by Henri Cohen. My new implementation in C# uses 64-bit integers and BigIntegers. Here are some results.

Comparison of Brent’s Modification of the Pollard rho Factoring Method and the Original Pollard rho Method C# Implementations by James Pate Williams, Jr.

27-Decimal Digit Number 123456789012345678901234567

31-Digit Decimal Number 1234567890123456789012345678901 Pollard Failed

The same factorizations by the Classical Shor-Pollard- James Pate Williams, Jr. algorithm.

Powering Algorithms from Heri Cohen’s Textbook Implemented by James Pate Williams, Jr.

// Algorithms from "A Course in Computational
// Algebraic Number Theory" by Henri Cohen
// Implemented by James Pate Williams, Jr.
// Copyright (c) 2023 All Rights Reserved

using System.Collections.Generic;
using System.Numerics;

namespace PoweringAlgorithms
{
    public interface IAlgorithms
    {
        // Extended Euclidean Algorithm
        BigInteger Algorithm_1_2_1(
            BigInteger g, BigInteger n);
        BigInteger Algorithm_1_2_2(
            int e, BigInteger g, BigInteger n);
        public BigInteger Algorithm_1_2_3(
            int e, BigInteger g, BigInteger n);
        void Algorithm_1_3_6(
            BigInteger a, BigInteger b,
            out BigInteger u, out BigInteger v,
            out BigInteger d);
        List<int> GetBits(BigInteger N);
        BigInteger Inverse(
            BigInteger a, BigInteger n);
    }
}
using System.Collections.Generic;
using System.Numerics;

namespace PoweringAlgorithms
{
    public class Algorithms : IAlgorithms
    { 
        public BigInteger Algorithm_1_2_1(
            BigInteger g, BigInteger n)
        {
            BigInteger y = 1;

            if (n == 0)
                return y;

            BigInteger N, z;

            if (n < 0)
            {
                N = -n;
                z = Inverse(g, n);
            }

            else
            {
                N = n;
                z = g;
            }

            while (N > 0)
            {
                if (N % 2 == 1)
                    y *= z;

                N /= 2;
                z *= z;
            }

            return y;
        }

        public BigInteger Algorithm_1_2_2(
            int e, BigInteger g, BigInteger n)
        {
            BigInteger E, N, y, z;

            if (n == 0)
                return BigInteger.One;

            if (n < 0)
            {
                N = -n;
                z = Inverse(g, n);
            }

            else
            {
                N = n;
                z = g;
            }
            
            y = z;
            E = BigInteger.Pow(2, e);
            N -= E;

            while (E > 1)
            {
                E /= 2;
                y *= y;

                if (N >= E)
                {
                    N -= E;
                    y *= z;
                }
            }

            return y;
        }

        public BigInteger Algorithm_1_2_3(
            int e, BigInteger g, BigInteger n)
        {
            BigInteger y = 1;

            if (n == 0)
                return y;

            BigInteger N, z;

            if (n < 0)
            {
                N = -n;
                z = Inverse(g, n);
            }

            else
            {
                N = n;
                z = g;
            }

            y = z;

            int f = e;
            List<int> bits = GetBits(N);

            while (f > 0)
            {
                f--;
                y *= y;
                if (bits[f] == 1)
                    y *= z;
            }

            return y;
        }

        public void Algorithm_1_3_6(
                    BigInteger a, BigInteger b,
                    out BigInteger u, out BigInteger v,
                    out BigInteger d)
        {
            u = 1;
            d = a;

            if (b == 0)
            {
                v = 0;
                return;
            }

            BigInteger q, t1, t3, v1 = 0, v3 = b;

            while (v3 > 0)
            {
                q = d / v3;
                t3 = d % v3;
                t1 = u - q * v1;
                u = v1;
                d = v3;
                v1 = t1;
                v3 = t3;
            }

            v = (d - a * u) / b;
        }

        public List<int> GetBits(BigInteger N)
        {
            int b;
            List<int> temp = new();

            while (N > 0)
            {
                b = (int)(N % 2);
                N /= 2;
                temp.Add(b);
            }

            List<int> bits = new();

            for (int i = temp.Count; i >= 0; i--)
                bits.Add(temp[i]);

            return bits;
        }

        public BigInteger Inverse(BigInteger a, BigInteger n)
        {
            BigInteger d, u, v;

            Algorithm_1_3_6(
                a, n, out u, out v, out d);
            
            return u;
        }
    }
}
using System;
using System.Numerics;
using System.Windows.Forms;

namespace PoweringAlgorithms
{
    public partial class TestForm : Form
    {
        public TestForm()
        {
            InitializeComponent();
        }

        private static BigInteger Horner(string text)
        {
            BigInteger a = text[0] - '0';

            for (int i = 1; i < text.Length; i++)
                a = 10 * a + text[i] - '0';

            return a;
        }

        private void button1_Click(object sender, EventArgs e)
        {
            int expon = 0;
            Algorithms algos = new Algorithms();
            BigInteger g = Horner(textBox1.Text);
            BigInteger n = Horner(textBox2.Text);
            BigInteger a = algos.Algorithm_1_2_1(g, n);
            BigInteger m = BigInteger.Abs(n);

            while (m > 0)
            {
                expon++;
                m /= 2;
            }
            
            BigInteger b = algos.Algorithm_1_2_2(expon - 1, g, n);
            BigInteger c = algos.Algorithm_1_2_2(expon - 1, g, n);

            textBox3.Text = a.ToString();
            textBox4.Text = b.ToString();
            textBox5.Text = c.ToString();
        }

        private void button2_Click(object sender, EventArgs e)
        {
            Close();
        }
    }
}

Infix Notation to Postfix Notation and Postfix Evaluator Implemented by James Pate Williams, Jr.

I translated a Pascal program that is found in “Applied Data Structures Using Pascal” by Guy J. Hale and Richard J. Easton. The original Pascal program only used single digits and four arithmetic operators: ‘*’, ‘/’, ‘+’, and ‘-‘. I extended the code to multiple digit positive integers and added an exponentiation operator ‘^’. The priority of the operators is ‘^’, ‘*’, and ‘/’ highest value and ‘+’ and ‘-‘ lowest priority. I could easily add a modulus operator ‘%’ and Boolean bit operators. Extension to negative integers should be a facile operation. Below is the output from my C++ application.

3 + 7 * 8 - 5
3 7 8 * + 5 - 
Positive integer value = 54
3 * 7 - 4 / 2
3 7 * 4 2 / - 
Positive integer value = 19
(3 + 7) * 8 - 5
3 7 + 8 * 5 - 
Positive integer value = 75
(3 + 4) * 8 - (7 * 3 - 4)
3 4 + 8 * 7 3 * 4 - - 
Positive integer value = 39
(100 + 50) * 20 - 100 / 2
100 50 + 20 * 100 2 / - 
Positive integer value = 2950
2 ^ 16 - 5 * 100
2 16 ^ 5 100 * - 
Positive integer value = 65036

#pragma once
#include <list>
#include <stack>
#include <string>
#include <vector>
using namespace std;

const char Exp = '^';
const char Mul = '*';
const char Div = '/';
const char Add = '+';
const char Sub = '-';
const char LtParen = '(';
const char RtParen = ')';

class InfixToPostFix
{
public:
	stack<char> numberStack;
	
	int Convert(
		string infixStr, string& postfixStr);
	int EvaluatePostFix(string postfixStr);
	int Priority(char opcode);
};
#include "pch.h"
#include "InfixToPostFix.h"
#include <vector>
using namespace std;

int InfixToPostFix::Convert(
	string infixStr, string& postfixStr)
{
	char ch = 0;
	int number = 0, opcode = 0, opcode1 = 0, parenLevel = 0;
	string numberStr;

	for (size_t i = 0; i < infixStr.size();)
	{
		//while (i < infixStr.size() && infixStr[i] == ' ')
			//i++;

		while (i < infixStr.size() && infixStr[i] >= '0' &&
			infixStr[i] <= '9')
			numberStr.push_back(infixStr[i++]);

		if (numberStr.size() != 0)
		{
			for (size_t j = 0; j < numberStr.size(); j++)
				postfixStr.push_back(numberStr[j]);

			postfixStr.push_back(' ');
			numberStr.clear();
		}

		//while (i < infixStr.size() && infixStr[i] == ' ')
			//i++;

		if (infixStr[i] == '(')
		{
			numberStack.push(infixStr[i]);
			parenLevel++;
		}

		//while (i < infixStr.size() && infixStr[i] == ' ')
			//i++;

		if (infixStr[i] == ')')
		{
			ch = numberStack.top();
			numberStack.pop();

			parenLevel--;

			//while (i < infixStr.size() && infixStr[i] == ' ')
				//i++;

			while (ch != '(')
			{
				postfixStr.push_back(ch);
				postfixStr.push_back(' ');

				parenLevel++;

				ch = numberStack.top();
				numberStack.pop();
			}
		}

		//while (i < infixStr.size() && infixStr[i] == ' ')
			//i++;

		if (infixStr[i] == '^' || infixStr[i] == '*' ||
			infixStr[i] == '/' || infixStr[i] == '+' ||
			infixStr[i] == '-')
		{
			if (numberStack.empty())
				numberStack.push(infixStr[i]);
			else
			{
				ch = numberStack.top();
				numberStack.pop();

				//while (i < infixStr.size() && infixStr[i] == ' ')
					//i++;

				while (Priority(ch) >= Priority(infixStr[i]) &&
					!numberStack.empty() && ch != '(')
				{
					postfixStr.push_back(ch);
					postfixStr.push_back(' ');

					ch = numberStack.top();
					numberStack.pop();
				}

				//while (i < infixStr.size() && infixStr[i] == ' ')
					//i++;

				if (ch != '(')
				{
					if (Priority(infixStr[i]) <= Priority(ch))
					{
						postfixStr.push_back(ch);
						postfixStr.push_back(' ');

						numberStack.push(infixStr[i]);
					}

					else
					{
						numberStack.push(ch);
						numberStack.push(infixStr[i]);
					}
				}
				else
				{
					numberStack.push(ch);
					numberStack.push(infixStr[i]);
				}
			}
		}

		i++;
	}

	while (!numberStack.empty())
	{
		ch = numberStack.top();
		numberStack.pop();

		postfixStr.push_back(ch);
		postfixStr.push_back(' ');
	}

	return 0;
}

int InfixToPostFix::EvaluatePostFix(string postfixStr)
{
	char opcode = 0;
	int charValue = 0, result = 0, value1 = 0, value2 = 0;
	stack<int> intStack;

	for (size_t i = 0; i < postfixStr.size();)
	{
		string numberStr;

		if (postfixStr[i] != ' ')
		{
			while (postfixStr[i] >= '0' && postfixStr[i] <= '9')
				numberStr.push_back(postfixStr[i++]);

			if (!numberStr.empty())
				intStack.push(atoi(numberStr.c_str()));

			else
			{
				value2 = intStack.top();
				intStack.pop();
				value1 = intStack.top();
				intStack.pop();

				opcode = postfixStr[i++];

				if (opcode == '^')
					result = (int)pow(value1, value2);
				else if (opcode == '*')
					result = value1 * value2;
				else if (opcode == '/')
					result = value1 / value2;
				else if (opcode == '+')
					result = value1 + value2;
				else if (opcode == '-')
					result = value1 - value2;

				intStack.push(result);
			}
		}

		else
			i++;
	}

	result = intStack.top();
	intStack.pop();

	return result;
}

int InfixToPostFix::Priority(char opcode)
{
	int result = 0;

	switch (opcode)
	{
	case Exp:
		result = 2;
		break;
	case Mul:
		result = 2;
		break;
	case Div:
		result = 2;
		break;
	case Add:
		result = 1;
		break;
	case Sub:
		result = 1;
		break;
	case LtParen:
		result = 0;
	}

	return result;
}
#include "pch.h"
#include "InfixToPostFix.h"
#include <fstream>
#include <iostream>
#include <string>
using namespace std;

int main()
{
    fstream inps, outs;
    char line[256];
    
    inps.open("TestFile.txt", fstream::in);
    outs.open("ResuFile.txt", fstream::out | fstream::ate);

    while (!inps.eof())
    {
        string postfixStr;

        inps.getline(line, 256);

        if (strlen(line) == 0)
            break;

        string infixStr(line, strlen(line));
        InfixToPostFix translate;

        int con = translate.Convert(
            infixStr, postfixStr);

        if (con != 0)
        {
            cout << "Conversion error!" << endl;
            cout << "Error code = " << con << endl;
        }

        else
        {
            char newline[] = { '\n' };

            outs.write(infixStr.c_str(), infixStr.size());
            outs.write(newline, 1);
            outs.write(postfixStr.c_str(), postfixStr.size());
            outs.write(newline, 1);

            int val = translate.EvaluatePostFix(postfixStr);

            if (val < 0)
            {
                cout << "Evaluation error!" << endl;
                cout << "Error code = " << val << endl;
            }

            else
            {
                char buffer[256] = { '\0' };
                string str = "Positive integer value = ";

                _itoa_s(val, buffer, 10);
                
                outs.write(str.c_str(), strlen(str.c_str()));
                outs.write(buffer, strlen(buffer));
                outs.write(newline, 1);
            }
        }
    }

    inps.close();
    outs.close();

    return 0;
}