Is it possible to implement bitwise operators using integer arithmetic?

First solutions for shifting (shift is the shift distance, must not be negative, a is the operand to be shifted and contains also the result when done). The power table is used by all three shift operations.

// table used for shift operations
powtab = { 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, -32768 };

// logical shift left
if (shift > 15) {
     a = 0; // if shifting more than 15 bits to the left, value is always zero
} else {
     a *= powtab[shift];
}

// logical shift right (unsigned)
if (shift > 15) {
    a = 0; // more than 15, becomes zero
} else if (shift > 0) {
    if (a < 0) {
        // deal with the sign bit (15)
        a += -32768;
        a /= powtab[shift];
        a += powtab[15 - shift];
    } else {
        a /= powtab[shift];
    }
}

// arithmetic shift right (signed)
if (shift >= 15) {
    if (a < 0) {
        a = -1;
    } else {
        a = 0;
    }
} else if (shift > 0) {
    if (a < 0) {
        // deal with the sign bit
        a += -32768;
        a /= powtab[shift];
        a -= powtab[15 - shift];
    } else {
        // same as unsigned shift
        a /= powtab[shift];
    }
}

For AND, OR and XOR i could not come up with a simple solution, so i’ll do it with looping over each single bit. There might be a better trick to do this. Pseudocode assumes a and b are input operands, c is the result value, x is the loop counter (each loop must run exactly 16 times):

// XOR (^)
c = 0;
for (x = 0; x <= 15; ++x) {
    c += c;
    if (a < 0) {
        if (b >= 0) {
            c += 1;
        }
    } else if (b < 0) {
        c += 1;
    }
    a += a;
    b += b;
}

// AND (&)
c = 0;
for (x = 0; x <= 15; ++x) {
    c += c;
    if (a < 0) {
        if (b < 0) {
            c += 1;
        }
    }
    a += a;
    b += b;
}

// OR (|)
c = 0;
for (x = 0; x <= 15; ++x) {
    c += c;
    if (a < 0) {
        c += 1;
    } else if (b < 0) {
        c += 1;
    }
    a += a;
    b += b;
}

Thats assuming that all variables are 16 bits and all operations behave as signed (so a<0 actually is true when bit 15 is set).

EDIT: i actually tested all possible operand values (-32768 to 32767) for shifts ranging from 0 to 31 for correctness and it works correctly (assuming integer divides). For the AND/OR/XOR code an exhaustive test takes too long on my machine, but since the code for these is pretty simple there should be no edge cases anyway.