How to find the answer to a XOR hash if I have the key & one number

I have a little routine that makes up a cumulative XOR hash. It's as if it is a savings account which gets bigger, cumulatively daily.. but in this sense we're saying the answer is being generated cumulatively and the key is always present.

I have taken a string of chars

pseudo code:

char H[10] = { "ABCDEFGHI", "\0" };

and I used 9 32-bit numeric keys to hash them in XOR encryption.

I did it like this:

for (i;i<10;i++)
    bitset<32> K ^= H[i] ^ NUMKEY[i];

Now this makes it impervious without the calculus plotting I did (see what I did there?) So K is an accumulation of calculus points, which are completely predictable according to calculus.

as far as I know, to undo it, I do

for (i;i<10;i++) {
    X=0;
    X ^= K ^ NUMKEY[i];
}

Is there other math involved? I think I have to take that X and do a little K - X to find the true derivative.

Here's the current routine I have for it. But I'm not getting what I'm looking for.

for_each (std::istreambuf_iterator<char>(in), \
    std::istreambuf_iterator<char>(), \
    [&] (long x) {
    t=s_nop(t,0);
    cred.push_back(t);
    alpha = static_cast<long>(cred[size]);
    delta = static_cast<long>(x);
    lambda ^= (alpha ^ delta);
    size++;
});

for (;i<bn;i++) {
    alpha =  static_cast<unsigned long>(cred[bn-1-i]);
    int m = lambda.to_ulong(), n = alpha.to_ulong();

    long hash1 = abs((m-n-1)%256-1);
    delta = static_cast<unsigned long>(hash1);
    btrace.push_back(hash1);
    cout << hash1 << " ";
}

Please have a safe and Merry Christmas. Thank you in advance!

Solution

I think what you might really want is a one time pad. (the snippet is javascript as pseudocode moving in that direction)

//ignore these top 3 functions (they're just for printing output)
function appendLine(text, target, space, color) {
  var line = document.createElement("div");
  line.style.border = "1px solid " + color;
  line.style.padding = line.style.margin = space;
  line.style["font-family"] = "monospace";
  line.appendChild(document.createTextNode(text));
  target.appendChild(line);
  return line;
}
function makeSection(title) {
  var d = appendLine(title, document.body, "5px", "#dddddd");
  var results = document.createElement("div");
  d.appendChild(results);
  return function(result) {
    appendLine(result, results, "2px", "#eeeeee");
  };
}
function toHex(arr) {
  return arr.map(function(n){
    var h = (n >>> 0).toString(16).toUpperCase();
    while(h.length < 8) h = "0" + h;
    return h;
  }).join(",");
}

//your message
var H = "ABCDEFGHI".split("").map(function(c){return c.charCodeAt(0)});
//your secret encoding key
var NUMKEY = Array.apply(null, Array(9)).map(function(){return Math.random() * Math.pow(2, 32)});

//what you're doing
(function() {
  var section = makeSection("what you're doing:");
  section("ABCDEFGHI as array of 32bit numbers: " + toHex(H));
  section("NUMKEY: " + toHex(NUMKEY));
  var K = 0;
  for (var i = 0; i < 10; i++) {
    K ^= H[i] ^ NUMKEY[i];
  }
  section("K: " + toHex([K]));
  var X = 0;
  for (var i = 0; i < 10; i++) {
    X ^= K ^ NUMKEY[i];
  }
  section("X: " + toHex([X]));
})();

//what you're trying to do
(function() {
  var section = makeSection("what you're trying to do:");
  section("ABCDEFGHI as array of 32bit numbers: " + toHex(H));
  section("NUMKEY: " + toHex(NUMKEY));
  var Hs_XORd = 0;
  for (var i = 0; i < 10; i++) {
    Hs_XORd ^= H[i];
  }
  section("H's XOR'd together: " + toHex([Hs_XORd]));
  var NUMKEYs_XORd = 0;
  for (var i = 0; i < H.length; i++) {
    NUMKEYs_XORd ^= NUMKEY[i];
  }
  section("NUMKEY's XOR'd together: " + toHex([NUMKEYs_XORd]));
  var K = NUMKEYs_XORd ^ Hs_XORd;
  section("K: " + toHex([K]));
  var X = K ^ NUMKEYs_XORd;
  section("X (should be the same as H's XOR'd together): " + toHex([X]));
})();


//what I think you mean to be doing (one time pad)
(function() {
  var section = makeSection("what I think you mean to be doing (one time pad):");
  section("ABCDEFGHI as array of 32bit numbers: " + toHex(H));
  section("NUMKEY: " + toHex(NUMKEY));
  var K = [];
  for (var i = 0; i < H.length; i++) {
    K[i] = H[i] ^ NUMKEY[i];
  }
  section("K (encoded message using NUMKEY as one time pad): " + toHex(K));
  var X = [];
  for (var i = 0; i < K.length; i++) {
    X[i] = K[i] ^ NUMKEY[i];
  }
  section("X (decoded message, should be the same as ABCDEFGHI): " + toHex(X));
})();