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+// Copyright 2009 The Closure Library Authors. All Rights Reserved.
+//
+// Licensed under the Apache License, Version 2.0 (the "License");
+// you may not use this file except in compliance with the License.
+// You may obtain a copy of the License at
+//
+//      http://www.apache.org/licenses/LICENSE-2.0
+//
+// Unless required by applicable law or agreed to in writing, software
+// distributed under the License is distributed on an "AS-IS" BASIS,
+// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+// See the License for the specific language governing permissions and
+// limitations under the License.
+
+/**
+ * @fileoverview Defines an Integer class for representing (potentially)
+ * infinite length two's-complement integer values.
+ *
+ * For the specific case of 64-bit integers, use goog.math.Long, which is more
+ * efficient.
+ *
+ */
+
+goog.provide('goog.math.Integer');
+
+
+
+/**
+ * Constructs a two's-complement integer an array containing bits of the
+ * integer in 32-bit (signed) pieces, given in little-endian order (i.e.,
+ * lowest-order bits in the first piece), and the sign of -1 or 0.
+ *
+ * See the from* functions below for other convenient ways of constructing
+ * Integers.
+ *
+ * The internal representation of an integer is an array of 32-bit signed
+ * pieces, along with a sign (0 or -1) that indicates the contents of all the
+ * other 32-bit pieces out to infinity.  We use 32-bit pieces because these are
+ * the size of integers on which Javascript performs bit-operations.  For
+ * operations like addition and multiplication, we split each number into 16-bit
+ * pieces, which can easily be multiplied within Javascript's floating-point
+ * representation without overflow or change in sign.
+ *
+ * @struct
+ * @constructor
+ * @param {Array<number>} bits Array containing the bits of the number.
+ * @param {number} sign The sign of the number: -1 for negative and 0 positive.
+ * @final
+ */
+goog.math.Integer = function(bits, sign) {
+  /**
+   * @type {!Array<number>}
+   * @private
+   */
+  this.bits_ = [];
+
+  /**
+   * @type {number}
+   * @private
+   */
+  this.sign_ = sign;
+
+  // Copy the 32-bit signed integer values passed in.  We prune out those at the
+  // top that equal the sign since they are redundant.
+  var top = true;
+  for (var i = bits.length - 1; i >= 0; i--) {
+    var val = bits[i] | 0;
+    if (!top || val != sign) {
+      this.bits_[i] = val;
+      top = false;
+    }
+  }
+};
+
+
+// NOTE: Common constant values ZERO, ONE, NEG_ONE, etc. are defined below the
+// from* methods on which they depend.
+
+
+/**
+ * A cache of the Integer representations of small integer values.
+ * @type {!Object}
+ * @private
+ */
+goog.math.Integer.IntCache_ = {};
+
+
+/**
+ * Returns an Integer representing the given (32-bit) integer value.
+ * @param {number} value A 32-bit integer value.
+ * @return {!goog.math.Integer} The corresponding Integer value.
+ */
+goog.math.Integer.fromInt = function(value) {
+  if (-128 <= value && value < 128) {
+    var cachedObj = goog.math.Integer.IntCache_[value];
+    if (cachedObj) {
+      return cachedObj;
+    }
+  }
+
+  var obj = new goog.math.Integer([value | 0], value < 0 ? -1 : 0);
+  if (-128 <= value && value < 128) {
+    goog.math.Integer.IntCache_[value] = obj;
+  }
+  return obj;
+};
+
+
+/**
+ * Returns an Integer representing the given value, provided that it is a finite
+ * number.  Otherwise, zero is returned.
+ * @param {number} value The value in question.
+ * @return {!goog.math.Integer} The corresponding Integer value.
+ */
+goog.math.Integer.fromNumber = function(value) {
+  if (isNaN(value) || !isFinite(value)) {
+    return goog.math.Integer.ZERO;
+  } else if (value < 0) {
+    return goog.math.Integer.fromNumber(-value).negate();
+  } else {
+    var bits = [];
+    var pow = 1;
+    for (var i = 0; value >= pow; i++) {
+      bits[i] = (value / pow) | 0;
+      pow *= goog.math.Integer.TWO_PWR_32_DBL_;
+    }
+    return new goog.math.Integer(bits, 0);
+  }
+};
+
+
+/**
+ * Returns a Integer representing the value that comes by concatenating the
+ * given entries, each is assumed to be 32 signed bits, given in little-endian
+ * order (lowest order bits in the lowest index), and sign-extending the highest
+ * order 32-bit value.
+ * @param {Array<number>} bits The bits of the number, in 32-bit signed pieces,
+ *     in little-endian order.
+ * @return {!goog.math.Integer} The corresponding Integer value.
+ */
+goog.math.Integer.fromBits = function(bits) {
+  var high = bits[bits.length - 1];
+  return new goog.math.Integer(bits, high & (1 << 31) ? -1 : 0);
+};
+
+
+/**
+ * Returns an Integer representation of the given string, written using the
+ * given radix.
+ * @param {string} str The textual representation of the Integer.
+ * @param {number=} opt_radix The radix in which the text is written.
+ * @return {!goog.math.Integer} The corresponding Integer value.
+ */
+goog.math.Integer.fromString = function(str, opt_radix) {
+  if (str.length == 0) {
+    throw Error('number format error: empty string');
+  }
+
+  var radix = opt_radix || 10;
+  if (radix < 2 || 36 < radix) {
+    throw Error('radix out of range: ' + radix);
+  }
+
+  if (str.charAt(0) == '-') {
+    return goog.math.Integer.fromString(str.substring(1), radix).negate();
+  } else if (str.indexOf('-') >= 0) {
+    throw Error('number format error: interior "-" character');
+  }
+
+  // Do several (8) digits each time through the loop, so as to
+  // minimize the calls to the very expensive emulated div.
+  var radixToPower = goog.math.Integer.fromNumber(Math.pow(radix, 8));
+
+  var result = goog.math.Integer.ZERO;
+  for (var i = 0; i < str.length; i += 8) {
+    var size = Math.min(8, str.length - i);
+    var value = parseInt(str.substring(i, i + size), radix);
+    if (size < 8) {
+      var power = goog.math.Integer.fromNumber(Math.pow(radix, size));
+      result = result.multiply(power).add(goog.math.Integer.fromNumber(value));
+    } else {
+      result = result.multiply(radixToPower);
+      result = result.add(goog.math.Integer.fromNumber(value));
+    }
+  }
+  return result;
+};
+
+
+/**
+ * A number used repeatedly in calculations.  This must appear before the first
+ * call to the from* functions below.
+ * @type {number}
+ * @private
+ */
+goog.math.Integer.TWO_PWR_32_DBL_ = (1 << 16) * (1 << 16);
+
+
+/** @type {!goog.math.Integer} */
+goog.math.Integer.ZERO = goog.math.Integer.fromInt(0);
+
+
+/** @type {!goog.math.Integer} */
+goog.math.Integer.ONE = goog.math.Integer.fromInt(1);
+
+
+/**
+ * @type {!goog.math.Integer}
+ * @private
+ */
+goog.math.Integer.TWO_PWR_24_ = goog.math.Integer.fromInt(1 << 24);
+
+
+/**
+ * Returns the value, assuming it is a 32-bit integer.
+ * @return {number} The corresponding int value.
+ */
+goog.math.Integer.prototype.toInt = function() {
+  return this.bits_.length > 0 ? this.bits_[0] : this.sign_;
+};
+
+
+/** @return {number} The closest floating-point representation to this value. */
+goog.math.Integer.prototype.toNumber = function() {
+  if (this.isNegative()) {
+    return -this.negate().toNumber();
+  } else {
+    var val = 0;
+    var pow = 1;
+    for (var i = 0; i < this.bits_.length; i++) {
+      val += this.getBitsUnsigned(i) * pow;
+      pow *= goog.math.Integer.TWO_PWR_32_DBL_;
+    }
+    return val;
+  }
+};
+
+
+/**
+ * @param {number=} opt_radix The radix in which the text should be written.
+ * @return {string} The textual representation of this value.
+ * @override
+ */
+goog.math.Integer.prototype.toString = function(opt_radix) {
+  var radix = opt_radix || 10;
+  if (radix < 2 || 36 < radix) {
+    throw Error('radix out of range: ' + radix);
+  }
+
+  if (this.isZero()) {
+    return '0';
+  } else if (this.isNegative()) {
+    return '-' + this.negate().toString(radix);
+  }
+
+  // Do several (6) digits each time through the loop, so as to
+  // minimize the calls to the very expensive emulated div.
+  var radixToPower = goog.math.Integer.fromNumber(Math.pow(radix, 6));
+
+  var rem = this;
+  var result = '';
+  while (true) {
+    var remDiv = rem.divide(radixToPower);
+    // The right shifting fixes negative values in the case when
+    // intval >= 2^31; for more details see
+    // https://github.com/google/closure-library/pull/498
+    var intval = rem.subtract(remDiv.multiply(radixToPower)).toInt() >>> 0;
+    var digits = intval.toString(radix);
+
+    rem = remDiv;
+    if (rem.isZero()) {
+      return digits + result;
+    } else {
+      while (digits.length < 6) {
+        digits = '0' + digits;
+      }
+      result = '' + digits + result;
+    }
+  }
+};
+
+
+/**
+ * Returns the index-th 32-bit (signed) piece of the Integer according to
+ * little-endian order (i.e., index 0 contains the smallest bits).
+ * @param {number} index The index in question.
+ * @return {number} The requested 32-bits as a signed number.
+ */
+goog.math.Integer.prototype.getBits = function(index) {
+  if (index < 0) {
+    return 0;  // Allowing this simplifies bit shifting operations below...
+  } else if (index < this.bits_.length) {
+    return this.bits_[index];
+  } else {
+    return this.sign_;
+  }
+};
+
+
+/**
+ * Returns the index-th 32-bit piece as an unsigned number.
+ * @param {number} index The index in question.
+ * @return {number} The requested 32-bits as an unsigned number.
+ */
+goog.math.Integer.prototype.getBitsUnsigned = function(index) {
+  var val = this.getBits(index);
+  return val >= 0 ? val : goog.math.Integer.TWO_PWR_32_DBL_ + val;
+};
+
+
+/** @return {number} The sign bit of this number, -1 or 0. */
+goog.math.Integer.prototype.getSign = function() {
+  return this.sign_;
+};
+
+
+/** @return {boolean} Whether this value is zero. */
+goog.math.Integer.prototype.isZero = function() {
+  if (this.sign_ != 0) {
+    return false;
+  }
+  for (var i = 0; i < this.bits_.length; i++) {
+    if (this.bits_[i] != 0) {
+      return false;
+    }
+  }
+  return true;
+};
+
+
+/** @return {boolean} Whether this value is negative. */
+goog.math.Integer.prototype.isNegative = function() {
+  return this.sign_ == -1;
+};
+
+
+/** @return {boolean} Whether this value is odd. */
+goog.math.Integer.prototype.isOdd = function() {
+  return (this.bits_.length == 0) && (this.sign_ == -1) ||
+      (this.bits_.length > 0) && ((this.bits_[0] & 1) != 0);
+};
+
+
+/**
+ * @param {goog.math.Integer} other Integer to compare against.
+ * @return {boolean} Whether this Integer equals the other.
+ */
+goog.math.Integer.prototype.equals = function(other) {
+  if (this.sign_ != other.sign_) {
+    return false;
+  }
+  var len = Math.max(this.bits_.length, other.bits_.length);
+  for (var i = 0; i < len; i++) {
+    if (this.getBits(i) != other.getBits(i)) {
+      return false;
+    }
+  }
+  return true;
+};
+
+
+/**
+ * @param {goog.math.Integer} other Integer to compare against.
+ * @return {boolean} Whether this Integer does not equal the other.
+ */
+goog.math.Integer.prototype.notEquals = function(other) {
+  return !this.equals(other);
+};
+
+
+/**
+ * @param {goog.math.Integer} other Integer to compare against.
+ * @return {boolean} Whether this Integer is greater than the other.
+ */
+goog.math.Integer.prototype.greaterThan = function(other) {
+  return this.compare(other) > 0;
+};
+
+
+/**
+ * @param {goog.math.Integer} other Integer to compare against.
+ * @return {boolean} Whether this Integer is greater than or equal to the other.
+ */
+goog.math.Integer.prototype.greaterThanOrEqual = function(other) {
+  return this.compare(other) >= 0;
+};
+
+
+/**
+ * @param {goog.math.Integer} other Integer to compare against.
+ * @return {boolean} Whether this Integer is less than the other.
+ */
+goog.math.Integer.prototype.lessThan = function(other) {
+  return this.compare(other) < 0;
+};
+
+
+/**
+ * @param {goog.math.Integer} other Integer to compare against.
+ * @return {boolean} Whether this Integer is less than or equal to the other.
+ */
+goog.math.Integer.prototype.lessThanOrEqual = function(other) {
+  return this.compare(other) <= 0;
+};
+
+
+/**
+ * Compares this Integer with the given one.
+ * @param {goog.math.Integer} other Integer to compare against.
+ * @return {number} 0 if they are the same, 1 if the this is greater, and -1
+ *     if the given one is greater.
+ */
+goog.math.Integer.prototype.compare = function(other) {
+  var diff = this.subtract(other);
+  if (diff.isNegative()) {
+    return -1;
+  } else if (diff.isZero()) {
+    return 0;
+  } else {
+    return +1;
+  }
+};
+
+
+/**
+ * Returns an integer with only the first numBits bits of this value, sign
+ * extended from the final bit.
+ * @param {number} numBits The number of bits by which to shift.
+ * @return {!goog.math.Integer} The shorted integer value.
+ */
+goog.math.Integer.prototype.shorten = function(numBits) {
+  var arr_index = (numBits - 1) >> 5;
+  var bit_index = (numBits - 1) % 32;
+  var bits = [];
+  for (var i = 0; i < arr_index; i++) {
+    bits[i] = this.getBits(i);
+  }
+  var sigBits = bit_index == 31 ? 0xFFFFFFFF : (1 << (bit_index + 1)) - 1;
+  var val = this.getBits(arr_index) & sigBits;
+  if (val & (1 << bit_index)) {
+    val |= 0xFFFFFFFF - sigBits;
+    bits[arr_index] = val;
+    return new goog.math.Integer(bits, -1);
+  } else {
+    bits[arr_index] = val;
+    return new goog.math.Integer(bits, 0);
+  }
+};
+
+
+/** @return {!goog.math.Integer} The negation of this value. */
+goog.math.Integer.prototype.negate = function() {
+  return this.not().add(goog.math.Integer.ONE);
+};
+
+
+/**
+ * Returns the sum of this and the given Integer.
+ * @param {goog.math.Integer} other The Integer to add to this.
+ * @return {!goog.math.Integer} The Integer result.
+ */
+goog.math.Integer.prototype.add = function(other) {
+  var len = Math.max(this.bits_.length, other.bits_.length);
+  var arr = [];
+  var carry = 0;
+
+  for (var i = 0; i <= len; i++) {
+    var a1 = this.getBits(i) >>> 16;
+    var a0 = this.getBits(i) & 0xFFFF;
+
+    var b1 = other.getBits(i) >>> 16;
+    var b0 = other.getBits(i) & 0xFFFF;
+
+    var c0 = carry + a0 + b0;
+    var c1 = (c0 >>> 16) + a1 + b1;
+    carry = c1 >>> 16;
+    c0 &= 0xFFFF;
+    c1 &= 0xFFFF;
+    arr[i] = (c1 << 16) | c0;
+  }
+  return goog.math.Integer.fromBits(arr);
+};
+
+
+/**
+ * Returns the difference of this and the given Integer.
+ * @param {goog.math.Integer} other The Integer to subtract from this.
+ * @return {!goog.math.Integer} The Integer result.
+ */
+goog.math.Integer.prototype.subtract = function(other) {
+  return this.add(other.negate());
+};
+
+
+/**
+ * Returns the product of this and the given Integer.
+ * @param {goog.math.Integer} other The Integer to multiply against this.
+ * @return {!goog.math.Integer} The product of this and the other.
+ */
+goog.math.Integer.prototype.multiply = function(other) {
+  if (this.isZero()) {
+    return goog.math.Integer.ZERO;
+  } else if (other.isZero()) {
+    return goog.math.Integer.ZERO;
+  }
+
+  if (this.isNegative()) {
+    if (other.isNegative()) {
+      return this.negate().multiply(other.negate());
+    } else {
+      return this.negate().multiply(other).negate();
+    }
+  } else if (other.isNegative()) {
+    return this.multiply(other.negate()).negate();
+  }
+
+  // If both numbers are small, use float multiplication
+  if (this.lessThan(goog.math.Integer.TWO_PWR_24_) &&
+      other.lessThan(goog.math.Integer.TWO_PWR_24_)) {
+    return goog.math.Integer.fromNumber(this.toNumber() * other.toNumber());
+  }
+
+  // Fill in an array of 16-bit products.
+  var len = this.bits_.length + other.bits_.length;
+  var arr = [];
+  for (var i = 0; i < 2 * len; i++) {
+    arr[i] = 0;
+  }
+  for (var i = 0; i < this.bits_.length; i++) {
+    for (var j = 0; j < other.bits_.length; j++) {
+      var a1 = this.getBits(i) >>> 16;
+      var a0 = this.getBits(i) & 0xFFFF;
+
+      var b1 = other.getBits(j) >>> 16;
+      var b0 = other.getBits(j) & 0xFFFF;
+
+      arr[2 * i + 2 * j] += a0 * b0;
+      goog.math.Integer.carry16_(arr, 2 * i + 2 * j);
+      arr[2 * i + 2 * j + 1] += a1 * b0;
+      goog.math.Integer.carry16_(arr, 2 * i + 2 * j + 1);
+      arr[2 * i + 2 * j + 1] += a0 * b1;
+      goog.math.Integer.carry16_(arr, 2 * i + 2 * j + 1);
+      arr[2 * i + 2 * j + 2] += a1 * b1;
+      goog.math.Integer.carry16_(arr, 2 * i + 2 * j + 2);
+    }
+  }
+
+  // Combine the 16-bit values into 32-bit values.
+  for (var i = 0; i < len; i++) {
+    arr[i] = (arr[2 * i + 1] << 16) | arr[2 * i];
+  }
+  for (var i = len; i < 2 * len; i++) {
+    arr[i] = 0;
+  }
+  return new goog.math.Integer(arr, 0);
+};
+
+
+/**
+ * Carries any overflow from the given index into later entries.
+ * @param {Array<number>} bits Array of 16-bit values in little-endian order.
+ * @param {number} index The index in question.
+ * @private
+ */
+goog.math.Integer.carry16_ = function(bits, index) {
+  while ((bits[index] & 0xFFFF) != bits[index]) {
+    bits[index + 1] += bits[index] >>> 16;
+    bits[index] &= 0xFFFF;
+  }
+};
+
+
+/**
+ * Returns "this" Integer divided by the given one. Both "this" and the given
+ * Integer MUST be positive.
+ *
+ * This method is only needed for very large numbers (>10^308),
+ * for which the original division algorithm gets into an infinite
+ * loop (see https://github.com/google/closure-library/issues/500).
+ *
+ * The algorithm has some possible performance enhancements (or
+ * could be rewritten entirely), it's just an initial solution for
+ * the issue linked above.
+ *
+ * @param {!goog.math.Integer} other The Integer to divide "this" by.
+ * @return {!goog.math.Integer} "this" value divided by the given one.
+ * @private
+ */
+goog.math.Integer.prototype.slowDivide_ = function(other) {
+  if (this.isNegative() || other.isNegative()) {
+    throw Error('slowDivide_ only works with positive integers.');
+  }
+
+  var twoPower = goog.math.Integer.ONE;
+  var multiple = other;
+
+  // First we have to figure out what the highest bit of the result
+  // is, so we increase "twoPower" and "multiple" until "multiple"
+  // exceeds "this".
+  while (multiple.lessThanOrEqual(this)) {
+    twoPower = twoPower.shiftLeft(1);
+    multiple = multiple.shiftLeft(1);
+  }
+
+  // Rewind by one power of two, giving us the highest bit of the
+  // result.
+  var res = twoPower.shiftRight(1);
+  var total = multiple.shiftRight(1);
+
+  // Now we starting decreasing "multiple" and "twoPower" to find the
+  // rest of the bits of the result.
+  var total2;
+  multiple = multiple.shiftRight(2);
+  twoPower = twoPower.shiftRight(2);
+  while (!multiple.isZero()) {
+    // whenever we can add "multiple" to the total and not exceed
+    // "this", that means we've found a 1 bit. Else we've found a 0
+    // and don't need to add to the result.
+    total2 = total.add(multiple);
+    if (total2.lessThanOrEqual(this)) {
+      res = res.add(twoPower);
+      total = total2;
+    }
+    multiple = multiple.shiftRight(1);
+    twoPower = twoPower.shiftRight(1);
+  }
+  return res;
+};
+
+
+/**
+ * Returns this Integer divided by the given one.
+ * @param {!goog.math.Integer} other The Integer to divide this by.
+ * @return {!goog.math.Integer} This value divided by the given one.
+ */
+goog.math.Integer.prototype.divide = function(other) {
+  if (other.isZero()) {
+    throw Error('division by zero');
+  } else if (this.isZero()) {
+    return goog.math.Integer.ZERO;
+  }
+
+  if (this.isNegative()) {
+    if (other.isNegative()) {
+      return this.negate().divide(other.negate());
+    } else {
+      return this.negate().divide(other).negate();
+    }
+  } else if (other.isNegative()) {
+    return this.divide(other.negate()).negate();
+  }
+
+  // Have to degrade to slowDivide for Very Large Numbers, because
+  // they're out of range for the floating-point approximation
+  // technique used below.
+  if (this.bits_.length > 30) {
+    return this.slowDivide_(other);
+  }
+
+  // Repeat the following until the remainder is less than other:  find a
+  // floating-point that approximates remainder / other *from below*, add this
+  // into the result, and subtract it from the remainder.  It is critical that
+  // the approximate value is less than or equal to the real value so that the
+  // remainder never becomes negative.
+  var res = goog.math.Integer.ZERO;
+  var rem = this;
+  while (rem.greaterThanOrEqual(other)) {
+    // Approximate the result of division. This may be a little greater or
+    // smaller than the actual value.
+    var approx = Math.max(1, Math.floor(rem.toNumber() / other.toNumber()));
+
+    // We will tweak the approximate result by changing it in the 48-th digit or
+    // the smallest non-fractional digit, whichever is larger.
+    var log2 = Math.ceil(Math.log(approx) / Math.LN2);
+    var delta = (log2 <= 48) ? 1 : Math.pow(2, log2 - 48);
+
+    // Decrease the approximation until it is smaller than the remainder.  Note
+    // that if it is too large, the product overflows and is negative.
+    var approxRes = goog.math.Integer.fromNumber(approx);
+    var approxRem = approxRes.multiply(other);
+    while (approxRem.isNegative() || approxRem.greaterThan(rem)) {
+      approx -= delta;
+      approxRes = goog.math.Integer.fromNumber(approx);
+      approxRem = approxRes.multiply(other);
+    }
+
+    // We know the answer can't be zero... and actually, zero would cause
+    // infinite recursion since we would make no progress.
+    if (approxRes.isZero()) {
+      approxRes = goog.math.Integer.ONE;
+    }
+
+    res = res.add(approxRes);
+    rem = rem.subtract(approxRem);
+  }
+  return res;
+};
+
+
+/**
+ * Returns this Integer modulo the given one.
+ * @param {!goog.math.Integer} other The Integer by which to mod.
+ * @return {!goog.math.Integer} This value modulo the given one.
+ */
+goog.math.Integer.prototype.modulo = function(other) {
+  return this.subtract(this.divide(other).multiply(other));
+};
+
+
+/** @return {!goog.math.Integer} The bitwise-NOT of this value. */
+goog.math.Integer.prototype.not = function() {
+  var len = this.bits_.length;
+  var arr = [];
+  for (var i = 0; i < len; i++) {
+    arr[i] = ~this.bits_[i];
+  }
+  return new goog.math.Integer(arr, ~this.sign_);
+};
+
+
+/**
+ * Returns the bitwise-AND of this Integer and the given one.
+ * @param {goog.math.Integer} other The Integer to AND with this.
+ * @return {!goog.math.Integer} The bitwise-AND of this and the other.
+ */
+goog.math.Integer.prototype.and = function(other) {
+  var len = Math.max(this.bits_.length, other.bits_.length);
+  var arr = [];
+  for (var i = 0; i < len; i++) {
+    arr[i] = this.getBits(i) & other.getBits(i);
+  }
+  return new goog.math.Integer(arr, this.sign_ & other.sign_);
+};
+
+
+/**
+ * Returns the bitwise-OR of this Integer and the given one.
+ * @param {goog.math.Integer} other The Integer to OR with this.
+ * @return {!goog.math.Integer} The bitwise-OR of this and the other.
+ */
+goog.math.Integer.prototype.or = function(other) {
+  var len = Math.max(this.bits_.length, other.bits_.length);
+  var arr = [];
+  for (var i = 0; i < len; i++) {
+    arr[i] = this.getBits(i) | other.getBits(i);
+  }
+  return new goog.math.Integer(arr, this.sign_ | other.sign_);
+};
+
+
+/**
+ * Returns the bitwise-XOR of this Integer and the given one.
+ * @param {goog.math.Integer} other The Integer to XOR with this.
+ * @return {!goog.math.Integer} The bitwise-XOR of this and the other.
+ */
+goog.math.Integer.prototype.xor = function(other) {
+  var len = Math.max(this.bits_.length, other.bits_.length);
+  var arr = [];
+  for (var i = 0; i < len; i++) {
+    arr[i] = this.getBits(i) ^ other.getBits(i);
+  }
+  return new goog.math.Integer(arr, this.sign_ ^ other.sign_);
+};
+
+
+/**
+ * Returns this value with bits shifted to the left by the given amount.
+ * @param {number} numBits The number of bits by which to shift.
+ * @return {!goog.math.Integer} This shifted to the left by the given amount.
+ */
+goog.math.Integer.prototype.shiftLeft = function(numBits) {
+  var arr_delta = numBits >> 5;
+  var bit_delta = numBits % 32;
+  var len = this.bits_.length + arr_delta + (bit_delta > 0 ? 1 : 0);
+  var arr = [];
+  for (var i = 0; i < len; i++) {
+    if (bit_delta > 0) {
+      arr[i] = (this.getBits(i - arr_delta) << bit_delta) |
+          (this.getBits(i - arr_delta - 1) >>> (32 - bit_delta));
+    } else {
+      arr[i] = this.getBits(i - arr_delta);
+    }
+  }
+  return new goog.math.Integer(arr, this.sign_);
+};
+
+
+/**
+ * Returns this value with bits shifted to the right by the given amount.
+ * @param {number} numBits The number of bits by which to shift.
+ * @return {!goog.math.Integer} This shifted to the right by the given amount.
+ */
+goog.math.Integer.prototype.shiftRight = function(numBits) {
+  var arr_delta = numBits >> 5;
+  var bit_delta = numBits % 32;
+  var len = this.bits_.length - arr_delta;
+  var arr = [];
+  for (var i = 0; i < len; i++) {
+    if (bit_delta > 0) {
+      arr[i] = (this.getBits(i + arr_delta) >>> bit_delta) |
+          (this.getBits(i + arr_delta + 1) << (32 - bit_delta));
+    } else {
+      arr[i] = this.getBits(i + arr_delta);
+    }
+  }
+  return new goog.math.Integer(arr, this.sign_);
+};