Initial check in of big integer library

third_party/bigint/BigUnsigned.cc

+#include "BigUnsigned.hh"
+
+// Memory management definitions have moved to the bottom of NumberlikeArray.hh.
+
+// The templates used by these constructors and converters are at the bottom of
+// BigUnsigned.hh.
+
+BigUnsigned::BigUnsigned(unsigned long  x) { initFromPrimitive      (x); }
+BigUnsigned::BigUnsigned(unsigned int   x) { initFromPrimitive      (x); }
+BigUnsigned::BigUnsigned(unsigned short x) { initFromPrimitive      (x); }
+BigUnsigned::BigUnsigned(         long  x) { initFromSignedPrimitive(x); }
+BigUnsigned::BigUnsigned(         int   x) { initFromSignedPrimitive(x); }
+BigUnsigned::BigUnsigned(         short x) { initFromSignedPrimitive(x); }
+
+unsigned long  BigUnsigned::toUnsignedLong () const { return convertToPrimitive      <unsigned long >(); }
+unsigned int   BigUnsigned::toUnsignedInt  () const { return convertToPrimitive      <unsigned int  >(); }
+unsigned short BigUnsigned::toUnsignedShort() const { return convertToPrimitive      <unsigned short>(); }
+long           BigUnsigned::toLong         () const { return convertToSignedPrimitive<         long >(); }
+int            BigUnsigned::toInt          () const { return convertToSignedPrimitive<         int  >(); }
+short          BigUnsigned::toShort        () const { return convertToSignedPrimitive<         short>(); }
+
+// BIT/BLOCK ACCESSORS
+
+void BigUnsigned::setBlock(Index i, Blk newBlock) {
+        if (newBlock == 0) {
+                if (i < len) {
+                        blk[i] = 0;
+                        zapLeadingZeros();
+                }
+                // If i >= len, no effect.
+        } else {
+                if (i >= len) {
+                        // The nonzero block extends the number.
+                        allocateAndCopy(i+1);
+                        // Zero any added blocks that we aren't setting.
+                        for (Index j = len; j < i; j++)
+                                blk[j] = 0;
+                        len = i+1;
+                }
+                blk[i] = newBlock;
+        }
+}
+
+/* Evidently the compiler wants BigUnsigned:: on the return type because, at
+ * that point, it hasn't yet parsed the BigUnsigned:: on the name to get the
+ * proper scope. */
+BigUnsigned::Index BigUnsigned::bitLength() const {
+        if (isZero())
+                return 0;
+        else {
+                Blk leftmostBlock = getBlock(len - 1);
+                Index leftmostBlockLen = 0;
+                while (leftmostBlock != 0) {
+                        leftmostBlock >>= 1;
+                        leftmostBlockLen++;
+                }
+                return leftmostBlockLen + (len - 1) * N;
+        }
+}
+
+void BigUnsigned::setBit(Index bi, bool newBit) {
+        Index blockI = bi / N;
+        Blk block = getBlock(blockI), mask = Blk(1) << (bi % N);
+        block = newBit ? (block | mask) : (block & ~mask);
+        setBlock(blockI, block);
+}
+
+// COMPARISON
+BigUnsigned::CmpRes BigUnsigned::compareTo(const BigUnsigned &x) const {
+        // A bigger length implies a bigger number.
+        if (len < x.len)
+                return less;
+        else if (len > x.len)
+                return greater;
+        else {
+                // Compare blocks one by one from left to right.
+                Index i = len;
+                while (i > 0) {
+                        i--;
+                        if (blk[i] == x.blk[i])
+                                continue;
+                        else if (blk[i] > x.blk[i])
+                                return greater;
+                        else
+                                return less;
+                }
+                // If no blocks differed, the numbers are equal.
+                return equal;
+        }
+}
+
+// COPY-LESS OPERATIONS
+
+/*
+ * On most calls to copy-less operations, it's safe to read the inputs little by
+ * little and write the outputs little by little.  However, if one of the
+ * inputs is coming from the same variable into which the output is to be
+ * stored (an "aliased" call), we risk overwriting the input before we read it.
+ * In this case, we first compute the result into a temporary BigUnsigned
+ * variable and then copy it into the requested output variable *this.
+ * Each put-here operation uses the DTRT_ALIASED macro (Do The Right Thing on
+ * aliased calls) to generate code for this check.
+ * 
+ * I adopted this approach on 2007.02.13 (see Assignment Operators in
+ * BigUnsigned.hh).  Before then, put-here operations rejected aliased calls
+ * with an exception.  I think doing the right thing is better.
+ * 
+ * Some of the put-here operations can probably handle aliased calls safely
+ * without the extra copy because (for example) they process blocks strictly
+ * right-to-left.  At some point I might determine which ones don't need the
+ * copy, but my reasoning would need to be verified very carefully.  For now
+ * I'll leave in the copy.
+ */
+#define DTRT_ALIASED(cond, op) \
+        if (cond) { \
+                BigUnsigned tmpThis; \
+                tmpThis.op; \
+                *this = tmpThis; \
+                return; \
+        }
+
+
+
+void BigUnsigned::add(const BigUnsigned &a, const BigUnsigned &b) {
+        DTRT_ALIASED(this == &a || this == &b, add(a, b));
+        // If one argument is zero, copy the other.
+        if (a.len == 0) {
+                operator =(b);
+                return;
+        } else if (b.len == 0) {
+                operator =(a);
+                return;
+        }
+        // Some variables...
+        // Carries in and out of an addition stage
+        bool carryIn, carryOut;
+        Blk temp;
+        Index i;
+        // a2 points to the longer input, b2 points to the shorter
+        const BigUnsigned *a2, *b2;
+        if (a.len >= b.len) {
+                a2 = &a;
+                b2 = &b;
+        } else {
+                a2 = &b;
+                b2 = &a;
+        }
+        // Set prelimiary length and make room in this BigUnsigned
+        len = a2->len + 1;
+        allocate(len);
+        // For each block index that is present in both inputs...
+        for (i = 0, carryIn = false; i < b2->len; i++) {
+                // Add input blocks
+                temp = a2->blk[i] + b2->blk[i];
+                // If a rollover occurred, the result is less than either input.
+                // This test is used many times in the BigUnsigned code.
+                carryOut = (temp < a2->blk[i]);
+                // If a carry was input, handle it
+                if (carryIn) {
+                        temp++;
+                        carryOut |= (temp == 0);
+                }
+                blk[i] = temp; // Save the addition result
+                carryIn = carryOut; // Pass the carry along
+        }
+        // If there is a carry left over, increase blocks until
+        // one does not roll over.
+        for (; i < a2->len && carryIn; i++) {
+                temp = a2->blk[i] + 1;
+                carryIn = (temp == 0);
+                blk[i] = temp;
+        }
+        // If the carry was resolved but the larger number
+        // still has blocks, copy them over.
+        for (; i < a2->len; i++)
+                blk[i] = a2->blk[i];
+        // Set the extra block if there's still a carry, decrease length otherwise
+        if (carryIn)
+                blk[i] = 1;
+        else
+                len--;
+}
+
+void BigUnsigned::subtract(const BigUnsigned &a, const BigUnsigned &b) {
+        DTRT_ALIASED(this == &a || this == &b, subtract(a, b));
+        if (b.len == 0) {
+                // If b is zero, copy a.
+                operator =(a);
+                return;
+        } else if (a.len < b.len)
+                // If a is shorter than b, the result is negative.
+                throw "BigUnsigned::subtract: "
+                        "Negative result in unsigned calculation";
+        // Some variables...
+        bool borrowIn, borrowOut;
+        Blk temp;
+        Index i;
+        // Set preliminary length and make room
+        len = a.len;
+        allocate(len);
+        // For each block index that is present in both inputs...
+        for (i = 0, borrowIn = false; i < b.len; i++) {
+                temp = a.blk[i] - b.blk[i];
+                // If a reverse rollover occurred,
+                // the result is greater than the block from a.
+                borrowOut = (temp > a.blk[i]);
+                // Handle an incoming borrow
+                if (borrowIn) {
+                        borrowOut |= (temp == 0);
+                        temp--;
+                }
+                blk[i] = temp; // Save the subtraction result
+                borrowIn = borrowOut; // Pass the borrow along
+        }
+        // If there is a borrow left over, decrease blocks until
+        // one does not reverse rollover.
+        for (; i < a.len && borrowIn; i++) {
+                borrowIn = (a.blk[i] == 0);
+                blk[i] = a.blk[i] - 1;
+        }
+        /* If there's still a borrow, the result is negative.
+         * Throw an exception, but zero out this object so as to leave it in a
+         * predictable state. */
+        if (borrowIn) {
+                len = 0;
+                throw "BigUnsigned::subtract: Negative result in unsigned calculation";
+        } else
+                // Copy over the rest of the blocks
+                for (; i < a.len; i++)
+                        blk[i] = a.blk[i];
+        // Zap leading zeros
+        zapLeadingZeros();
+}
+
+/*
+ * About the multiplication and division algorithms:
+ *
+ * I searched unsucessfully for fast C++ built-in operations like the `b_0'
+ * and `c_0' Knuth describes in Section 4.3.1 of ``The Art of Computer
+ * Programming'' (replace `place' by `Blk'):
+ *
+ *    ``b_0[:] multiplication of a one-place integer by another one-place
+ *      integer, giving a two-place answer;
+ *
+ *    ``c_0[:] division of a two-place integer by a one-place integer,
+ *      provided that the quotient is a one-place integer, and yielding
+ *      also a one-place remainder.''
+ *
+ * I also missed his note that ``[b]y adjusting the word size, if
+ * necessary, nearly all computers will have these three operations
+ * available'', so I gave up on trying to use algorithms similar to his.
+ * A future version of the library might include such algorithms; I
+ * would welcome contributions from others for this.
+ *
+ * I eventually decided to use bit-shifting algorithms.  To multiply `a'
+ * and `b', we zero out the result.  Then, for each `1' bit in `a', we
+ * shift `b' left the appropriate amount and add it to the result.
+ * Similarly, to divide `a' by `b', we shift `b' left varying amounts,
+ * repeatedly trying to subtract it from `a'.  When we succeed, we note
+ * the fact by setting a bit in the quotient.  While these algorithms
+ * have the same O(n^2) time complexity as Knuth's, the ``constant factor''
+ * is likely to be larger.
+ *
+ * Because I used these algorithms, which require single-block addition
+ * and subtraction rather than single-block multiplication and division,
+ * the innermost loops of all four routines are very similar.  Study one
+ * of them and all will become clear.
+ */
+
+/*
+ * This is a little inline function used by both the multiplication
+ * routine and the division routine.
+ *
+ * `getShiftedBlock' returns the `x'th block of `num << y'.
+ * `y' may be anything from 0 to N - 1, and `x' may be anything from
+ * 0 to `num.len'.
+ *
+ * Two things contribute to this block:
+ *
+ * (1) The `N - y' low bits of `num.blk[x]', shifted `y' bits left.
+ *
+ * (2) The `y' high bits of `num.blk[x-1]', shifted `N - y' bits right.
+ *
+ * But we must be careful if `x == 0' or `x == num.len', in
+ * which case we should use 0 instead of (2) or (1), respectively.
+ *
+ * If `y == 0', then (2) contributes 0, as it should.  However,
+ * in some computer environments, for a reason I cannot understand,
+ * `a >> b' means `a >> (b % N)'.  This means `num.blk[x-1] >> (N - y)'
+ * will return `num.blk[x-1]' instead of the desired 0 when `y == 0';
+ * the test `y == 0' handles this case specially.
+ */
+inline BigUnsigned::Blk getShiftedBlock(const BigUnsigned &num,
+        BigUnsigned::Index x, unsigned int y) {
+        BigUnsigned::Blk part1 = (x == 0 || y == 0) ? 0 : (num.blk[x - 1] >> (BigUnsigned::N - y));
+        BigUnsigned::Blk part2 = (x == num.len) ? 0 : (num.blk[x] << y);
+        return part1 | part2;
+}
+
+void BigUnsigned::multiply(const BigUnsigned &a, const BigUnsigned &b) {
+        DTRT_ALIASED(this == &a || this == &b, multiply(a, b));
+        // If either a or b is zero, set to zero.
+        if (a.len == 0 || b.len == 0) {
+                len = 0;
+                return;
+        }
+        /*
+         * Overall method:
+         *
+         * Set this = 0.
+         * For each 1-bit of `a' (say the `i2'th bit of block `i'):
+         *    Add `b << (i blocks and i2 bits)' to *this.
+         */
+        // Variables for the calculation
+        Index i, j, k;
+        unsigned int i2;
+        Blk temp;
+        bool carryIn, carryOut;
+        // Set preliminary length and make room
+        len = a.len + b.len;
+        allocate(len);
+        // Zero out this object
+        for (i = 0; i < len; i++)
+                blk[i] = 0;
+        // For each block of the first number...
+        for (i = 0; i < a.len; i++) {
+                // For each 1-bit of that block...
+                for (i2 = 0; i2 < N; i2++) {
+                        if ((a.blk[i] & (Blk(1) << i2)) == 0)
+                                continue;
+                        /*
+                         * Add b to this, shifted left i blocks and i2 bits.
+                         * j is the index in b, and k = i + j is the index in this.
+                         *
+                         * `getShiftedBlock', a short inline function defined above,
+                         * is now used for the bit handling.  It replaces the more
+                         * complex `bHigh' code, in which each run of the loop dealt
+                         * immediately with the low bits and saved the high bits to
+                         * be picked up next time.  The last run of the loop used to
+                         * leave leftover high bits, which were handled separately.
+                         * Instead, this loop runs an additional time with j == b.len.
+                         * These changes were made on 2005.01.11.
+                         */
+                        for (j = 0, k = i, carryIn = false; j <= b.len; j++, k++) {
+                                /*
+                                 * The body of this loop is very similar to the body of the first loop
+                                 * in `add', except that this loop does a `+=' instead of a `+'.
+                                 */
+                                temp = blk[k] + getShiftedBlock(b, j, i2);
+                                carryOut = (temp < blk[k]);
+                                if (carryIn) {
+                                        temp++;
+                                        carryOut |= (temp == 0);
+                                }
+                                blk[k] = temp;
+                                carryIn = carryOut;
+                        }
+                        // No more extra iteration to deal with `bHigh'.
+                        // Roll-over a carry as necessary.
+                        for (; carryIn; k++) {
+                                blk[k]++;
+                                carryIn = (blk[k] == 0);
+                        }
+                }
+        }
+        // Zap possible leading zero
+        if (blk[len - 1] == 0)
+                len--;
+}
+
+/*
+ * DIVISION WITH REMAINDER
+ * This monstrous function mods *this by the given divisor b while storing the
+ * quotient in the given object q; at the end, *this contains the remainder.
+ * The seemingly bizarre pattern of inputs and outputs was chosen so that the
+ * function copies as little as possible (since it is implemented by repeated
+ * subtraction of multiples of b from *this).
+ * 
+ * "modWithQuotient" might be a better name for this function, but I would
+ * rather not change the name now.
+ */
+void BigUnsigned::divideWithRemainder(const BigUnsigned &b, BigUnsigned &q) {
+        /* Defending against aliased calls is more complex than usual because we
+         * are writing to both *this and q.
+         * 
+         * It would be silly to try to write quotient and remainder to the
+         * same variable.  Rule that out right away. */
+        if (this == &q)
+                throw "BigUnsigned::divideWithRemainder: Cannot write quotient and remainder into the same variable";
+        /* Now *this and q are separate, so the only concern is that b might be
+         * aliased to one of them.  If so, use a temporary copy of b. */
+        if (this == &b || &q == &b) {
+                BigUnsigned tmpB(b);
+                divideWithRemainder(tmpB, q);
+                return;
+        }
+
+        /*
+         * Knuth's definition of mod (which this function uses) is somewhat
+         * different from the C++ definition of % in case of division by 0.
+         *
+         * We let a / 0 == 0 (it doesn't matter much) and a % 0 == a, no
+         * exceptions thrown.  This allows us to preserve both Knuth's demand
+         * that a mod 0 == a and the useful property that
+         * (a / b) * b + (a % b) == a.
+         */
+        if (b.len == 0) {
+                q.len = 0;
+                return;
+        }
+
+        /*
+         * If *this.len < b.len, then *this < b, and we can be sure that b doesn't go into
+         * *this at all.  The quotient is 0 and *this is already the remainder (so leave it alone).
+         */
+        if (len < b.len) {
+                q.len = 0;
+                return;
+        }
+
+        // At this point we know (*this).len >= b.len > 0.  (Whew!)
+
+        /*
+         * Overall method:
+         *
+         * For each appropriate i and i2, decreasing:
+         *    Subtract (b << (i blocks and i2 bits)) from *this, storing the
+         *      result in subtractBuf.
+         *    If the subtraction succeeds with a nonnegative result:
+         *        Turn on bit i2 of block i of the quotient q.
+         *        Copy subtractBuf back into *this.
+         *    Otherwise bit i2 of block i remains off, and *this is unchanged.
+         * 
+         * Eventually q will contain the entire quotient, and *this will
+         * be left with the remainder.
+         *
+         * subtractBuf[x] corresponds to blk[x], not blk[x+i], since 2005.01.11.
+         * But on a single iteration, we don't touch the i lowest blocks of blk
+         * (and don't use those of subtractBuf) because these blocks are
+         * unaffected by the subtraction: we are subtracting
+         * (b << (i blocks and i2 bits)), which ends in at least `i' zero
+         * blocks. */
+        // Variables for the calculation
+        Index i, j, k;
+        unsigned int i2;
+        Blk temp;
+        bool borrowIn, borrowOut;
+
+        /*
+         * Make sure we have an extra zero block just past the value.
+         *
+         * When we attempt a subtraction, we might shift `b' so
+         * its first block begins a few bits left of the dividend,
+         * and then we'll try to compare these extra bits with
+         * a nonexistent block to the left of the dividend.  The
+         * extra zero block ensures sensible behavior; we need
+         * an extra block in `subtractBuf' for exactly the same reason.
+         */
+        Index origLen = len; // Save real length.
+        /* To avoid an out-of-bounds access in case of reallocation, allocate
+         * first and then increment the logical length. */
+        allocateAndCopy(len + 1);
+        len++;
+        blk[origLen] = 0; // Zero the added block.
+
+        // subtractBuf holds part of the result of a subtraction; see above.
+        Blk *subtractBuf = new Blk[len];
+
+        // Set preliminary length for quotient and make room
+        q.len = origLen - b.len + 1;
+        q.allocate(q.len);
+        // Zero out the quotient
+        for (i = 0; i < q.len; i++)
+                q.blk[i] = 0;
+
+        // For each possible left-shift of b in blocks...
+        i = q.len;
+        while (i > 0) {
+                i--;
+                // For each possible left-shift of b in bits...
+                // (Remember, N is the number of bits in a Blk.)
+                q.blk[i] = 0;
+                i2 = N;
+                while (i2 > 0) {
+                        i2--;
+                        /*
+                         * Subtract b, shifted left i blocks and i2 bits, from *this,
+                         * and store the answer in subtractBuf.  In the for loop, `k == i + j'.
+                         *
+                         * Compare this to the middle section of `multiply'.  They
+                         * are in many ways analogous.  See especially the discussion
+                         * of `getShiftedBlock'.
+                         */
+                        for (j = 0, k = i, borrowIn = false; j <= b.len; j++, k++) {
+                                temp = blk[k] - getShiftedBlock(b, j, i2);
+                                borrowOut = (temp > blk[k]);
+                                if (borrowIn) {
+                                        borrowOut |= (temp == 0);
+                                        temp--;
+                                }
+                                // Since 2005.01.11, indices of `subtractBuf' directly match those of `blk', so use `k'.
+                                subtractBuf[k] = temp; 
+                                borrowIn = borrowOut;
+                        }
+                        // No more extra iteration to deal with `bHigh'.
+                        // Roll-over a borrow as necessary.
+                        for (; k < origLen && borrowIn; k++) {
+                                borrowIn = (blk[k] == 0);
+                                subtractBuf[k] = blk[k] - 1;
+                        }
+                        /*
+                         * If the subtraction was performed successfully (!borrowIn),
+                         * set bit i2 in block i of the quotient.
+                         *
+                         * Then, copy the portion of subtractBuf filled by the subtraction
+                         * back to *this.  This portion starts with block i and ends--
+                         * where?  Not necessarily at block `i + b.len'!  Well, we
+                         * increased k every time we saved a block into subtractBuf, so
+                         * the region of subtractBuf we copy is just [i, k).
+                         */
+                        if (!borrowIn) {
+                                q.blk[i] |= (Blk(1) << i2);
+                                while (k > i) {
+                                        k--;
+                                        blk[k] = subtractBuf[k];
+                                }
+                        } 
+                }
+        }
+        // Zap possible leading zero in quotient
+        if (q.blk[q.len - 1] == 0)
+                q.len--;
+        // Zap any/all leading zeros in remainder
+        zapLeadingZeros();
+        // Deallocate subtractBuf.
+        // (Thanks to Brad Spencer for noticing my accidental omission of this!)
+        delete [] subtractBuf;
+}
+
+/* BITWISE OPERATORS
+ * These are straightforward blockwise operations except that they differ in
+ * the output length and the necessity of zapLeadingZeros. */
+
+void BigUnsigned::bitAnd(const BigUnsigned &a, const BigUnsigned &b) {
+        DTRT_ALIASED(this == &a || this == &b, bitAnd(a, b));
+        // The bitwise & can't be longer than either operand.
+        len = (a.len >= b.len) ? b.len : a.len;
+        allocate(len);
+        Index i;
+        for (i = 0; i < len; i++)
+                blk[i] = a.blk[i] & b.blk[i];
+        zapLeadingZeros();
+}
+
+void BigUnsigned::bitOr(const BigUnsigned &a, const BigUnsigned &b) {
+        DTRT_ALIASED(this == &a || this == &b, bitOr(a, b));
+        Index i;
+        const BigUnsigned *a2, *b2;
+        if (a.len >= b.len) {
+                a2 = &a;
+                b2 = &b;
+        } else {
+                a2 = &b;
+                b2 = &a;
+        }
+        allocate(a2->len);
+        for (i = 0; i < b2->len; i++)
+                blk[i] = a2->blk[i] | b2->blk[i];
+        for (; i < a2->len; i++)
+                blk[i] = a2->blk[i];
+        len = a2->len;
+        // Doesn't need zapLeadingZeros.
+}
+
+void BigUnsigned::bitXor(const BigUnsigned &a, const BigUnsigned &b) {
+        DTRT_ALIASED(this == &a || this == &b, bitXor(a, b));
+        Index i;
+        const BigUnsigned *a2, *b2;
+        if (a.len >= b.len) {
+                a2 = &a;
+                b2 = &b;
+        } else {
+                a2 = &b;
+                b2 = &a;
+        }
+        allocate(a2->len);
+        for (i = 0; i < b2->len; i++)
+                blk[i] = a2->blk[i] ^ b2->blk[i];
+        for (; i < a2->len; i++)
+                blk[i] = a2->blk[i];
+        len = a2->len;
+        zapLeadingZeros();
+}
+
+void BigUnsigned::bitShiftLeft(const BigUnsigned &a, int b) {
+        DTRT_ALIASED(this == &a, bitShiftLeft(a, b));
+        if (b < 0) {
+                if (b << 1 == 0)
+                        throw "BigUnsigned::bitShiftLeft: "
+                                "Pathological shift amount not implemented";
+                else {
+                        bitShiftRight(a, -b);
+                        return;
+                }
+        }
+        Index shiftBlocks = b / N;
+        unsigned int shiftBits = b % N;
+        // + 1: room for high bits nudged left into another block
+        len = a.len + shiftBlocks + 1;
+        allocate(len);
+        Index i, j;
+        for (i = 0; i < shiftBlocks; i++)
+                blk[i] = 0;
+        for (j = 0, i = shiftBlocks; j <= a.len; j++, i++)
+                blk[i] = getShiftedBlock(a, j, shiftBits);
+        // Zap possible leading zero
+        if (blk[len - 1] == 0)
+                len--;
+}
+
+void BigUnsigned::bitShiftRight(const BigUnsigned &a, int b) {
+        DTRT_ALIASED(this == &a, bitShiftRight(a, b));
+        if (b < 0) {
+                if (b << 1 == 0)
+                        throw "BigUnsigned::bitShiftRight: "
+                                "Pathological shift amount not implemented";
+                else {
+                        bitShiftLeft(a, -b);
+                        return;
+                }
+        }
+        // This calculation is wacky, but expressing the shift as a left bit shift
+        // within each block lets us use getShiftedBlock.
+        Index rightShiftBlocks = (b + N - 1) / N;
+        unsigned int leftShiftBits = N * rightShiftBlocks - b;
+        // Now (N * rightShiftBlocks - leftShiftBits) == b
+        // and 0 <= leftShiftBits < N.
+        if (rightShiftBlocks >= a.len + 1) {
+                // All of a is guaranteed to be shifted off, even considering the left
+                // bit shift.
+                len = 0;
+                return;
+        }
+        // Now we're allocating a positive amount.
+        // + 1: room for high bits nudged left into another block
+        len = a.len + 1 - rightShiftBlocks;
+        allocate(len);
+        Index i, j;
+        for (j = rightShiftBlocks, i = 0; j <= a.len; j++, i++)
+                blk[i] = getShiftedBlock(a, j, leftShiftBits);
+        // Zap possible leading zero
+        if (blk[len - 1] == 0)
+                len--;
+}
+
+// INCREMENT/DECREMENT OPERATORS
+
+// Prefix increment
+void BigUnsigned::operator ++() {
+        Index i;
+        bool carry = true;
+        for (i = 0; i < len && carry; i++) {
+                blk[i]++;
+                carry = (blk[i] == 0);
+        }
+        if (carry) {
+                // Allocate and then increase length, as in divideWithRemainder
+                allocateAndCopy(len + 1);
+                len++;
+                blk[i] = 1;
+        }
+}
+
+// Postfix increment: same as prefix
+void BigUnsigned::operator ++(int) {
+        operator ++();
+}
+
+// Prefix decrement
+void BigUnsigned::operator --() {
+        if (len == 0)
+                throw "BigUnsigned::operator --(): Cannot decrement an unsigned zero";
+        Index i;
+        bool borrow = true;
+        for (i = 0; borrow; i++) {
+                borrow = (blk[i] == 0);
+                blk[i]--;
+        }
+        // Zap possible leading zero (there can only be one)
+        if (blk[len - 1] == 0)
+                len--;
+}
+
+// Postfix decrement: same as prefix
+void BigUnsigned::operator --(int) {
+        operator --();
+}