Massive cleanup of the entire codebase. Notable changes include:

[bigint/bigint.git] / BigUnsigned.cc

#include "BigUnsigned.hh"

// Memory management definitions have moved to the bottom of NumberlikeArray.hh.

// CONSTRUCTION FROM PRIMITIVE INTEGERS

/* Initialize this BigUnsigned from the given primitive integer.  The same
 * pattern works for all primitive integer types, so I put it into a template to
 * reduce code duplication.  (Don't worry: this is protected and we instantiate
 * it only with primitive integer types.)  Type X could be signed, but x is
 * known to be nonnegative. */
template <class X>
void BigUnsigned::initFromPrimitive(X x) {
        if (x == 0)
                ; // NumberlikeArray already initialized us to zero.
        else {
                // Create a single block.  blk is NULL; no need to delete it.
                cap = 1;
                blk = new Blk[1];
                len = 1;
                blk[0] = Blk(x);
        }
}

/* Ditto, but first check that x is nonnegative.  I could have put the check in
 * initFromPrimitive and let the compiler optimize it out for unsigned-type
 * instantiations, but I wanted to avoid the warning stupidly issued by g++ for
 * a condition that is constant in *any* instantiation, even if not in all. */
template <class X>
void BigUnsigned::initFromSignedPrimitive(X x) {
        if (x < 0)
                throw "BigUnsigned constructor: "
                        "Cannot construct a BigUnsigned from a negative number";
        else
                initFromPrimitive(x);
}

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); }

// CONVERSION TO PRIMITIVE INTEGERS

/* Template with the same idea as initFromPrimitive.  This might be slightly
 * slower than the previous version with the masks, but it's much shorter and
 * clearer, which is the library's stated goal. */
template <class X>
X BigUnsigned::convertToPrimitive() const {
        if (len == 0)
                // The number is zero; return zero.
                return 0;
        else if (len == 1) {
                // The single block might fit in an X.  Try the conversion.
                X x = X(blk[0]);
                // Make sure the result accurately represents the block.
                if (Blk(x) == blk[0])
                        // Successful conversion.
                        return x;
                // Otherwise fall through.
        }
        throw "BigUnsigned::to<Primitive>: "
                "Value is too big to fit in the requested type";
}

/* Wrap the above in an x >= 0 test to make sure we got a nonnegative result,
 * not a negative one that happened to convert back into the correct nonnegative
 * one.  (E.g., catch incorrect conversion of 2^31 to the long -2^31.)  Again,
 * separated to avoid a g++ warning. */
template <class X>
X BigUnsigned::convertToSignedPrimitive() const {
        X x = convertToPrimitive<X>();
        if (x >= 0)
                return x;
        else
                throw "BigUnsigned::to(Primitive): "
                        "Value is too big to fit in the requested type";
}

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>();
}

// 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, unsigned int b) {
        DTRT_ALIASED(this == &a, bitShiftLeft(a, b));
        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, unsigned int b) {
        DTRT_ALIASED(this == &a, bitShiftRight(a, b));
        // 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 --();
}