#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 int BigUnsigned::toUnsignedInt () const { return convertToPrimitive (); }
unsigned short BigUnsigned::toUnsignedShort() const { return convertToPrimitive (); }
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 --();
}