diff --git a/src/share/classes/java/math/BigInteger.java b/src/share/classes/java/math/BigInteger.java
--- a/src/share/classes/java/math/BigInteger.java
+++ b/src/share/classes/java/math/BigInteger.java
@@ -93,6 +93,7 @@
* @see BigDecimal
* @author Josh Bloch
* @author Michael McCloskey
+ * @author Alan Eliasen
* @since JDK1.1
*/
@@ -173,6 +174,38 @@
*/
final static long LONG_MASK = 0xffffffffL;
+ /**
+ * The threshold value for using Karatsuba multiplication. If the number
+ * of ints in both mag arrays are greater than this number, then
+ * Karatsuba multiplication will be used. This value is found
+ * experimentally to work well.
+ */
+ private static final int KARATSUBA_THRESHOLD = 50;
+
+ /**
+ * The threshold value for using 3-way Toom-Cook multiplication.
+ * If the number of ints in both mag arrays are greater than this number,
+ * then Toom-Cook multiplication will be used. This value is found
+ * experimentally to work well.
+ */
+ private static final int TOOM_COOK_THRESHOLD = 75;
+
+ /**
+ * The threshold value for using Karatsuba squaring. If the number
+ * of ints in the number are larger than this value,
+ * Karatsuba squaring will be used. This value is found
+ * experimentally to work well.
+ */
+ private static final int KARATSUBA_SQUARE_THRESHOLD = 90;
+
+ /**
+ * The threshold value for using Toom-Cook squaring. If the number
+ * of ints in the number are larger than this value,
+ * Karatsuba squaring will be used. This value is found
+ * experimentally to work well.
+ */
+ private static final int TOOM_COOK_SQUARE_THRESHOLD = 140;
+
//Constructors
/**
@@ -533,7 +566,8 @@
if (bitLength < 2)
throw new ArithmeticException("bitLength < 2");
// The cutoff of 95 was chosen empirically for best performance
- prime = (bitLength < 95 ? smallPrime(bitLength, certainty, rnd)
+ prime = (bitLength < SMALL_PRIME_THRESHOLD
+ ? smallPrime(bitLength, certainty, rnd)
: largePrime(bitLength, certainty, rnd));
signum = 1;
mag = prime.mag;
@@ -1025,6 +1059,11 @@
private static final BigInteger TWO = valueOf(2);
/**
+ * The BigInteger constant -1. (Not exported.)
+ */
+ private static final BigInteger NEGATIVE_ONE = valueOf(-1);
+
+ /**
* The BigInteger constant ten.
*
* @since 1.5
@@ -1166,10 +1205,21 @@
if (val.signum == 0 || signum == 0)
return ZERO;
- int[] result = multiplyToLen(mag, mag.length,
- val.mag, val.mag.length, null);
- result = trustedStripLeadingZeroInts(result);
- return new BigInteger(result, signum == val.signum ? 1 : -1);
+ int xlen = mag.length;
+ int ylen = val.mag.length;
+
+ if ((xlen < KARATSUBA_THRESHOLD) || (ylen < KARATSUBA_THRESHOLD))
+ {
+ int[] result = multiplyToLen(mag, xlen,
+ val.mag, ylen, null);
+ result = trustedStripLeadingZeroInts(result);
+ return new BigInteger(result, signum == val.signum ? 1 : -1);
+ }
+ else
+ if ((xlen < TOOM_COOK_THRESHOLD) && (ylen < TOOM_COOK_THRESHOLD))
+ return multiplyKaratsuba(this, val);
+ else
+ return multiplyToomCook3(this, val);
}
/**
@@ -1249,6 +1299,280 @@
}
/**
+ * Multiplies two BigIntegers using the Karatsuba multiplication
+ * algorithm. This is a recursive divide-and-conquer algorithm which is
+ * more efficient for large numbers than what is commonly called the
+ * "grade-school" algorithm used in multiplyToLen. If the numbers to be
+ * multiplied have length n, the "grade-school" algorithm has an
+ * asymptotic complexity of O(n^2). In contrast, the Karatsuba algorithm
+ * has complexity of O(n^(log2(3))), or O(n^1.585). It achieves this
+ * increased performance by doing 3 multiplies instead of 4 when
+ * evaluating the product. As it has some overhead, should be used when
+ * both numbers are larger than a certain threshold (found
+ * experimentally).
+ *
+ */
+ private static BigInteger multiplyKaratsuba(BigInteger x, BigInteger y)
+ {
+ int xlen = x.mag.length;
+ int ylen = y.mag.length;
+
+ // The number of ints in each half of the number.
+ int half = (Math.max(xlen, ylen)+1) / 2;
+
+ // xl and yl are the lower halves of x and y respectively,
+ // xh and yh are the upper halves.
+ BigInteger xl = x.getLower(half);
+ BigInteger xh = x.getUpper(half);
+ BigInteger yl = y.getLower(half);
+ BigInteger yh = y.getUpper(half);
+
+ BigInteger p1 = xh.multiply(yh); // p1 = xh*yh
+ BigInteger p2 = xl.multiply(yl); // p2 = xl*yl
+
+ // The following is p3=(xh+xl)*(yh+yl)
+ BigInteger p3 = xh.add(xl).multiply(yh.add(yl));
+
+ // p1 * 2^(32*2*half) + (p3 - p1 - p2) * 2^(32*half) + p2
+ BigInteger result = p1.shiftLeft(32*half).add(p3.subtract(p1).subtract(p2)).shiftLeft(32*half).add(p2);
+
+ if (x.signum * y.signum == -1)
+ return result.negate();
+ else
+ return result;
+ }
+
+ /**
+ * Multiplies two BigIntegers using a 3-way Toom-Cook multiplication
+ * algorithm. This is a recursive divide-and-conquer algorithm which is
+ * more efficient for large numbers than what is commonly called the
+ * "grade-school" algorithm used in multiplyToLen. If the numbers to be
+ * multiplied have length n, the "grade-school" algorithm has an
+ * asymptotic complexity of O(n^2). In contrast, 3-way Toom-Cook has a
+ * complexity of about O(n^1.465). It achieves this increased asymptotic
+ * performance by breaking each number into three parts and by doing 5
+ * multiplies instead of 9 when evaluating the product. Due to overhead
+ * (additions, shifts, and one division) in the Toom-Cook algorithm, it
+ * should only be used when both numbers are larger than a certain
+ * threshold (found experimentally). This threshold is generally larger
+ * than that for Karatsuba multiplication, so this algorithm is generally
+ * only used when numbers become significantly larger.
+ *
+ * The algorithm used is the "optimal" 3-way Toom-Cook algorithm outlined
+ * by Marco Bodrato.
+ *
+ * See: http://bodrato.it/papers/#WAIFI2007
+ * http://bodrato.it/toom-cook/
+ *
+ * "Towards Optimal Toom-Cook Multiplication for Univariate and
+ * Multivariate Polynomials in Characteristic 2 and 0." by Marco BODRATO;
+ * In C.Carlet and B.Sunar, Eds., "WAIFI'07 proceedings", p. 116-133,
+ * LNCS #4547. Springer, Madrid, Spain, June 21-22, 2007.
+ *
+ */
+ private static BigInteger multiplyToomCook3(BigInteger a, BigInteger b)
+ {
+ int alen = a.mag.length;
+ int blen = b.mag.length;
+
+ int largest = Math.max(alen, blen);
+
+ // k is the size (in ints) of the lower-order slices.
+ int k = (largest+2)/3; // Equal to ceil(largest/3)
+
+ // r is the size (in ints) of the highest-order slice.
+ int r = largest - 2*k;
+
+ // Obtain slices of the numbers. a2 and b2 are the most significant
+ // bits of the number, and a0 and b0 the least significant.
+ BigInteger a0, a1, a2, b0, b1, b2;
+ a2 = a.getToomSlice(k, r, 0, largest);
+ a1 = a.getToomSlice(k, r, 1, largest);
+ a0 = a.getToomSlice(k, r, 2, largest);
+ b2 = b.getToomSlice(k, r, 0, largest);
+ b1 = b.getToomSlice(k, r, 1, largest);
+ b0 = b.getToomSlice(k, r, 2, largest);
+
+ BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1, db1;
+
+ v0 = a0.multiply(b0);
+ da1 = a2.add(a0);
+ db1 = b2.add(b0);
+ vm1 = da1.subtract(a1).multiply(db1.subtract(b1));
+ da1 = da1.add(a1);
+ db1 = db1.add(b1);
+ v1 = da1.multiply(db1);
+ v2 = da1.add(a2).shiftLeft(1).subtract(a0).multiply(
+ db1.add(b2).shiftLeft(1).subtract(b0));
+ vinf = a2.multiply(b2);
+
+ /* The algorithm requires two divisions by 2 and one by 3.
+ All divisions are known to be exact, that is, they do not produce
+ remainders, and all results are positive. The divisions by 2 are
+ implemented as right shifts which are relatively efficient, leaving
+ only an exact division by 3, which is done by a specialized
+ linear-time algorithm. */
+ t2 = v2.subtract(vm1).exactDivideBy3();
+ tm1 = v1.subtract(vm1).shiftRight(1);
+ t1 = v1.subtract(v0);
+ t2 = t2.subtract(t1).shiftRight(1);
+ t1 = t1.subtract(tm1).subtract(vinf);
+ t2 = t2.subtract(vinf.shiftLeft(1));
+ tm1 = tm1.subtract(t2);
+
+ // Number of bits to shift left.
+ int ss = k*32;
+
+ BigInteger result = vinf.shiftLeft(ss).add(t2).shiftLeft(ss).add(t1).shiftLeft(ss).add(tm1).shiftLeft(ss).add(v0);
+
+ if (a.signum * b.signum == -1)
+ return result.negate();
+ else
+ return result;
+ }
+
+
+ /** Returns a slice of a BigInteger for use in Toom-Cook multiplication.
+ @param lowerSize The size of the lower-order bit slices.
+ @param upperSize The size of the higher-order bit slices.
+ @param slice The index of which slice is requested, which must be a
+ number from 0 to size-1. Slice 0 is the highest-order bits,
+ and slice size-1 are the lowest-order bits.
+ Slice 0 may be of different size than the other slices. */
+ private BigInteger getToomSlice(int lowerSize, int upperSize, int slice, int fullsize)
+ {
+ int start, end, sliceSize, len, offset;
+
+ len = mag.length;
+ offset = fullsize - len;
+
+ if (slice == 0)
+ {
+ start = 0 - offset;
+ end = upperSize - 1 - offset;
+ }
+ else
+ {
+ start = upperSize + (slice-1)*lowerSize - offset;
+ end = start + lowerSize - 1;
+ }
+
+ if (start < 0)
+ start = 0;
+ if (end < 0)
+ return ZERO;
+
+ sliceSize = (end-start) + 1;
+
+ if (sliceSize <= 0)
+ return ZERO;
+ if (start==0 && sliceSize >= len)
+ return this.abs();
+
+ int intSlice[] = new int[sliceSize];
+ System.arraycopy(mag, start, intSlice, 0, sliceSize);
+
+ return new BigInteger(trustedStripLeadingZeroInts(intSlice), 1);
+ }
+
+ /** Does an exact division (that is, the remainder is known to be zero)
+ of the specified number by 3. This is used in Toom-Cook
+ multiplication. This is an efficient algorithm that runs in linear
+ time. If the argument is not exactly divisible by 3, results are
+ undefined. */
+ private BigInteger exactDivideBy3()
+ {
+ int len = mag.length;
+ int[] result = new int[len];
+ long w, q, carry;
+ carry = 0L;
+ for (int i=len-1; i>=0; i--)
+ {
+ w = (mag[i] & LONG_MASK) - carry;
+ carry = 0L;
+
+ // 0xAAAAAAAB is the modular inverse of 3 (mod 2^32). Thus,
+ // the effect of this is to divide by 3 (mod 2^32).
+ // This is much faster than division on most architectures.
+ q = (w * 0xAAAAAAABL) & LONG_MASK;
+ result[i] = (int) q;
+
+ // Now check the carry. The carry may be 0, 1, or 2 after these
+ // two checks. The second check can of course be eliminated if
+ // the first fails.
+ if (q >= 0x55555556L)
+ {
+ carry++;
+ if (q >= 0xAAAAAAABL)
+ carry++;
+ }
+ }
+ result = trustedStripLeadingZeroInts(result);
+ return new BigInteger(result, signum);
+ }
+
+ /** Returns a slice of a BigInteger for use in Karatsuba multiplication.
+ @param size The approximate size of each slice, in number of ints.
+ @param numSlices The number of pieces to slice the number into.
+ @param slice The index of which slice is requested, which must be a
+ number from 0 to size-1. Slice 0 is the highest-order bits,
+ and slice size-1 are the lowest-order bits.
+ Slice 0 may be of different size than the other slices. */
+ public BigInteger getSlice(int size, int numSlices, int slice)
+ {
+ int len = mag.length;
+ int end = (len-1) - (numSlices - slice - 1) * size;
+ int start = end - size + 1;
+ if (slice == 0 || start < 0)
+ start = 0; // Expand slice 0 to contain remaining bits
+
+ int slicelen = (end-start) + 1;
+ if (slicelen <= 0)
+ return ZERO;
+ if (start==0 && slicelen >= len)
+ return this;
+
+ int intSlice[] = new int[slicelen];
+ System.arraycopy(mag, start, intSlice, 0, slicelen);
+
+ return new BigInteger(trustedStripLeadingZeroInts(intSlice), 1);
+ }
+
+ /**
+ * Returns a new BigInteger representing n lower ints of the number.
+ * This is used by Karatsuba multiplication and squaring.
+ */
+ private BigInteger getLower(int n) {
+ int len = mag.length;
+
+ if (len <= n)
+ return this;
+
+ int lowerInts[] = new int[n];
+ System.arraycopy(mag, len-n, lowerInts, 0, n);
+
+ return new BigInteger(trustedStripLeadingZeroInts(lowerInts), 1);
+ }
+
+ /**
+ * Returns a new BigInteger representing mag.length-n upper
+ * ints of the number. This is used by Karatsuba multiplication and
+ * squaring.
+ */
+ private BigInteger getUpper(int n) {
+ int len = mag.length;
+
+ if (len <= n)
+ return ZERO;
+
+ int upperLen = len - n;
+ int upperInts[] = new int[upperLen];
+ System.arraycopy(mag, 0, upperInts, 0, upperLen);
+
+ return new BigInteger(trustedStripLeadingZeroInts(upperInts), 1);
+ }
+
+ /**
* Returns a BigInteger whose value is {@code (this2)}.
*
* @return {@code this2}
@@ -1256,8 +1580,18 @@
private BigInteger square() {
if (signum == 0)
return ZERO;
- int[] z = squareToLen(mag, mag.length, null);
- return new BigInteger(trustedStripLeadingZeroInts(z), 1);
+ int len = mag.length;
+
+ if (len < KARATSUBA_SQUARE_THRESHOLD)
+ {
+ int[] z = squareToLen(mag, len, null);
+ return new BigInteger(trustedStripLeadingZeroInts(z), 1);
+ }
+ else
+ if (len < TOOM_COOK_SQUARE_THRESHOLD)
+ return squareKaratsuba();
+ else
+ return squareToomCook3();
}
/**
@@ -1328,6 +1662,81 @@
}
/**
+ * Squares a BigInteger using the Karatsuba squaring algorithm. It should
+ * be used when both numbers are larger than a certain threshold (found
+ * experimentally). It is a recursive divide-and-conquer algorithm that
+ * has better asymptotic performance than the algorithm used in
+ * squareToLen.
+ */
+ private BigInteger squareKaratsuba()
+ {
+ int half = (mag.length+1) / 2;
+
+ BigInteger xl = getLower(half);
+ BigInteger xh = getUpper(half);
+
+ BigInteger xhs = xh.square(); // xhs = xh^2
+ BigInteger xls = xl.square(); // xls = xl^2
+
+ // xh^2 << 64 + (((xl+xh)^2 - (xh^2 + xl^2)) << 32) + xl^2
+ return xhs.shiftLeft(half*32).add(xl.add(xh).square().subtract(xhs.add(xls))).shiftLeft(half*32).add(xls);
+ }
+
+ /**
+ * Squares a BigInteger using the 3-way Toom-Cook squaring algorithm. It
+ * should be used when both numbers are larger than a certain threshold
+ * (found experimentally). It is a recursive divide-and-conquer algorithm
+ * that has better asymptotic performance than the algorithm used in
+ * squareToLen or squareKaratsuba.
+ */
+ private BigInteger squareToomCook3()
+ {
+ int len = mag.length;
+
+ // k is the size (in ints) of the lower-order slices.
+ int k = (len+2)/3; // Equal to ceil(largest/3)
+
+ // r is the size (in ints) of the highest-order slice.
+ int r = len - 2*k;
+
+ // Obtain slices of the numbers. a2 is the most significant
+ // bits of the number, and a0 the least significant.
+ BigInteger a0, a1, a2;
+ a2 = getToomSlice(k, r, 0, len);
+ a1 = getToomSlice(k, r, 1, len);
+ a0 = getToomSlice(k, r, 2, len);
+ BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1;
+
+ v0 = a0.square();
+ da1 = a2.add(a0);
+ vm1 = da1.subtract(a1).square();
+ da1 = da1.add(a1);
+ v1 = da1.square();
+ vinf = a2.square();
+ v2 = da1.add(a2).shiftLeft(1).subtract(a0).square();
+
+ /* The algorithm requires two divisions by 2 and one by 3.
+ All divisions are known to be exact, that is, they do not produce
+ remainders, and all results are positive. The divisions by 2 are
+ implemented as right shifts which are relatively efficient, leaving
+ only a division by 3.
+ The division by 3 is done by an optimized algorithm for this case.
+ */
+ t2 = v2.subtract(vm1).exactDivideBy3();
+ tm1 = v1.subtract(vm1).shiftRight(1);
+ t1 = v1.subtract(v0);
+ t2 = t2.subtract(t1).shiftRight(1);
+ t1 = t1.subtract(tm1).subtract(vinf);
+ t2 = t2.subtract(vinf.shiftLeft(1));
+ tm1 = tm1.subtract(t2);
+
+ // Number of bits to shift left.
+ int ss = k*32;
+
+ return vinf.shiftLeft(ss).add(t2).shiftLeft(ss).add(t1).shiftLeft(ss).add(tm1).shiftLeft(ss).add(v0);
+ }
+
+ /**
* Returns a BigInteger whose value is {@code (this / val)}.
*
* @param val value by which this BigInteger is to be divided.