changeset 53140:620b31ed8807

8200659: Improve BigDecimal support Reviewed-by: darcy, rhalade, mschoene
author bpb
date Wed, 22 Aug 2018 15:55:04 -0700
parents 0060e9d7c450
children dff86e25073f
files src/java.base/share/classes/java/math/BigDecimal.java src/java.base/share/classes/java/math/BigInteger.java test/jdk/java/math/BigDecimal/AddTests.java test/jdk/java/math/BigDecimal/Constructor.java test/jdk/java/math/BigInteger/LargeValueExceptions.java
diffstat 5 files changed, 388 insertions(+), 44 deletions(-) [+]
line wrap: on
line diff
--- a/src/java.base/share/classes/java/math/BigDecimal.java	Sat Sep 29 10:08:42 2018 +0800
+++ b/src/java.base/share/classes/java/math/BigDecimal.java	Wed Aug 22 15:55:04 2018 -0700
@@ -1,5 +1,5 @@
 /*
- * Copyright (c) 1996, 2017, Oracle and/or its affiliates. All rights reserved.
+ * Copyright (c) 1996, 2018, Oracle and/or its affiliates. All rights reserved.
  * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
  *
  * This code is free software; you can redistribute it and/or modify it
@@ -31,6 +31,7 @@
 
 import static java.math.BigInteger.LONG_MASK;
 import java.util.Arrays;
+import java.util.Objects;
 
 /**
  * Immutable, arbitrary-precision signed decimal numbers.  A
@@ -424,9 +425,14 @@
      * @since  1.5
      */
     public BigDecimal(char[] in, int offset, int len, MathContext mc) {
-        // protect against huge length.
-        if (offset + len > in.length || offset < 0)
-            throw new NumberFormatException("Bad offset or len arguments for char[] input.");
+        // protect against huge length, negative values, and integer overflow
+        try {
+            Objects.checkFromIndexSize(offset, len, in.length);
+        } catch (IndexOutOfBoundsException e) {
+            throw new NumberFormatException
+                ("Bad offset or len arguments for char[] input.");
+        }
+
         // This is the primary string to BigDecimal constructor; all
         // incoming strings end up here; it uses explicit (inline)
         // parsing for speed and generates at most one intermediate
--- a/src/java.base/share/classes/java/math/BigInteger.java	Sat Sep 29 10:08:42 2018 +0800
+++ b/src/java.base/share/classes/java/math/BigInteger.java	Wed Aug 22 15:55:04 2018 -0700
@@ -307,10 +307,8 @@
     public BigInteger(byte[] val, int off, int len) {
         if (val.length == 0) {
             throw new NumberFormatException("Zero length BigInteger");
-        } else if ((off < 0) || (off >= val.length) || (len < 0) ||
-                   (len > val.length - off)) { // 0 <= off < val.length
-            throw new IndexOutOfBoundsException();
         }
+        Objects.checkFromIndexSize(off, len, val.length);
 
         if (val[off] < 0) {
             mag = makePositive(val, off, len);
@@ -395,12 +393,8 @@
     public BigInteger(int signum, byte[] magnitude, int off, int len) {
         if (signum < -1 || signum > 1) {
             throw(new NumberFormatException("Invalid signum value"));
-        } else if ((off < 0) || (len < 0) ||
-            (len > 0 &&
-                ((off >= magnitude.length) ||
-                 (len > magnitude.length - off)))) { // 0 <= off < magnitude.length
-            throw new IndexOutOfBoundsException();
         }
+        Objects.checkFromIndexSize(off, len, magnitude.length);
 
         // stripLeadingZeroBytes() returns a zero length array if len == 0
         this.mag = stripLeadingZeroBytes(magnitude, off, len);
@@ -1239,6 +1233,14 @@
     private static final double LOG_TWO = Math.log(2.0);
 
     static {
+        assert 0 < KARATSUBA_THRESHOLD
+            && KARATSUBA_THRESHOLD < TOOM_COOK_THRESHOLD
+            && TOOM_COOK_THRESHOLD < Integer.MAX_VALUE
+            && 0 < KARATSUBA_SQUARE_THRESHOLD
+            && KARATSUBA_SQUARE_THRESHOLD < TOOM_COOK_SQUARE_THRESHOLD
+            && TOOM_COOK_SQUARE_THRESHOLD < Integer.MAX_VALUE :
+            "Algorithm thresholds are inconsistent";
+
         for (int i = 1; i <= MAX_CONSTANT; i++) {
             int[] magnitude = new int[1];
             magnitude[0] = i;
@@ -1562,6 +1564,18 @@
      * @return {@code this * val}
      */
     public BigInteger multiply(BigInteger val) {
+        return multiply(val, false);
+    }
+
+    /**
+     * Returns a BigInteger whose value is {@code (this * val)}.  If
+     * the invocation is recursive certain overflow checks are skipped.
+     *
+     * @param  val value to be multiplied by this BigInteger.
+     * @param  isRecursion whether this is a recursive invocation
+     * @return {@code this * val}
+     */
+    private BigInteger multiply(BigInteger val, boolean isRecursion) {
         if (val.signum == 0 || signum == 0)
             return ZERO;
 
@@ -1589,6 +1603,63 @@
             if ((xlen < TOOM_COOK_THRESHOLD) && (ylen < TOOM_COOK_THRESHOLD)) {
                 return multiplyKaratsuba(this, val);
             } else {
+                //
+                // In "Hacker's Delight" section 2-13, p.33, it is explained
+                // that if x and y are unsigned 32-bit quantities and m and n
+                // are their respective numbers of leading zeros within 32 bits,
+                // then the number of leading zeros within their product as a
+                // 64-bit unsigned quantity is either m + n or m + n + 1. If
+                // their product is not to overflow, it cannot exceed 32 bits,
+                // and so the number of leading zeros of the product within 64
+                // bits must be at least 32, i.e., the leftmost set bit is at
+                // zero-relative position 31 or less.
+                //
+                // From the above there are three cases:
+                //
+                //     m + n    leftmost set bit    condition
+                //     -----    ----------------    ---------
+                //     >= 32    x <= 64 - 32 = 32   no overflow
+                //     == 31    x >= 64 - 32 = 32   possible overflow
+                //     <= 30    x >= 64 - 31 = 33   definite overflow
+                //
+                // The "possible overflow" condition cannot be detected by
+                // examning data lengths alone and requires further calculation.
+                //
+                // By analogy, if 'this' and 'val' have m and n as their
+                // respective numbers of leading zeros within 32*MAX_MAG_LENGTH
+                // bits, then:
+                //
+                //     m + n >= 32*MAX_MAG_LENGTH        no overflow
+                //     m + n == 32*MAX_MAG_LENGTH - 1    possible overflow
+                //     m + n <= 32*MAX_MAG_LENGTH - 2    definite overflow
+                //
+                // Note however that if the number of ints in the result
+                // were to be MAX_MAG_LENGTH and mag[0] < 0, then there would
+                // be overflow. As a result the leftmost bit (of mag[0]) cannot
+                // be used and the constraints must be adjusted by one bit to:
+                //
+                //     m + n >  32*MAX_MAG_LENGTH        no overflow
+                //     m + n == 32*MAX_MAG_LENGTH        possible overflow
+                //     m + n <  32*MAX_MAG_LENGTH        definite overflow
+                //
+                // The foregoing leading zero-based discussion is for clarity
+                // only. The actual calculations use the estimated bit length
+                // of the product as this is more natural to the internal
+                // array representation of the magnitude which has no leading
+                // zero elements.
+                //
+                if (!isRecursion) {
+                    // The bitLength() instance method is not used here as we
+                    // are only considering the magnitudes as non-negative. The
+                    // Toom-Cook multiplication algorithm determines the sign
+                    // at its end from the two signum values.
+                    if (bitLength(mag, mag.length) +
+                        bitLength(val.mag, val.mag.length) >
+                        32L*MAX_MAG_LENGTH) {
+                        reportOverflow();
+                    }
+                }
+
                 return multiplyToomCook3(this, val);
             }
         }
@@ -1674,7 +1745,7 @@
         int ystart = ylen - 1;
 
         if (z == null || z.length < (xlen+ ylen))
-            z = new int[xlen+ylen];
+             z = new int[xlen+ylen];
 
         long carry = 0;
         for (int j=ystart, k=ystart+1+xstart; j >= 0; j--, k--) {
@@ -1808,16 +1879,16 @@
 
         BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1, db1;
 
-        v0 = a0.multiply(b0);
+        v0 = a0.multiply(b0, true);
         da1 = a2.add(a0);
         db1 = b2.add(b0);
-        vm1 = da1.subtract(a1).multiply(db1.subtract(b1));
+        vm1 = da1.subtract(a1).multiply(db1.subtract(b1), true);
         da1 = da1.add(a1);
         db1 = db1.add(b1);
-        v1 = da1.multiply(db1);
+        v1 = da1.multiply(db1, true);
         v2 = da1.add(a2).shiftLeft(1).subtract(a0).multiply(
-             db1.add(b2).shiftLeft(1).subtract(b0));
-        vinf = a2.multiply(b2);
+             db1.add(b2).shiftLeft(1).subtract(b0), true);
+        vinf = a2.multiply(b2, true);
 
         // The algorithm requires two divisions by 2 and one by 3.
         // All divisions are known to be exact, that is, they do not produce
@@ -1983,6 +2054,17 @@
      * @return {@code this<sup>2</sup>}
      */
     private BigInteger square() {
+        return square(false);
+    }
+
+    /**
+     * Returns a BigInteger whose value is {@code (this<sup>2</sup>)}. If
+     * the invocation is recursive certain overflow checks are skipped.
+     *
+     * @param isRecursion whether this is a recursive invocation
+     * @return {@code this<sup>2</sup>}
+     */
+    private BigInteger square(boolean isRecursion) {
         if (signum == 0) {
             return ZERO;
         }
@@ -1995,6 +2077,15 @@
             if (len < TOOM_COOK_SQUARE_THRESHOLD) {
                 return squareKaratsuba();
             } else {
+                //
+                // For a discussion of overflow detection see multiply()
+                //
+                if (!isRecursion) {
+                    if (bitLength(mag, mag.length) > 16L*MAX_MAG_LENGTH) {
+                        reportOverflow();
+                    }
+                }
+
                 return squareToomCook3();
             }
         }
@@ -2146,13 +2237,13 @@
         a0 = getToomSlice(k, r, 2, len);
         BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1;
 
-        v0 = a0.square();
+        v0 = a0.square(true);
         da1 = a2.add(a0);
-        vm1 = da1.subtract(a1).square();
+        vm1 = da1.subtract(a1).square(true);
         da1 = da1.add(a1);
-        v1 = da1.square();
-        vinf = a2.square();
-        v2 = da1.add(a2).shiftLeft(1).subtract(a0).square();
+        v1 = da1.square(true);
+        vinf = a2.square(true);
+        v2 = da1.add(a2).shiftLeft(1).subtract(a0).square(true);
 
         // The algorithm requires two divisions by 2 and one by 3.
         // All divisions are known to be exact, that is, they do not produce
@@ -2323,10 +2414,11 @@
         // The remaining part can then be exponentiated faster.  The
         // powers of two will be multiplied back at the end.
         int powersOfTwo = partToSquare.getLowestSetBit();
-        long bitsToShift = (long)powersOfTwo * exponent;
-        if (bitsToShift > Integer.MAX_VALUE) {
+        long bitsToShiftLong = (long)powersOfTwo * exponent;
+        if (bitsToShiftLong > Integer.MAX_VALUE) {
             reportOverflow();
         }
+        int bitsToShift = (int)bitsToShiftLong;
 
         int remainingBits;
 
@@ -2336,9 +2428,9 @@
             remainingBits = partToSquare.bitLength();
             if (remainingBits == 1) {  // Nothing left but +/- 1?
                 if (signum < 0 && (exponent&1) == 1) {
-                    return NEGATIVE_ONE.shiftLeft(powersOfTwo*exponent);
+                    return NEGATIVE_ONE.shiftLeft(bitsToShift);
                 } else {
-                    return ONE.shiftLeft(powersOfTwo*exponent);
+                    return ONE.shiftLeft(bitsToShift);
                 }
             }
         } else {
@@ -2383,13 +2475,16 @@
                 if (bitsToShift + scaleFactor <= 62) { // Fits in long?
                     return valueOf((result << bitsToShift) * newSign);
                 } else {
-                    return valueOf(result*newSign).shiftLeft((int) bitsToShift);
+                    return valueOf(result*newSign).shiftLeft(bitsToShift);
                 }
-            }
-            else {
+            } else {
                 return valueOf(result*newSign);
             }
         } else {
+            if ((long)bitLength() * exponent / Integer.SIZE > MAX_MAG_LENGTH) {
+                reportOverflow();
+            }
+
             // Large number algorithm.  This is basically identical to
             // the algorithm above, but calls multiply() and square()
             // which may use more efficient algorithms for large numbers.
@@ -2409,7 +2504,7 @@
             // Multiply back the (exponentiated) powers of two (quickly,
             // by shifting left)
             if (powersOfTwo > 0) {
-                answer = answer.shiftLeft(powersOfTwo*exponent);
+                answer = answer.shiftLeft(bitsToShift);
             }
 
             if (signum < 0 && (exponent&1) == 1) {
@@ -3584,7 +3679,7 @@
                      for (int i=1; i< len && pow2; i++)
                          pow2 = (mag[i] == 0);
 
-                     n = (pow2 ? magBitLength -1 : magBitLength);
+                     n = (pow2 ? magBitLength - 1 : magBitLength);
                  } else {
                      n = magBitLength;
                  }
--- a/test/jdk/java/math/BigDecimal/AddTests.java	Sat Sep 29 10:08:42 2018 +0800
+++ b/test/jdk/java/math/BigDecimal/AddTests.java	Wed Aug 22 15:55:04 2018 -0700
@@ -1,5 +1,5 @@
 /*
- * Copyright (c) 2006, Oracle and/or its affiliates. All rights reserved.
+ * Copyright (c) 2006, 2018, Oracle and/or its affiliates. All rights reserved.
  * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
  *
  * This code is free software; you can redistribute it and/or modify it
@@ -23,7 +23,7 @@
 
 /*
  * @test
- * @bug 6362557
+ * @bug 6362557 8200698
  * @summary Some tests of add(BigDecimal, mc)
  * @author Joseph D. Darcy
  */
@@ -290,12 +290,35 @@
         return failures;
     }
 
+    private static int arithmeticExceptionTest() {
+        int failures = 0;
+        BigDecimal x;
+        try {
+            //
+            // The string representation "1e2147483647", which is equivalent
+            // to 10^Integer.MAX_VALUE, is used to create an augend with an
+            // unscaled value of 1 and a scale of -Integer.MAX_VALUE. The
+            // addend "1" has an unscaled value of 1 with a scale of 0. The
+            // addition is performed exactly and is specified to have a
+            // preferred scale of max(-Integer.MAX_VALUE, 0). As the scale
+            // of the result is 0, a value with Integer.MAX_VALUE + 1 digits
+            // would need to be created. Therefore the next statement is
+            // expected to overflow with an ArithmeticException.
+            //
+            x = new BigDecimal("1e2147483647").add(new BigDecimal(1));
+            failures++;
+        } catch (ArithmeticException ae) {
+        }
+        return failures;
+    }
+
     public static void main(String argv[]) {
         int failures = 0;
 
         failures += extremaTests();
         failures += roundingGradationTests();
         failures += precisionConsistencyTest();
+        failures += arithmeticExceptionTest();
 
         if (failures > 0) {
             throw new RuntimeException("Incurred " + failures +
--- a/test/jdk/java/math/BigDecimal/Constructor.java	Sat Sep 29 10:08:42 2018 +0800
+++ b/test/jdk/java/math/BigDecimal/Constructor.java	Wed Aug 22 15:55:04 2018 -0700
@@ -1,5 +1,5 @@
 /*
- * Copyright (c) 1999, Oracle and/or its affiliates. All rights reserved.
+ * Copyright (c) 1999, 2018, Oracle and/or its affiliates. All rights reserved.
  * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
  *
  * This code is free software; you can redistribute it and/or modify it
@@ -23,20 +23,48 @@
 
 /*
  * @test
- * @bug 4259453
- * @summary Test string constructor of BigDecimal
+ * @bug 4259453 8200698
+ * @summary Test constructors of BigDecimal
+ * @library ..
+ * @run testng Constructor
  */
+
 import java.math.BigDecimal;
+import org.testng.annotations.Test;
 
 public class Constructor {
-    public static void main(String[] args) throws Exception {
-        boolean nfe = false;
+    @Test(expectedExceptions=NumberFormatException.class)
+    public void stringConstructor() {
+        BigDecimal bd = new BigDecimal("1.2e");
+    }
+
+    @Test(expectedExceptions=NumberFormatException.class)
+    public void charArrayConstructorNegativeOffset() {
+        BigDecimal bd = new BigDecimal(new char[5], -1, 4, null);
+    }
+
+    @Test(expectedExceptions=NumberFormatException.class)
+    public void charArrayConstructorNegativeLength() {
+        BigDecimal bd = new BigDecimal(new char[5], 0, -1, null);
+    }
+
+    @Test(expectedExceptions=NumberFormatException.class)
+    public void charArrayConstructorIntegerOverflow() {
         try {
-            BigDecimal bd = new BigDecimal("1.2e");
-        } catch (NumberFormatException e) {
-            nfe = true;
+            BigDecimal bd = new BigDecimal(new char[5], Integer.MAX_VALUE - 5,
+                6, null);
+        } catch (NumberFormatException nfe) {
+            if (nfe.getCause() instanceof IndexOutOfBoundsException) {
+                throw new RuntimeException
+                    ("NumberFormatException should not have a cause");
+            } else {
+                throw nfe;
+            }
         }
-        if (!nfe)
-            throw new Exception("Didn't throw NumberFormatException");
+    }
+
+    @Test(expectedExceptions=NumberFormatException.class)
+    public void charArrayConstructorIndexOutOfBounds() {
+        BigDecimal bd = new BigDecimal(new char[5], 1, 5, null);
     }
 }
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/test/jdk/java/math/BigInteger/LargeValueExceptions.java	Wed Aug 22 15:55:04 2018 -0700
@@ -0,0 +1,192 @@
+/*
+ * Copyright (c) 2018, Oracle and/or its affiliates. All rights reserved.
+ * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
+ *
+ * This code is free software; you can redistribute it and/or modify it
+ * under the terms of the GNU General Public License version 2 only, as
+ * published by the Free Software Foundation.
+ *
+ * This code is distributed in the hope that it will be useful, but WITHOUT
+ * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
+ * FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
+ * version 2 for more details (a copy is included in the LICENSE file that
+ * accompanied this code).
+ *
+ * You should have received a copy of the GNU General Public License version
+ * 2 along with this work; if not, write to the Free Software Foundation,
+ * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
+ *
+ * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
+ * or visit www.oracle.com if you need additional information or have any
+ * questions.
+ */
+
+/*
+ * @test
+ * @bug 8200698
+ * @summary Tests that exceptions are thrown for ops which would overflow
+ * @requires os.maxMemory >= 4g
+ * @run testng/othervm -Xmx4g LargeValueExceptions
+ */
+import java.math.BigInteger;
+import static java.math.BigInteger.ONE;
+import org.testng.annotations.Test;
+
+//
+// The intent of this test is to probe the boundaries between overflow and
+// non-overflow, principally for multiplication and squaring, specifically
+// the largest values which should not overflow and the smallest values which
+// should. The transition values used are not necessarily at the exact
+// boundaries but should be "close." Quite a few different values were used
+// experimentally before settling on the ones in this test. For multiplication
+// and squaring all cases are exercised: definite overflow and non-overflow
+// which can be detected "up front," and "indefinite" overflow, i.e., overflow
+// which cannot be detected up front so further calculations are required.
+//
+// Testing negative values is unnecessary. For both multiplication and squaring
+// the paths lead to the Toom-Cook algorithm where the signum is used only to
+// determine the sign of the result and not in the intermediate calculations.
+// This is also true for exponentiation.
+//
+// @Test annotations with optional element "enabled" set to "false" should
+// succeed when "enabled" is set to "true" but they take too to run in the
+// course of the typical regression test execution scenario.
+//
+public class LargeValueExceptions {
+    // BigInteger.MAX_MAG_LENGTH
+    private static final int MAX_INTS = 1 << 26;
+
+    // Number of bits corresponding to MAX_INTS
+    private static final long MAX_BITS = (0xffffffffL & MAX_INTS) << 5L;
+
+    // Half BigInteger.MAX_MAG_LENGTH
+    private static final int MAX_INTS_HALF = MAX_INTS / 2;
+
+    // --- squaring ---
+
+    // Largest no overflow determined by examining data lengths alone.
+    @Test(enabled=false)
+    public void squareNoOverflow() {
+        BigInteger x = ONE.shiftLeft(16*MAX_INTS - 1).subtract(ONE);
+        BigInteger y = x.multiply(x);
+    }
+
+    // Smallest no overflow determined by extra calculations.
+    @Test(enabled=false)
+    public void squareIndefiniteOverflowSuccess() {
+        BigInteger x = ONE.shiftLeft(16*MAX_INTS - 1);
+        BigInteger y = x.multiply(x);
+    }
+
+    // Largest overflow detected by extra calculations.
+    @Test(expectedExceptions=ArithmeticException.class,enabled=false)
+    public void squareIndefiniteOverflowFailure() {
+        BigInteger x = ONE.shiftLeft(16*MAX_INTS).subtract(ONE);
+        BigInteger y = x.multiply(x);
+    }
+
+    // Smallest overflow detected by examining data lengths alone.
+    @Test(expectedExceptions=ArithmeticException.class)
+    public void squareDefiniteOverflow() {
+        BigInteger x = ONE.shiftLeft(16*MAX_INTS);
+        BigInteger y = x.multiply(x);
+    }
+
+    // --- multiplication ---
+
+    // Largest no overflow determined by examining data lengths alone.
+    @Test(enabled=false)
+    public void multiplyNoOverflow() {
+        final int halfMaxBits = MAX_INTS_HALF << 5;
+
+        BigInteger x = ONE.shiftLeft(halfMaxBits).subtract(ONE);
+        BigInteger y = ONE.shiftLeft(halfMaxBits - 1).subtract(ONE);
+        BigInteger z = x.multiply(y);
+    }
+
+    // Smallest no overflow determined by extra calculations.
+    @Test(enabled=false)
+    public void multiplyIndefiniteOverflowSuccess() {
+        BigInteger x = ONE.shiftLeft((int)(MAX_BITS/2) - 1);
+        long m = MAX_BITS - x.bitLength();
+
+        BigInteger y = ONE.shiftLeft((int)(MAX_BITS/2) - 1);
+        long n = MAX_BITS - y.bitLength();
+
+        if (m + n != MAX_BITS) {
+            throw new RuntimeException("Unexpected leading zero sum");
+        }
+
+        BigInteger z = x.multiply(y);
+    }
+
+    // Largest overflow detected by extra calculations.
+    @Test(expectedExceptions=ArithmeticException.class,enabled=false)
+    public void multiplyIndefiniteOverflowFailure() {
+        BigInteger x = ONE.shiftLeft((int)(MAX_BITS/2)).subtract(ONE);
+        long m = MAX_BITS - x.bitLength();
+
+        BigInteger y = ONE.shiftLeft((int)(MAX_BITS/2)).subtract(ONE);
+        long n = MAX_BITS - y.bitLength();
+
+        if (m + n != MAX_BITS) {
+            throw new RuntimeException("Unexpected leading zero sum");
+        }
+
+        BigInteger z = x.multiply(y);
+    }
+
+    // Smallest overflow detected by examining data lengths alone.
+    @Test(expectedExceptions=ArithmeticException.class)
+    public void multiplyDefiniteOverflow() {
+        // multiply by 4 as MAX_INTS_HALF refers to ints
+        byte[] xmag = new byte[4*MAX_INTS_HALF];
+        xmag[0] = (byte)0xff;
+        BigInteger x = new BigInteger(1, xmag);
+
+        byte[] ymag = new byte[4*MAX_INTS_HALF + 1];
+        ymag[0] = (byte)0xff;
+        BigInteger y = new BigInteger(1, ymag);
+
+        BigInteger z = x.multiply(y);
+    }
+
+    // --- exponentiation ---
+
+    @Test(expectedExceptions=ArithmeticException.class)
+    public void powOverflow() {
+        BigInteger.TEN.pow(Integer.MAX_VALUE);
+    }
+
+    @Test(expectedExceptions=ArithmeticException.class)
+    public void powOverflow1() {
+        int shift = 20;
+        int exponent = 1 << shift;
+        BigInteger x = ONE.shiftLeft((int)(MAX_BITS / exponent));
+        BigInteger y = x.pow(exponent);
+    }
+
+    @Test(expectedExceptions=ArithmeticException.class)
+    public void powOverflow2() {
+        int shift = 20;
+        int exponent = 1 << shift;
+        BigInteger x = ONE.shiftLeft((int)(MAX_BITS / exponent)).add(ONE);
+        BigInteger y = x.pow(exponent);
+    }
+
+    @Test(expectedExceptions=ArithmeticException.class,enabled=false)
+    public void powOverflow3() {
+        int shift = 20;
+        int exponent = 1 << shift;
+        BigInteger x = ONE.shiftLeft((int)(MAX_BITS / exponent)).subtract(ONE);
+        BigInteger y = x.pow(exponent);
+    }
+
+    @Test(enabled=false)
+    public void powOverflow4() {
+        int shift = 20;
+        int exponent = 1 << shift;
+        BigInteger x = ONE.shiftLeft((int)(MAX_BITS / exponent - 1)).add(ONE);
+        BigInteger y = x.pow(exponent);
+    }
+}