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📁 Inside the java virtualMachine,深入研究java虚拟机
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<TITLE>Understanding Digital Signatures: Inside the Java Virtual Machine
 by Bill Venners - Beta Version</TITLE>
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<P><H1>Chapter Twelve</H1></P>
<P><H2>Integer Arithmetic</H2></P>
<P>This chapter describes integer arithmetic in the Java Virtual Machine. It explains two韘 complement arithmetic (the mechanism used by the Java Virtual Machine to perform integer arithmetic) and describes the instructions that perform integer arithmetic.</P>
<P>Accompanying this chapter on the CD-ROM are two applets that interactively illustrate the material presented in the chapter. One applet, named <I>Inner Int</I>, allows you to view and manipulate the individual components that make up a two韘 complement number. The other applet, named <I>Prime Time</I>, simulates the Java Virtual Machine executing a method that generates prime numbers. At the end of this chapter, you will find a description of this applet and the bytecodes it executes.</P>
<I><P>Two韘 Complement Arithmetic</P>
</I><P>All integer types supported by the Java Virtual Machine--<FONT FACE="Courier New">byte</FONT>s, <FONT FACE="Courier New">short</FONT>s, <FONT FACE="Courier New">int</FONT>s, and <FONT FACE="Courier New">long</FONT>s--are signed two's-complement numbers. The two's-complement scheme allows both positive and negative integers to be represented. The most significant bit of a two's-complement number is its sign bit. The sign bit is one for negative numbers and zero for positive numbers and for the number zero. </P>
<P>The number of unique values that can be represented by the two's-complement scheme is two raised to the power of the total number of bits. For example, the <FONT FACE="Courier New">short</FONT> type in Java is a 16-bit signed two's-complement integer. The number of unique integers that can be represented by this scheme is 2<SUP>16</SUP>, or 65,536. Half of the <FONT FACE="Courier New">short</FONT> type's range of values are used to represent zero and positive numbers; the other half of the <FONT FACE="Courier New">short</FONT> type's range are used to represent negative numbers. The range of negative values for a 16-bit two's-complement number is -32,768 (0x8000) to -1 (0xffff). Zero is 0x0000. The range of positive values is one (0x0001) to 32,767 (0x7fff). </P>
<P>Positive numbers are intuitive in that they are merely the base two representation of the number. Negative numbers can be calculated by adding the negative number to two raised to the power of the total number of bits. For example, the total number of bits in a <FONT FACE="Courier New">short</FONT> is 16, so the two's-complement representation of a negative number in the valid range for a <FONT FACE="Courier New">short</FONT> (-32,768 to -1) can be calculated by adding the negative number to 2<SUP>16</SUP>, or 65,536. The two's-complement representation for -1 is 65,536 + (-1) or 65,535 (0xffff). The two's-complement representation for -2 is 65,536 + (-2) or 65,534 (0xfffe). </P>
<P>Addition is performed on two's-complement signed numbers in the same way it would be performed on unsigned binary numbers. The two numbers are added, overflow is ignored, and the result is interpreted as a signed two's-complement number. This will work as long as the result is actually within the range of valid values for the type. For example, to add 4 + (-2), just add 0x00000004 and 0xfffffffe. The result is actually 0x100000002, but because there are only 32 bits in an <FONT FACE="Courier New">int</FONT>, the overflow is ignored and the result becomes 0x00000002. </P>
<P>Overflow in integer operations does not throw any exception in the Java Virtual Machine. The result is merely truncated to fit into the result type (either <FONT FACE="Courier New">int</FONT> or <FONT FACE="Courier New">long</FONT>). For example, adding <FONT FACE="Courier New">int</FONT>s 0x7fffffff and 1 yields 0x80000000. This means that the Java Virtual Machine will report that 2,147,483,647 + 1 = -2,147,483,648, if the type of the values being added are <FONT FACE="Courier New">int</FONT>s and not <FONT FACE="Courier New">long</FONT>s. As you program in Java, you must keep in mind that overflow can happen and make sure you choose the appropriate type, <FONT FACE="Courier New">int</FONT> or <FONT FACE="Courier New">long</FONT>, in each situation. Integer division by zero does throw an <FONT FACE="Courier New">ArithmeticException</FONT>, so you should also keep in mind that this exception could be thrown and catch it if necessary.</P>
<P>If you encounter a situation in which <FONT FACE="Courier New">long</FONT> just isn't long enough, you can use the <FONT FACE="Courier New">BigInteger</FONT> class of the <FONT FACE="Courier New">java.math</FONT> package. Instances of this class are arbitrary-length integers. The <FONT FACE="Courier New">BigInteger</FONT> class supports all arithmetic operations on arbitrary-length integers that are provided for the primitive types by the Java Virtual Machine and the <FONT FACE="Courier New">java.lang.Math</FONT> package.</P>
<I><P>Inner Int: A Java int Reveals its Inner Nature </P>
</I><P>The <I>Inner Int</I> applet, shown in Figure 12-1, lets you play around with the two's-complement format of integers in the Java Virtual Machine. The Max and Min buttons will give you the maximum and minimum values of the <FONT FACE="Courier New">int</FONT> type. By clicking Max followed by ++ you can increment beyond the maximum integer and see what happens. Clicking Min followed by -- lets you decrement beyond the minimum integer. Both of these result in overflow, but no exceptions are thrown by the Java Virtual Machine. The applet is embedded in a web page on the CD-ROM in file <FONT FACE="Courier New">applets/InnerInt.html</FONT>.</P>
<P><IMG SRC="fig12-1.gif" tppabs="http://www.pbg.mcgraw-hill.com/betabooks/venners/images/fig12-1.gif" ALT="Figure 12-1"></P>

<I><STRONG><P>Arithmetic Opcodes</P>
</I><P></STRONG>The Java Virtual Machine offers several opcodes that perform integer arithmetic operations on <FONT FACE="Courier New">int</FONT>s and <FONT FACE="Courier New">long</FONT>s. As mentioned in earlier chapters, values of type <FONT FACE="Courier New">byte</FONT>, <FONT FACE="Courier New">short</FONT>, and <FONT FACE="Courier New">char</FONT> are converted to <FONT FACE="Courier New">int</FONT> before they take part in arithmetic operations. For each opcode that performs arithmetic on <FONT FACE="Courier New">int</FONT>s, there is a corresponding opcode that performs the same operation on <FONT FACE="Courier New">long</FONT>s. </P>
<P>Integer addition can be performed on <FONT FACE="Courier New">int</FONT>s and <FONT FACE="Courier New">long</FONT>s. Table 12-1 shows the opcodes that pop the top two values on the stack, add them, and push the resulting sum. The values to be added must have been pushed onto the stack by previous instructions. The type of the values is indicated by the opcode itself, and the result always has the same type as the numbers being added. No exceptions are thrown for any of these opcodes. Overflow is just ignored.</P>
<P>Table 12-1. Integer addition</P>
<TABLE WIDTH="500">
<TR><TD VALIGN="TOP"><STRONG>Opcode</STRONG></TD><TD VALIGN="TOP"><STRONG>Operand(s)</STRONG></TD><TD VALIGN="TOP"><STRONG>Description</STRONG></TD></TR>
<TR><TD VALIGN="TOP"><FONT FACE="Courier New">iadd</FONT></TD><TD VALIGN="TOP">(none)</TD><TD VALIGN="TOP">pops two <FONT FACE="Courier New">int</FONT>s, adds them, and pushes the <FONT FACE="Courier New">int</FONT> result</TD></TR>
<TR><TD VALIGN="TOP"><FONT FACE="Courier New">ladd</FONT></TD><TD VALIGN="TOP">(none)</TD><TD VALIGN="TOP">pops two <FONT FACE="Courier New">long</FONT>s, adds them, and pushes the <FONT FACE="Courier New">long</FONT> result</TD></TR>
</TABLE>
<P>Table 12-2 shows the exception to the rule that arithmetic opcodes take their operands from the stack. The <FONT FACE="Courier New">iinc</FONT> opcode performs an addition on a local variable of type <FONT FACE="Courier New">int</FONT>. The local variable to increment is specified by the first byte that follows the <FONT FACE="Courier New">iinc</FONT> instruction in the bytecode stream. The amount to add to the local variable is taken from the second byte following the <FONT FACE="Courier New">iinc</FONT> instruction. The second byte is interpreted as an 8-bit signed two's-complement number. The local variable and 8-bit signed value are added, and the result is written back to the local variable. This opcode can be used to change a local variable value by any number between and including -128 through 127. This opcode makes for more efficient incrementing and decrementing of variables that are used to control execution of loops, such as <FONT FACE="Courier New">for</FONT> or <FONT FACE="Courier New">while</FONT>. As with the add instruction, no exceptions are thrown. Overflow is ignored.</P>
<P>The second row in Table 12-2 shows the <FONT FACE="Courier New">wide</FONT> variant of the <FONT FACE="Courier New">iinc</FONT> instruction. As mentioned in Chapter 10, &quot;Stack and Local Variable Operations,&quot; the <FONT FACE="Courier New">wide</FONT> instruction is used to extend unsigned local variable indexes from eight bits to sixteen. Sixteen bits allows instructions to address up to 65,536 local variable slots. In the <FONT FACE="Courier New">iinc</FONT> case, the <FONT FACE="Courier New">wide</FONT> instruction also extends the size of the signed increment value from eight bits to sixteen. This means the <FONT FACE="Courier New">wide</FONT> variant of the <FONT FACE="Courier New">iinc</FONT> opcode can be used to change a local variable value by any number between and including -32,768 through 32,767.</P>
<P>Table 12-2. Increment a local variable by a constant</P>
<TABLE WIDTH="500">
<TR><TD VALIGN="TOP"><STRONG>Opcode</STRONG></TD><TD VALIGN="TOP"><STRONG>Operand(s)</STRONG></TD><TD VALIGN="TOP"><STRONG>Description</STRONG></TD></TR>
<TR><TD VALIGN="TOP"><FONT FACE="Courier New">iinc</FONT></TD><TD VALIGN="TOP">vindex, const</TD><TD VALIGN="TOP">adds const to an <FONT FACE="Courier New">int</FONT> at local variable position vindex</TD></TR>
<TR><TD VALIGN="TOP"><FONT FACE="Courier New">wide</FONT></TD><TD VALIGN="TOP"><FONT FACE="Courier New">iinc</FONT>, indexbyte1, indexbyte2, constbyte1, constbyte2</TD><TD VALIGN="TOP">adds const to an <FONT FACE="Courier New">int</FONT> at local variable position index <FONT FACE="Courier New">end table</FONT></TD></TR>
</TABLE>
<P>Integer subtraction is performed on <FONT FACE="Courier New">int</FONT>s and <FONT FACE="Courier New">long</FONT>s via the opcodes shown in Table 12-3. Each opcode causes the top two values of the appropriate type to be popped off the stack. The topmost value is subtracted from the value just beneath it. The result is pushed back onto the stack. No exceptions are thrown by these opcodes.</P>
<P>Table 12-3. Integer subtraction</P>
<TABLE WIDTH="500">
<TR><TD VALIGN="TOP"><STRONG>Opcode</STRONG></TD><TD VALIGN="TOP"><STRONG>Operand(s)</STRONG></TD><TD VALIGN="TOP"><STRONG>Description</STRONG></TD></TR>
<TR><TD VALIGN="TOP"><FONT FACE="Courier New">isub</FONT></TD><TD VALIGN="TOP">(none)</TD><TD VALIGN="TOP">pops two <FONT FACE="Courier New">int</FONT>s, subtracts them, and pushes the <FONT FACE="Courier New">int</FONT> result</TD></TR>
<TR><TD VALIGN="TOP"><FONT FACE="Courier New">lsub</FONT></TD><TD VALIGN="TOP">(none)</TD><TD VALIGN="TOP">pops two <FONT FACE="Courier New">long</FONT>s, subtracts them, and pushes the <FONT FACE="Courier New">long</FONT> result</TD></TR>
</TABLE>
<P>Integer multiplication of <FONT FACE="Courier New">int</FONT>s and <FONT FACE="Courier New">long</FONT>s is accomplished via the opcodes shown in Table 12-4. Each opcode causes two values of the same type to be popped off the stack and multiplied. The result, of the same type as the numbers being multiplied, is pushed back onto the stack. No exceptions are thrown.</P>
<P>Table 12-4. Integer multiplication</P>
<TABLE WIDTH="500">
<TR><TD VALIGN="TOP"><STRONG>Opcode</STRONG></TD><TD VALIGN="TOP"><STRONG>Operand(s)</STRONG></TD><TD VALIGN="TOP"><STRONG>Description</STRONG></TD></TR>
<TR><TD VALIGN="TOP"><FONT FACE="Courier New">imul</FONT></TD><TD VALIGN="TOP">(none)</TD><TD VALIGN="TOP">pops two <FONT FACE="Courier New">int</FONT>s, multiplies them, and pushes the <FONT FACE="Courier New">int</FONT> result</TD></TR>
<TR><TD VALIGN="TOP"><FONT FACE="Courier New">lmul</FONT></TD><TD VALIGN="TOP">(none)</TD><TD VALIGN="TOP">pops two <FONT FACE="Courier New">long</FONT>s, multiplies them, and pushes the <FONT FACE="Courier New">long</FONT> result</TD></TR>
</TABLE>
<P>The opcodes that perform division on <FONT FACE="Courier New">int</FONT>s and <FONT FACE="Courier New">long</FONT>s are shown in Table 12-5. The division opcodes cause the top two values of the appropriate type to be popped off the stack. The value just beneath the topmost value is divided by the topmost value. (In other words, the first value pushed onto the stack is used as the dividend or numerator. The second value pushed--the top of the stack--is used as the divisor or denominator.) The result of the division is pushed back onto the stack. Integer division yields a result that is truncated down to the nearest integer value between it and zero. Integer division by zero throws an <FONT FACE="Courier New">ArithmeticException</FONT>.</P>
<P>Table 12-5. Integer division</P>
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