Programs must not allow mathematical operations to exceed the integer ranges provided by their primitive integer data types. According to The Java Language Specification (JLS), §4.2.2, "Integer Operations" [JLS 2015]:
The builtin integer operators do not indicate overflow or underflow in any way. Integer operators can throw a
NullPointerException
if unboxing conversion of anull
reference is required. Other than that, the only integer operators that can throw an exception are the integer divide operator/
and the integer remainder operator%
, which throw anArithmeticException
if the righthand operand is zero, and the increment and decrement operators ++ and  which can throw anOutOfMemoryError
if boxing conversion is required and there is insufficient memory to perform the conversion.
The integral types in Java, representation, and inclusive ranges are shown in the following table taken from the JLS, §4.2.1, "Integral Types and Values" [JLS 2015]:
Type  Representation  Inclusive Range 

 8bit signed two'scomplement  −128 to 127 
 16bit signed two'scomplement  −32,768 to 32,767 
 32bit signed two'scomplement  −2,147,483,648 to 2,147,483,647 
 64bit signed two'scomplement  −9,223,372,036,854,775,808 to 9,223,372,036,854,775,807 
 16bit unsigned integers representing UTF16 code units 

The following table shows the integer overflow behavior of the integral operators.
Operator  Overflow 
 Operator  Overflow 
 Operator  Overflow 
 Operator  Overflow 

 Yes 

 Yes 

 No 

 No 
 Yes 

 Yes 

 No 

 No 
 Yes 

 Yes 

 No 

 No 
 Yes 

 No 

 No 

 No 
 No 

 No 

 No 

 No 
 Yes 

 No 

 No 

 No 
 Yes 

 No 

 No  
 No 

 No 
 Unary  No  
 Yes 

 No 
 Unary  Yes 
Because the ranges of Java types are not symmetric (the negation of each minimum value is one more than each maximum value), even operations such as unary negation can overflow if applied to a minimum value. Because the java.lang.math.abs()
method returns the absolute value of any number, it can also overflow if given the minimum int
or long
as an argument.
When a mathematical operation cannot be represented using the supplied integer types, Java's builtin integer operators silently wrap the result without indicating overflow. The silent wrap can result in incorrect computations and unanticipated outcomes. Failure to account for integer overflow has resulted in failures of real systems, for example, when implementing the compareTo()
method. The meaning of the return value of the compareTo()
method is defined only in terms of its sign and whether it is zero; the magnitude of the return value is irrelevant. Consequently, an apparent but incorrect optimization would be to subtract the operands and return the result. For operands of opposite signs, this approach can result in integer overflow, consequently violating the compareTo()
contract [Bloch 2008].
Comparison of Compliant Techniques
Following are the three main techniques for detecting unintended integer overflow:
 Precondition testing. Check the inputs to each arithmetic operator to ensure that overflow cannot occur. Throw an
ArithmeticException
when the operation would overflow if it were performed; otherwise, perform the operation.
 Upcasting. Cast the inputs to the next larger primitive integer type and perform the arithmetic in the larger size. Check each intermediate result for overflow of the original smaller type and throw an
ArithmeticException
if the range check fails. Note that the range check must be performed after each arithmetic operation; larger expressions without peroperation bounds checking can overflow the larger type. Downcast the final result to the original smaller type before assigning to a variable of the original smaller type. This approach cannot be used for typelong
becauselong
is already the largest primitive integer type.
BigInteger
. Convert the inputs into objects of typeBigInteger
and perform all arithmetic usingBigInteger
methods. TypeBigInteger
is the standard arbitraryprecision integer type provided by the Java standard libraries. The arithmetic operations implemented as methods of this type cannot overflow; instead, they produce the numerically correct result. Consequently, compliant code performs only a single range check just before converting the final result to the original smaller type and throws anArithmeticException
if the final result is outside the range of the original smaller type.
The precondition testing technique requires different precondition tests for each arithmetic operation. This approach can be somewhat more difficult to implement and to audit than either of the other two approaches.
The upcast technique is the preferred approach when applicable. The checks it requires are simpler than those of the previous technique; it is substantially more efficient than using BigInteger
. Unfortunately, it cannot be applied to operations involving type long
, as there is no bigger type to upcast to.
The BigInteger
technique is conceptually the simplest of the three techniques because arithmetic operations on BigInteger
cannot overflow. However, it requires the use of method calls for each operation in place of primitive arithmetic operators, which can obscure the intended meaning of the code. Operations on objects of type BigInteger
can also be significantly less efficient than operations on the original primitive integer type.
Precondition Testing
The following code example shows the necessary precondition checks required for each arithmetic operation on arguments of type int
. The checks for the other integral types are analogous. These methods throw an exception when an integer overflow would otherwise occur; any other conforming error handling is also acceptable. Since ArithmeticException
inherits from RuntimeException
, we do not need to declare it in a throws
clause.
These method calls are likely to be inlined by most justintime (JIT) systems.
These checks can be simplified when the original type is char
. Because the range of type char
includes only positive values, all comparisons with negative values may be omitted.
Noncompliant Code Example
Either operation in this noncompliant code example could result in an overflow. When overflow occurs, the result will be incorrect.
Compliant Solution (Precondition Testing)
This compliant solution uses the safeAdd()
and safeMultiply()
methods defined in the "Precondition Testing" section to perform secure integral operations or throw ArithmeticException
on overflow:
Compliant Solution (Java 8, Math.*Exact()
)
This compliant solution uses the addExact()
and multiplyExact()
methods defined in the Math
class. These methods were added to Java as part of the Java 8 release, and they also either return a mathematically correct value or throw ArithmeticException
. The Math
class also provides SubtractExact()
and negateExact()
but does not provide any methods for safe division or absolute value.
Compliant Solution (Upcasting)
This compliant solution shows the implementation of a method for checking whether a value of type long
falls within the representable range of an int
using the upcasting technique. The implementations of range checks for the smaller primitive integer types are similar.
Note that this approach cannot be applied to values of type long
because long
is the largest primitive integral type. Use the BigInteger
technique instead when the original variables are of type long
.
Compliant Solution (BigInteger
)
This compliant solution uses the BigInteger
technique to detect overflow:
Noncompliant Code Example (AtomicInteger
)
Operations on objects of type AtomicInteger
suffer from the same overflow issues as other integer types. The solutions are generally similar to the solutions already presented; however, concurrency issues add additional complications. First, potential issues with timeofcheck, timeofuse (TOCTOU) must be avoided (see VNA02J. Ensure that compound operations on shared variables are atomic for more information). Second, use of an AtomicInteger
creates happensbefore relationships between the various threads that access it. Consequently, changes to the number of accesses or order of accesses can alter the execution of the overall program. In such cases, you must either choose to accept the altered execution or carefully craft your implementation to preserve the exact number of accesses and order of accesses to the AtomicInteger
.
This noncompliant code example uses an AtomicInteger
, which is part of the concurrency utilities. The concurrency utilities lack integer overflow checks.
Consequently, itemsInInventory
can wrap around to Integer.MIN_VALUE
when the nextItem()
method is invoked when itemsInInventory == Integer.MAX_VALUE
.
Compliant Solution (AtomicInteger
)
This compliant solution uses the get()
and compareAndSet()
methods provided by AtomicInteger
to guarantee successful manipulation of the shared value of itemsInInventory
. This solution has the following characteristics:
 The number and order of accesses to
itemsInInventory
remain unchanged from the noncompliant code example.  All operations on the value of
itemsInInventory
are performed on a temporary local copy of its value.  The overflow check in this example is performed in inline code rather than encapsulated in a method call. This is an acceptable alternative implementation. The choice of method call versus inline code should be made according to your organization's standards and needs.
The two arguments to the compareAndSet()
method are the expected value of the variable when the method is invoked and the intended new value. The variable's value is updated only when the current value and the expected value are equal [API 2006] (refer to VNA02J. Ensure that compound operations on shared variables are atomic for more details).
Exceptions
NUM00JEX0: Depending on circumstances, integer overflow could be benign. For example, many algorithms for computing hash codes use modular arithmetic, intentionally allowing overflow to occur. Such benign uses must be carefully documented.
NUM00JEX1: Prevention of integer overflow is unnecessary for numeric fields that undergo bitwise operations and not arithmetic operations (see NUM01J. Do not perform bitwise and arithmetic operations on the same data for more information).
Risk Assessment
Failure to perform appropriate range checking can lead to integer overflows, which can cause unexpected program control flow or unanticipated program behavior.
Rule  Severity  Likelihood  Remediation Cost  Priority  Level 

NUM00J  Medium  Unlikely  Medium  P4  L3 
Automated Detection
Automated detection of integer operations that can potentially overflow is straightforward. Automatic determination of which potential overflows are true errors and which are intended by the programmer is infeasible. Heuristic warnings might be helpful.
Tool  Version  Checker  Description 

Coverity  7.5  BAD_SHIFT  Implemented 
Parasoft Jtest  9.5  PB.NUM.{ICO,BSA,CACO} 
Related Guidelines
INT32C. Ensure that operations on signed integers do not result in overflow  
Wraparound Error [XYY]  
CWE682, Incorrect Calculation 
Android Implementation Details
Mezzofanti for Android contained an integer overflow that prevented the use of a big SD card. Mezzofanti contained an expression:
(int) StatFs.getAvailableBlocks() * (int) StatFs.getBlockSize()
to calculate the available memory in an SD card, which could result in a negative value when the available memory is larger than Integer.MAX_VALUE
. Note that these methods are deprecated in API level 18 and replaced by getAvailableBlocksLong()
and getBlockSizeLong()
.
Bibliography
[API 2006]  Class 
Puzzle 27, "Shifty i's"  
[Bloch 2008]  Item 12, "Minimize the Accessibility of Classes and Members" 
[JLS 2015]  §4.2.1, "Integral Types and Values" 
Chapter 5, "Integers"  
[Seacord 2015]  IDS17J. Prevent XML External Entity Attacks LiveLesson 
29 Comments
David Svoboda
Efstathios, this is starting to look good. Nice quote from SUN at the top. Does SUN's document there state anything about silent wrapping on integers?
The title is way too long...suggest taking the BigInteger out of the title, BigInteger is part of the solution, not the problem.
Please also clean up the formatting and language.
Saying that a vul is "almost impossible to exploit" is dangerous. Java vulnerabilities are quite exploitable; they just aren't as easy to exploit as C/C++ (for the reasons you state). For instance, if a Java program is computing my bank account, I don't want any damn integer overflows!
This rule should refer to INT32C when comparing Java and C. It should also refer to INT32CPP when comparing Java and C+, since these two rules are also about signed integer overflow in C and C+.
Provide noncompliant code samples for addition and subtraction...they will ease the descriptions for compliant solutions. Three compliant solutions for each form is a good plan: one compliant solution that prevents overflow (before doing the operation), one solution that does the operation on longs, and one solution that uses BigInteger.
If Integer.MIN_VALUE % 1 == 0, then % is actually safe wrt overflow, as 0 is the mathematically correct answer, and Java is indeed safer than C/C++ here.
Where are the shift operations?
Provide a 'Risk Assessment' section, a 'References' section (listing the SUN document linked above), and an 'Other Languages' section.
Thomas Hawtin
I would like to see more details in the risk assesment. Integer overflows are a real problem in C because they are a prominent cause of buffer overflows. That does apply to Java when it is being used either directly or indirectly to guard native code, but rarely otherwise.
Perhaps some examples could be added. It has been speculated that Comair had an integer overflow bug in December 2004 that had a major business impact. I could quite believe that integer overflows could easily cause an infinite loop leading to availability/DoS issues.
I have never seen an exploitable vulnerability in Java caused by an integer overflow. I therefore feel that the risk assesment is inflated.
IMO, a better solution to checking is to put up with the fugly code and use BigInteger. Only in area determined to be performance criticial should difficult bounded integer arithmetic be used.
As well as operations that happen to have language syntax available, library methods should be considered. Notable is Math.abs has behaviour similar to unary minus (Math.abs(Integer.MIN_VALUE) == Integer.MIN_VALUE).
Efstathios Mertikas
Hi Thomas, thank you for your comment.
I have pointed out in several areas that overflows are very hard to exploit in Java than C\C++ and have explained the reason. I have not claimed that they lead to vulnerabilities easily. I will modify the Risk Assesment section, to reflect this difference as you suggest.
Regardless of how easy it is to exploit the overflow though, there is the problem of overflow in Java. Not only in rare cases but there is this potential in any case of arithmetic operation. And overflows lead at least, in wrong results or undefined behavior.
The guideline for overflows in general are to detect than to prevent so an explicit range checking would be prudent approach.
Regards,
Efstathios
Kirk Sayre
Should the sentence:
"Because of the asymmetry between the representation of negative and positive integer values (there is an extra minimum negative value 128), there is no equivalent positive value (+128) for Integer.MIN_VALUE."
Actually be something like:
"Because of the asymmetry between the representation of negative and positive integer values (Integer.MAX_VALUE is 2147483647 and Integer.MIN_VALUE is 2147483648, which means there is one more negative integer than positive integers), there is no equivalent positive value (+2147483648) for Integer.MIN_VALUE."
References to 128 and +128 appear to be an error.
Philip Miller
There is a lot of material in this one rule. Should we consider breaking this into smaller pieces?
Robert Seacord (Manager)
This rule says that:
Addition (as with all arithmetic operations) in Java is performed on signed numbers only as unsigned numbers are unsupported.
This bothers me, because I understand that char is an unsigned type in Java. Isn't it possible to add two chars or a char and a short?
Dhruv Mohindra
Yes, we do have a guideline on those lines. However, it is still quite unusual to use
char
in arithmetic expressions. Perhaps I can change that to:Typically, addition (as with all arithmetic operations) ....
Storing a signed value in a char is forbidden by INT04J. Do not attempt to store signed values in the char integral type.
Robert Seacord (Manager)
This rule currently states:
Ignoring char for now, it seems to me that addition, subtraction, multiplication, and division of byte, short, operands always succeeds without overflow because these operands are first widened to type int by numeric promotion. These operations on int type can be safely performed as long, and long types as BigInteger.
The obvious mistake in the rule is that arithmetic on small types needs to be performed using variables of type long. This is unnecessarily inefficient.
Dhruv Mohindra
I think the words "or smaller" should be eliminated because this condition applies only to
int
. Addition of two or moreint
values may overflow. This can be prevented by usinglong
.Robert Seacord (Manager)
I wouldn't worry about making a bunch of small changes to this rule; I'm fixing to overhaul it once I understand the topic better.
Robert Seacord
i'm not sure i believe this test is for overflow is correct
if b > 0 and a < 0 it is going to test if a (a negative number) is greater than Integer.MAX_VALUE  b (it is not) which evaluates to false
if b > 0 and a > 0 it is going to perform the same test which also looks fine.
if b < 0 and a > 0 it is going to test if a < Integer.MIN_VALUE  b. Since a is positive, this test will always fail and an exception will be thrown even though this should always succeed.
if b < 0 and a < 0 then it is going to test if a < Integer.MIN_VALUE  b which should be fine.
I think we need to toss this solution and look more closely at the remaining solutions (or just copy from the C wiki for signed integer overflow).
Dhruv Mohindra
I agree that the addition is guaranteed to succeed when b > 0 and a < 0 and the example (rightly) does not throw an exception for this case. I am not sure what the problem is...(the test fails and the condition evaluates to false; the exception is not thrown)
EDIT (in reply to your edit): In your case 3, b < 0 and a > 0:
Suppose
a
is 5 andb
is 5.5 < large number (5)
5 < large number + 5
5 < still a large number (5 is not less than this negative large number)
So the condition expression evaluates to false so the exception is not thrown.
Dhruv Mohindra
Note that the above integer overflow (precondition) check for + is almost the same as
SafeInt
class for C++ by LeBlanc.Dean Sutherland
Fred Long and I just added INT00EX2, observing that checking EVERY integer operation for overflow may be more trouble than it is worth – except for the most critical programs (or portions thereof). A possible alternative is to rangecheck inputs to portions of the code to ensure that overflow is impossible therein. We note that this is quite difficult in practice!
This particular problem makes me wish I was writing the Ada Secure Coding Standard, as this whole issue would be a nonproblem. The whole overflow check thing really belongs in the hands of the compiler, rather than the programmer.
David Svoboda
Here is a common integer vul in C:
size_t x = /* comes from the user */
struct foo* array = malloc(x * sizeof( struct foo));
Clearly if an attacker can set the value of x, they can set it to so large a value that malloc() is bound to fail. But there is a more insidious vul; an attacker can set x such that the multiplication overflows, yet produces a small value. For instance if x = SIZE_MAX / sizeof(struct foo) + 1, then the product will be 1 * sizeof( struct foo), which malloc will prob succeed. Then you have an array that the program thinks is much larger than it actually is, and heap overflow is imminent.
Which is to say that you have to be VERY careful and precise when doing your taint analysis.
Robert Seacord (Manager)
For the C standard, we focus on the following applications of integers as requiring overflow checks:
I'm sure the list of criteria for Java is different.
You can write the Ada secure coding standard when you finish this one. 8^)
Robert Seacord (Manager)
The following is not quite enough for me to understand the example:
Can you turn this into a NCE/CS pair?
Robert Seacord (Manager)
"Performing arithmetic operations that use operands of type char is strongly discouraged."
How discouraged? Should there be a guideline that requires occurrences to be diagnosed?
Why is this comment in the "Addition" section, when it appears to apply to all arithmetic operations?
Robert Seacord (Manager)
Why are there two Bounds Checking Compliant Solutions for addition? What is the difference between the two?
Dean Sutherland
Major reorganization to clearly describe and demonstrate the three different compliant techniques. The section on preconditioning the inputs now shows each of the necessary checks. The other two approaches both demonstrate and discuss the location of their necessary checks.
Still need to find a home for the prior content that is interesting, but has nothing to do with integer overflow.
Robert Seacord (Manager)
made changes to resolve JPCERT comments.
Dhruv Mohindra
We should get the following lines to compile:
David Svoboda
Done
Masaki Kubo
1. In NUM00EX1, "Prevention of integer overflow is not necessary for numeric type...", its not about type but rather some value(data) with numeric type which is of our interest?
2. What is "heuristic warnings" and how they are helpful?
David Svoboda
1. You're right. I s/type/field/ in that exception.
2. Heuristic warnings are generated by heuristic methods, which, by definition, are not guaranteed to be perfect...they may generate false positives & false negatives. IOW the text is saying we don't know a way to perfectly identify overflow vulnerabilities, but we can employ several imperfect methods.
joshua
Is this something that can use Math.multiplyExact and Math.addExact (java 8) instead of these safe operations?
David Svoboda
Yes. I've added a Compliant Solution that address the *Exact() methods in the Math class.
Axel Hauschulte
Is there a specific reason why many methods in the examples explicitly declare
ArithmeticException
to be thrown? E.g.Since
ArithmeticException
is a runtime exception there is no need to declare it in the throws clause. Item 62 of Effective Java actually advises against including unchecked exceptions in the method declaration.I tend to remove
ArithmeticException
from the method declarations' throw clause. Any opinions on this?Axel Hauschulte
I have removed the
throws
ArithmeticException
declaration from the method signatures.