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Common Weakness Enumeration

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ID

CWE-1339: Insufficient Precision or Accuracy of a Real Number

Weakness ID: 1339
Abstraction: Base
Structure: Simple
Status: Draft
Presentation Filter:
+ Description
The program processes a real number with an implementation in which the number’s representation does not preserve required accuracy and precision in its fractional part, causing an incorrect result.
+ Extended Description

When a security decision or calculation requires highly precise, accurate numbers – such as financial calculations or prices – then small variations in the number could be exploited by an attacker.

There are multiple ways to store the fractional part of a real number in a computer. In all of these cases, there is a limit to the accuracy of recording a fraction. If the fraction can be represented in a fixed number of digits (binary or decimal), there might not be enough digits assigned to represent the number. In other cases the number cannot be represented in a fixed number of digits due to repeating in decimal or binary notation (e.g. 0.333333...) or due to a transcendental number such as Π or √2. Rounding of numbers can lead to situations where the computer results do not adequately match the result of sufficiently accurate math.

+ Relationships
Section HelpThis table shows the weaknesses and high level categories that are related to this weakness. These relationships are defined as ChildOf, ParentOf, MemberOf and give insight to similar items that may exist at higher and lower levels of abstraction. In addition, relationships such as PeerOf and CanAlsoBe are defined to show similar weaknesses that the user may want to explore.
+ Relevant to the view "Research Concepts" (CWE-1000)
NatureTypeIDName
ChildOfPillarPillar - a weakness that is the most abstract type of weakness and represents a theme for all class/base/variant weaknesses related to it. A Pillar is different from a Category as a Pillar is still technically a type of weakness that describes a mistake, while a Category represents a common characteristic used to group related things.682Incorrect Calculation
PeerOfBaseBase - a weakness that is still mostly independent of a resource or technology, but with sufficient details to provide specific methods for detection and prevention. Base level weaknesses typically describe issues in terms of 2 or 3 of the following dimensions: behavior, property, technology, language, and resource.190Integer Overflow or Wraparound
CanPrecedeClassClass - a weakness that is described in a very abstract fashion, typically independent of any specific language or technology. More specific than a Pillar Weakness, but more general than a Base Weakness. Class level weaknesses typically describe issues in terms of 1 or 2 of the following dimensions: behavior, property, and resource.119Improper Restriction of Operations within the Bounds of a Memory Buffer
CanPrecedeClassClass - a weakness that is described in a very abstract fashion, typically independent of any specific language or technology. More specific than a Pillar Weakness, but more general than a Base Weakness. Class level weaknesses typically describe issues in terms of 1 or 2 of the following dimensions: behavior, property, and resource.834Excessive Iteration
Section HelpThis table shows the weaknesses and high level categories that are related to this weakness. These relationships are defined as ChildOf, ParentOf, MemberOf and give insight to similar items that may exist at higher and lower levels of abstraction. In addition, relationships such as PeerOf and CanAlsoBe are defined to show similar weaknesses that the user may want to explore.
+ Relevant to the view "Software Development" (CWE-699)
NatureTypeIDName
MemberOfCategoryCategory - a CWE entry that contains a set of other entries that share a common characteristic.189Numeric Errors
+ Background Details
There are three major ways to store real numbers in computers. Each method is described along with the limitations of how they store their numbers.
  1. Fixed: Some implementations use a fixed number of binary bits to represent both the integer and the fraction. In the demonstrative example about Muller's Recurrence, the fraction 108.0 - ((815.0 - 1500.0 / z) / y) cannot be represented in 8 binary digits. The numeric accuracy within languages such as PL/1, COBOL and Ada is expressed in decimal digits rather than binary digits. In SQL and most databases, the length of the integer and the fraction are specified by the programmer in decimal. In the language C, fixed reals are implemented according to ISO/IEC TR18037
  2. Floating: The number is stored in a version of scientific notation with a fixed length for the base and the significand. This allows flexibility for more accuracy when the integer portion is smaller. When dealing with large integers, the fractional accuracy is less. Languages such as PL/1, COBOL and Ada set the accuracy by decimal digit representation rather than using binary digits. Python also implements decimal floating-point numbers using the IEEE 754-2008 encoding method.
  3. Ratio: The number is stored as the ratio of two integers. These integers also have their limits. These integers can be stored in a fixed number of bits or in a vector of digits. While the vector of digits method provides for very large integers, they cannot truly represent a repeating or transcendental number as those numbers do not ever have a fixed length.
+ Modes Of Introduction
Section HelpThe different Modes of Introduction provide information about how and when this weakness may be introduced. The Phase identifies a point in the life cycle at which introduction may occur, while the Note provides a typical scenario related to introduction during the given phase.
PhaseNote
ImplementationThis weakness is introduced when the developer picks a method to represent a real number. The weakness may only be visible with very specific numeric inputs.
+ Applicable Platforms
Section HelpThis listing shows possible areas for which the given weakness could appear. These may be for specific named Languages, Operating Systems, Architectures, Paradigms, Technologies, or a class of such platforms. The platform is listed along with how frequently the given weakness appears for that instance.

Languages

Class: Language-Independent (Undetermined Prevalence)

Operating Systems

Class: OS-Independent (Undetermined Prevalence)

Architectures

Class: Architecture-Independent (Undetermined Prevalence)

Technologies

Class: Technology-Independent (Undetermined Prevalence)

+ Common Consequences
Section HelpThis table specifies different individual consequences associated with the weakness. The Scope identifies the application security area that is violated, while the Impact describes the negative technical impact that arises if an adversary succeeds in exploiting this weakness. The Likelihood provides information about how likely the specific consequence is expected to be seen relative to the other consequences in the list. For example, there may be high likelihood that a weakness will be exploited to achieve a certain impact, but a low likelihood that it will be exploited to achieve a different impact.
ScopeImpactLikelihood
Availability

Technical Impact: DoS: Crash, Exit, or Restart

This weakness will generally lead to undefined results and therefore crashes. In some implementations the program will halt if the weakness causes an overflow during a calculation.
Integrity

Technical Impact: Execute Unauthorized Code or Commands

The results of the math are not as expected. This could cause issues where a value would not be properly calculated and provide an incorrect answer.
Confidentiality
Availability
Access Control

Technical Impact: Read Application Data; Modify Application Data

This weakness can sometimes trigger buffer overflows which can be used to execute arbitrary code. This is usually outside the scope of a program's implicit security policy.
+ Demonstrative Examples

Example 1

Muller's Recurrence is a series that is supposed to converge to the number 5. When running this series with the following code, different implementations of real numbers fail at specific iterations:

(bad code)
Example Language: Rust 
fn rec_float(y: f64, z: f64) -> f64
{
  108.0 - ((815.0 - 1500.0 / z) / y);
}

fn float_calc(turns: usize) -> f64
{
  let mut x: Vec<f64> = vec![4.0, 4.25];
  (2..turns + 1).for_each(|number|
  {
    x.push(rec_float(x[number - 1], x[number - 2]));
  });

  x[turns]
}

The chart below shows values for different data structures in the rust language when Muller’s recurrence is executed to 80 iterations. The data structure f64 is a 64 bit float. The data structures I<number>F<number> are fixed representations 128 bits in length that use the first number as the size of the integer and the second size as the size of the fraction (e.g. I16F112 uses 16 bits for the integer and 112 bits for the fraction). The data structure of Ratio comes in three different implementations: i32 uses a ratio of 32 bit signed integers, i64 uses a ratio of 64 bit signed integers and BigInt uses a ratio of signed integer with up to 2^32 digits of base 256. Notice how even with 112 bits of fractions or ratios of 64bit unsigned integers, this math still does not converge to an expected value of 5. Muller's Recurrence

(good code)
Example Language: Rust 
Use num_rational::BigRational;

fn rec_big(y: BigRational, z: BigRational) -> BigRational
{
  BigRational::from_integer(BigInt::from(108))
    - ((BigRational::from_integer(BigInt::from(815))
    - BigRational::from_integer(BigInt::from(1500)) / z)
    / y)
}

fn big_calc(turns: usize) -> BigRational
{
  let mut x: Vec<BigRational> = vec![BigRational::from_float(4.0).unwrap(), BigRational::from_float(4.25).unwrap(),];

  (2..turns + 1).for_each(|number|
  {
    x.push(rec_big(x[number - 1].clone(), x[number - 2].clone()));
  });
  x[turns].clone()
}

Example 2

On February 25, 1991, during the eve of the of an Iraqi invasion of Saudi Arabia, a Scud missile fired from Iraqi positions hit a US Army barracks in Dhahran, Saudi Arabia. It miscalculated time and killed 28 people [REF-1190].

Example 2 References:
[REF-1190] "An Improvement To Floating Point Numbers". 2015-10-22. <https://hackaday.com/2015/10/22/an-improvement-to-floating-point-numbers/>.

Example 3

Sleipner A, an offshore drilling platform in the North Sea was incorrectly constructed with an underestimate of 50% of strength in a critical cluster of buoyancy cells needed for construction. This led to a leak in buoyancy cells during lowering, causing a seismic event of 3.0 on the Richter Scale and about $700M loss [REF-1190].

Example 3 References:
[REF-1190] "An Improvement To Floating Point Numbers". 2015-10-22. <https://hackaday.com/2015/10/22/an-improvement-to-floating-point-numbers/>.
+ Observed Examples
ReferenceDescription
Chain: series of floating-point precision errors (CWE-1339) in a web browser rendering engine causes out-of-bounds read (CWE-125), giving access to cross-origin data
Chain: rounding error in floating-point calculations (CWE-1339) in image processor leads to infinite loop (CWE-835)
Chain: machine-learning product can have a heap-based buffer overflow (CWE-122) when some integer-oriented bounds are calculated by using ceiling() and floor() on floating point values (CWE-1339)
Chain: insufficient precision (CWE-1339) in random-number generator causes some zero bits to be reliably generated, reducing the amount of entropy (CWE-331)
Chain: web browser crashes due to infinite loop - "bad looping logic [that relies on] floating point math [CWE-1339] to exit the loop [CWE-835]"
+ Potential Mitigations

Phases: Implementation; Patching and Maintenance

The developer or maintainer can move to a more accurate representation of real numbers. In extreme cases, the programmer can move to representations such as ratios of BigInts which can represent real numbers to extremely fine precision. The programmer can also use the concept of an Unum real. The memory and CPU tradeoffs of this change must be examined. Since floating point reals are used in many programs and many locations, they are implemented in hardware and most format changes will cause the calculations to be moved into software resulting in slower programs.
+ References
[REF-1186] "Is COBOL holding you hostage with Math?". 2018-07-28. <https://medium.com/the-technical-archaeologist/is-cobol-holding-you-hostage-with-math-5498c0eb428b>.
[REF-1187] "Intermediate results and arithmetic precision". 2021-06-30. <https://www.ibm.com/docs/en/cobol-zos/6.2?topic=appendixes-intermediate-results-arithmetic-precision>.
[REF-1188] "8.1.2. Arbitrary Precision Numbers". 2021-06-24. <https://www.postgresql.org/docs/8.3/datatype-numeric.html#DATATYPE-NUMERIC-DECIMAL>.
[REF-1189] "Muller's Recurrence". 2017-11-11. <https://scipython.com/blog/mullers-recurrence/>.
[REF-1190] "An Improvement To Floating Point Numbers". 2015-10-22. <https://hackaday.com/2015/10/22/an-improvement-to-floating-point-numbers/>.
[REF-1191] "HIGH PERFORMANCE COMPUTING: ARE WE JUST GETTING WRONG ANSWERS FASTER?". 1999-06-23. <https://www3.nd.edu/~markst/cast-award-speech.pdf >.
+ Content History
+ Submissions
Submission DateSubmitterOrganization
2021-07-08CWE Content TeamMITRE
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Page Last Updated: July 20, 2021