Description
Return one coefficient of a carryless word product.
Carryless arithmetic treats bits as coefficients over the two-element field. Addition is XOR and coefficient multiplication uses parity, so polynomial products and division steps differ from ordinary integer multiplication, carrying and remainder operations.
Return one coefficient of a carryless word product.
When to use this tool
Use Bitwise: Carryless Product Coefficient Parity when evaluating the stated binary-polynomial coefficient or cancellation step.
How to use the tool and how it works
- Enter left, right, width, coefficientIndex. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read result. Expand result to inspect the returned fields.
Return one coefficient of a carryless word product. Validate bounded words and coefficientIndex 0..2*width-2; XOR all matching bit-pair products whose indexes sum to the requested coefficient.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| left | Left | Required. Text. Integer syntax: optional + or - and 1-128 decimal digits; leading zeros accepted. Apply the sign restriction above. Implementation limit: 8192 Unicode code points and 16384 UTF-16 code units. Word value must satisfy 0 <= value < 2^width. |
| right | Right | Required. Text. Integer syntax: optional + or - and 1-128 decimal digits; leading zeros accepted. Apply the sign restriction above. Implementation limit: 8192 Unicode code points and 16384 UTF-16 code units. Word value must satisfy 0 <= value < 2^width. |
| width | Width under the stated bit, interval or geometry convention; apply the field-specific units and bounds. | Required. Safe integer. Fixed-word implementation domain: whole width 1-128 bits. |
| coefficientIndex | Coefficient Index | Required. Safe integer. Calculation range: 0 through 2 * width - 2, inclusive. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| result | Returned calculated value or nested object under the stated output convention. | Required. Object. |
| result.coefficient | One GF(2) product coefficient, determined by parity rather than integer carrying. | Calculated result member: number. |
{
"input": {
"left": "3",
"right": "3",
"width": 2,
"coefficientIndex": 1
},
"output": {
"result": {
"coefficient": 0
}
}
}Limitations and assumptions
- Coefficients are over GF(2); this is one polynomial coefficient, not ordinary integer multiplication or a complete product.
- Outputs are nested inside result. Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.
Alternative or Complementary methods
Use a complete polynomial multiplication/division or code-verification method with a specified polynomial and bit-order convention when a full result is needed.
References
Similar or alternative tools
- Carryless Division Leading Cancellation Step
Perform one leading-term binary-polynomial division cancellation.
- Bitmask Lane Parity Contribution Tool
Compute bitmask lane parity contribution from explicit bounded inputs.