Carryless Division Leading Cancellation Step Tool

Description

Perform one leading-term binary-polynomial division cancellation.

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.

Perform one leading-term binary-polynomial division cancellation.

When to use this tool

Use Carryless Division Leading Cancellation Step when evaluating the stated binary-polynomial coefficient or cancellation step.

How to use the tool and how it works

Enter dividend, divisor. Use the units and conventions below.
Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
Read remainder, quotientTerm. Use the output convention below.

Perform one leading-term binary-polynomial division cancellation. Require unsigned128 dividend and nonzero unsigned128 divisor; align highest set bits and XOR when dividend degree is sufficient, otherwise preserve it and return zero quotientTerm.^[1]

Input field	Meaning and units	Accepted input
dividend	Dividend	Required. Text; maximum UTF-16 length 8192. Integer syntax: optional + or - and 1-128 decimal digits; leading zeros accepted. Apply the sign restriction above.
divisor	Divisor	Required. Text; maximum UTF-16 length 8192. Integer syntax: optional + or - and 1-128 decimal digits; leading zeros accepted. Apply the sign restriction above.

Output field	Interpretation	Returned shape
remainder	Remainder	Required. Text; maximum UTF-16 length 8192.
quotientTerm	Only the current leading cancellation monomial, not the complete quotient.	Required. Text; maximum UTF-16 length 8192.

json

{
  "input": {
    "dividend": "13",
    "divisor": "3"
  },
  "output": {
    "remainder": "1",
    "quotientTerm": "4"
  }
}

Limitations and assumptions

The returned remainder is after one step only; no complete division, irreducibility test or CRC validation is performed.
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.

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

Calculation and input contract: bitwise/carryless-division-leading-cancellation-step — tool.io

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