Finite Field Multiplication Calculator

1. Select Prime field or Binary extension.
2. Enter a and b in the chosen format.
3. Provide p for GF(p), or m and polynomial for GF(2^m).
4. Press Multiply to see the result above the form.
5. Use Download CSV or Download PDF to export tables.

Finite fields and controlled arithmetic

Finite fields define arithmetic on a fixed set where addition, multiplication, and inverses behave predictably. This tool supports GF(p) for prime moduli and GF(2^m) for binary extensions, so you can match course theory with working calculations. Every run records inputs, parameters, and a timestamped result.

GF(p) multiplication with modular reduction

In GF(p), multiply normally and reduce modulo p. If p=101, a=87, and b=59, the raw product is 5133 and the reduced product is 83. Because values are always reduced into 0…p−1, results remain bounded, making comparisons and manual checks straightforward.

GF(2^m) multiplication as polynomial math

Binary fields treat an m-bit number as a polynomial over {0,1}. Addition becomes XOR, and multiplication uses shifts and conditional XOR into an accumulator. Reduction occurs when a shift produces a carry beyond degree m−1, using an irreducible polynomial that includes the x^m term.

Parameter discipline matters. For GF(2^4), choosing poly 0x13 corresponds to x^4+x+1, and inputs are masked to 4 bits before processing. Changing the polynomial changes the field representation, so identical bit patterns can produce different products. The Plotly chart summarizes the run: GF(p) shows bars for a, b, a×b, and the reduced result, while GF(2^m) plots the accumulator after each iteration.

Step trace data for debugging and grading

The calculator outputs a structured trace: iteration index, a and b before shifting, least significant bit of b, accumulator before and after XOR, and updated a and b. For m=8, at most 8 iterations appear; for m=16, at most 16 iterations. This trace turns a final answer into an auditable derivation.

Exports and reproducible reporting

CSV exports preserve tables for spreadsheet review and bulk annotation. PDF exports capture the same tables in a clean page layout, suitable for submissions or lab notes. When you repeat a computation, your session history keeps up to 25 recent runs, helping you compare parameter choices quickly. Store exports with assignments to document assumptions, units, and chosen modulus values.

Validation examples and practical use cases

Known pairs are essential for validation. With m=8 and poly 0x11B, 0x57×0x83 yields 0xC1, a standard check used in cryptography coursework. Finite-field products also appear in error-correcting codes and algebraic algorithms, where consistent reduction rules prevent silent implementation mistakes.

1) Do I always need p to be prime?

For field properties, yes. If p is composite, some nonzero values lack inverses, so the structure is a ring rather than a field.

2) Why must the polynomial include x^m?

It defines a degree-m modulus for reduction. Without that leading term, the arithmetic no longer matches an m-bit field representation.

3) What does XOR represent in GF(2^m)?

XOR is addition modulo 2 applied bitwise. It matches coefficient-wise addition of polynomials whose coefficients are 0 or 1.

4) How many iterations will the trace show?

At most m iterations, because b shifts right once per step until it becomes zero.

5) Why do results change when I change the polynomial?

Different irreducible polynomials define different bases. The same bit patterns can represent different elements, so products can differ.

6) What should I export for assignments?

Export the steps table plus inputs and parameters. That provides a reproducible record that can be checked independently.