1. What is Polynomial Division Modulo?

Polynomial division modulo is an operation that divides one polynomial by another under a given modulus, producing a quotient polynomial. It's fundamental in many areas of mathematics and computer science, particularly in cryptography and error-correcting codes.

2. How Does the Calculator Work?

The calculator performs polynomial division under modulo arithmetic:

Where:

• Dividend — The polynomial being divided
• Divisor — The polynomial dividing the dividend
• Modulus — The integer modulus for the operation
• Quotient — The resulting polynomial

Explanation: The operation involves polynomial long division where all coefficients are reduced modulo m at each step.

3. Importance of Polynomial Division Modulo

Details: This operation is crucial in constructing finite fields, which are essential for cryptographic algorithms like AES and elliptic curve cryptography.

4. Using the Calculator

Tips: Enter polynomials as comma-separated coefficients (e.g., "1,0,1" for x² + 1). The modulus must be an integer ≥ 2.

5. Frequently Asked Questions (FAQ)

Q1: What polynomial formats are supported?
A: Enter polynomials as comma-separated coefficients from highest degree to lowest.

Q2: What's the difference from regular polynomial division?
A: All coefficient arithmetic is performed modulo m, affecting both the division process and final result.

Q3: What if the divisor isn't monic?
A: The calculator handles non-monic polynomials by using modular inverses where needed.

Q4: Are there limitations to this calculator?
A: The modulus must be prime for the division to always be possible (to ensure a field structure).

Q5: What applications use this operation?
A: Reed-Solomon codes, AES encryption, and many other cryptographic systems rely on polynomial arithmetic modulo.