**Division Algorithm to search for monic irreducible polynomials over extended Galois Field GF(p^q).**

*Sankhanil Dey and Amlan Chakrabarti and Ranjan Ghosh*

**Abstract: **In modern era of computer science there are many applications of the polynomials over finite fields especially of the polynomials over extended Galois fields GF(p^q) where p is the prime modulus and q is the extension of the said Galois field, in the generation of the modern algorithms in the computer science, the soft computation, the cryptology and the cryptanalysis and especially in generation of the S-boxes of the cryptographic block and stream ciphers. The procedure and the algorithms of the subtraction and the division of the two Galois field polynomials over the Galois field GF(p^q) was remained untouched to the researchers of the applications of finite field theory in the computer science. In this paper the procedure and algorithms to subtract and divide the two Galois field polynomials over Galois field GF(p^q) or the two Galois field numbers over the Galois field GF(p^q) are introduced in detail. If a monic basic polynomial over the Galois field GF(p^q) (BP) [1] is divisible by any of the monic elemental polynomials over the Galois field GF(p^q) (EP) [1] except the constant polynomials (CPs) [1] over the Galois field GF(p^q) then the monic BP over the Galois field GF(p^q) is termed as the monic reducible polynomial (RP) [1] over the Galois field GF(pq) and if a monic BP over the Galois field GF(p^q) is not divisible to any of the EPs over the Galois field GF(p^q) except the CPs over the Galois field GF(p^q) or more specifically to any monic EP over the Galois field GF(p^q) with half of the degree of the concerned monic BP over the Galois field GF(p^q) then the monic BP over Galois field GF(p^q) is called as the irreducible polynomial (IP) [1] over the Galois field GF(p^q). Here the common algorithm to generate all the monic IPs over the Galois field GF(p^q) is introduced. The time complexity analyses of the algorithms prove the said algorithms to be less time consuming and efficient

**Category / Keywords: **foundations / Irreducible polynomials, cryptology, S-boxes, Algorithms

**Original Publication**** (with minor differences): **International Journal of Tomography and Simulation

**Date: **received 26 Mar 2020

**Contact author: **sdrpe_rs at caluniv ac in

**Available format(s): **PDF | BibTeX Citation

**Note: **The publication is the division algorithm to generate irreducible polynomials that are used to generate 4-bit, 8-bit, 32 bit S-boxes.

**Version: **20200328:151632 (All versions of this report)

**Short URL: **ia.cr/2020/358

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