Factoring Polynomials Over Finite Fields

Berlekamp, Elwyn R.

doi:10.1002/j.1538-7305.1967.tb03174.x

Cited by 257 publications

(166 citation statements)

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“…Let F be a field of prime characteristic p , and let r > 1 be a power of p . Then the differential equation in question is (1) H(r-»(y)=y'…”

Section: Description Of the Algorithmmentioning

confidence: 99%

“…The central problem that remains to be discussed is how to actually solve the differential equation (1). Let F be an arbitrary field of prime characteristic p , let r > 1 be a power of p , and let / e F[x] be monic with deg(/) = d > 1.…”

Section: Description Of the Algorithmmentioning

confidence: 99%

“…Like the classical algorithm of Berlekamp [1], the recently developed deterministic algorithm of the author [12] for factoring polynomials over finite fields is based on a linearization of the factorization problem, i.e., on a reduction to a system of linear equations. However, the new algorithm uses a completely different procedure to achieve the linearization, a key step being the consideration of differential equations in rational function fields.…”

Section: Introductionmentioning

confidence: 99%

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Factoring Polynomials over Finite Fields Using Differential Equations and Normal Bases

Niederreiter¹

1994

Mathematics of Computation

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Abstract. The deterministic factorization algorithm for polynomials over finite fields that was recently introduced by the author is based on a new type of linearization of the factorization problem. The main ingredients are differential equations in rational function fields and normal bases of field extensions. For finite fields of characteristic 2, it is known that this algorithm has several advantages over the classical Berlekamp algorithm. We develop the algorithm in a general framework, and we show that it is feasible for arbitrary finite fields, in the sense that the linearization can be achieved in polynomial time.

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“…Let F be a field of prime characteristic p , and let r > 1 be a power of p . Then the differential equation in question is (1) H(r-»(y)=y'…”

Section: Description Of the Algorithmmentioning

confidence: 99%

Section: Description Of the Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Factoring Polynomials over Finite Fields Using Differential Equations and Normal Bases

Niederreiter¹

1994

Mathematics of Computation

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“…Next to it Elementary Techniques to compute over finite Fields or Galois Field GF(p q ) had been descried with proper modifications [TD63]. In next the factorization of Polynomials over Galois Field GF(p q ) had been elaborated [EB67]. Later appropriate coding techniques of Polynomials over Galois Field GF(p q ) had been illustrated with example [TK68].…”

mentioning

confidence: 99%

“…Later appropriate coding techniques of Polynomials over Galois Field GF(p q ) had been illustrated with example [TK68]. The previous idea of factorizing Polynomials over Galois Field GF(p q ) [EB67] had also been extended to Large value of P or Large Finite fields [EB70]. Later Few Probabilistic Algorithms to find IPs over Galois Field GF(p q ) for degree q had been elaborated with example [MR80].…”

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confidence: 99%

A review of cryptographic properties of S-boxes with generation and analysis of crypto secure S-boxes.

Dey

Ghosh

2018

Preprint

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.in 1 , rghosh47@yahoo.co.in 2 .Abstract. In modern as well as ancient ciphers of public key cryptography, substitution boxes find a permanent seat. Generation and cryptanalysis of 4-bit as well as 8-bit crypto S-boxes is of utmost importance in modern cryptography. In this paper, a detailed review of cryptographic properties of S-boxes has been illustrated. The generation of crypto S-boxes with 4-bit as well as 8-bit Boolean functions (BFs) and polynomials over Galois field GF(p q ) has also been of keen interest of this paper. The detailed analysis and comparison of generated 4-bit and 8-bit S-boxes with 4-bit as well as 8-bit S-boxes of Data Encryption Standard (DES) and Advance Encryption Standard (AES) respectively, has incorporated with example. Detailed analysis of generated Sboxes claims a better result than DES and AES in view of security of crypto S-boxes.

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Computer Algebra

Davenport

2015

Digital Encyclopedia of Applied Physics

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The article contains sections titled: Introduction Computer Algebra Systems “Elementary” Algorithms Representation of Polynomials Greatest Common Divisors Square‐free Decomposition Extended Euclidean Algorithm Advanced Algorithms Modular Algorithms–Integer Modular Algorithms–Polynomial The Challenge of Factorization p ‐adic Algorithms–Integer p ‐adic Algorithms–Polynomial Solving Polynomial Systems Solving One Polynomial Real Roots Linear Systems Multivariate Systems Gröbner Bases Regular Chains Integration Rational Functions More Complicated Functions Linear Ordinary Differential Equations Interpreting Formulae as Functions Fundamental Theorem of Calculus Revisited Simplification of Functions Real Problems Conclusion

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Factoring Polynomials Over Finite Fields

Abstract: We present here an algorithm for factoring a given polynomial over GF(q) into powers of irreducible polynomials. The method reduces the factorization of a polynomial of degree m over GF (q) to the solution of about m(q — 1)/q linear equations in as many unknowns over GF(q).

Cited by 257 publications

References 1 publication

Factoring Polynomials over Finite Fields Using Differential Equations and Normal Bases

Factoring Polynomials over Finite Fields Using Differential Equations and Normal Bases

A review of cryptographic properties of S-boxes with generation and analysis of crypto secure S-boxes.

Computer Algebra

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