# Improving the Time Complexity of the Computation of Irreducible and Primitive Polynomials in Finite Fields

@inproceedings{Rif1991ImprovingTT, title={Improving the Time Complexity of the Computation of Irreducible and Primitive Polynomials in Finite Fields}, author={Josep Rif{\`a} and Joan Borrell}, booktitle={AAECC}, year={1991} }

In this paper, we present a method to compute all the irreducible and primitive polynomials of degree m over a finite field. We also describe two concrete implementations of our method with respective time complexities O(m2 + m log m) and O(m2 + log m). These implementations, using in parallel different devices introduced to operate in these fields [1], [7], allows us to reduce the time complexity of our method below that of the best previously known methods [3]. Our algorithm is especially…

## 9 Citations

### A fast algorithm to compute irreducible and primitive polynomials in finite fields

- Computer Science, MathematicsMathematical systems theory
- 2005

This paper finds each new irreducible or primitive polynomial with a complexity ofO(m) arithmetic operations inGF(q) by using the Berlekamp-Massey algorithm.

### On Computing the Resultant of Generic Bivariate Polynomials

- Mathematics, Computer ScienceISSAC
- 2018

An algorithm is presented for computing the resultant of two generic bivariate polynomials over a field K using (n2 - 1/ømega d) 1+o(1) arithmetic operations in K, where two n x n matrices are multiplied using O(nømega) operations.

### Normal Bases over Finite Fields

- Mathematics, Computer Science
- 1993

The principal result in the thesis is the complete determination of all optimal normal bases in finite fields, which confirms a conjecture by Mullin, Onyszchuk, Vanstone and Wilson.

### Fast computation of special resultants

- Mathematics, Computer ScienceJ. Symb. Comput.
- 2006

### Cryptographic Counters and Applications to Electronic Voting

- Computer Science, MathematicsEUROCRYPT
- 2001

This work formalizes the notion of a cryptographic counter, which allows a group of participants to increment and decrement a cryptographic representation of a (hidden) numerical value privately and robustly, and shows a general and efficient reduction from any encryption scheme to a general cryptographic counter.

### Fast Computation With Two Algebraic Numbers

- Computer Science, Mathematics
- 2001

We propose fast algorithms for computing composed multiplications and composed sums, as well as «diamond products» of univariate polynomials.

### Fast construction of irreducible polynomials over finite fields

- Computer Science, MathematicsSODA '93
- 1993

The main result of this paper a new algorithm for constructing an irreducible polynomial of specified degree n over a finite field Fq . The algorithm is probabilistic, and is asymptotically faster…

## References

SHOWING 1-9 OF 9 REFERENCES

### Introduction to finite fields and their applications: List of Symbols

- Mathematics, Computer Science
- 1986

An introduction to the theory of finite fields, with emphasis on those aspects that are relevant for applications, especially information theory, algebraic coding theory and cryptology and a chapter on applications within mathematics, such as finite geometries.

### Systolic Multipliers for Finite Fields GF(2m)

- Computer ScienceIEEE Transactions on Computers
- 1984

Two systolic architectures are developed for performing the product–sum computation AB + C in the finite field GF(2m) of 2melements, where A, B, and C are arbitrary elements of GF(2m). The first…

### Note for computing the minimum polynomial of elements in large finite fields

- Computer Science
- 1989

Two methods for computing the minimun polynomial of an element in the finite field GF(qm) have pratically no storage constraint and may improve by a factor 2.6 the classical method.

### Efficient bit-serial multiplication and the discrete-time Wiener-Hopf equation over finite fields

- MathematicsIEEE Trans. Inf. Theory
- 1989

It is shown that solving the DTWHE is equivalent to performing division over finite fields, and the proof provides a new interpretation of the relationship between bit- serial multiplication and DTWHEs that enables bit-serial multiplication over GF(2/sup m/) to be understood more easily.

### Bit-serial Reed - Solomon encoders

- Computer ScienceIEEE Transactions on Information Theory
- 1982

New concepts and techniques for implementing encoders for Reed-Solomon codes, with or without interleaving are presented, including only fields of order 2”, where m m ight be any integer.

### Shift-register synthesis and BCH decoding

- Computer ScienceIEEE Trans. Inf. Theory
- 1969

It is shown in this paper that the iterative algorithm introduced by Berlekamp for decoding BCH codes actually provides a general solution to the problem of synthesizing the shortest linear feedback…

### Designing Efficient Algorithms for Parallel Computers

- Computer Science
- 1987

This is it, the designing efficient algorithms for parallel computers that will be your best choice for better reading book that will not spend wasted by reading this website.

### VLSI Architectures for Computing Multiplications and Inverses in GF(2m)

- Computer ScienceIEEE Transactions on Computers
- 1985

The designs developed for the Massey-Omura multiplier and the computation of inverse elements are regular, simple, expandable, and therefore, naturally suitable for VLSI implementation.