Counting Extended Irreducible Goppa Codes

Abstract

We produce an upper bound on the number of extended irreducible Goppa codes over any finite field.

1. Introduction

The advent of quantum computing has brought Goppa codes to the forefront. Most cryptosystems which are in general use today are asymmetric cryptosystems which are based on the integer factorization problem or the discrete logarithm problem and it is conjectured that these cryptosystems may become insecure when quantum computing is further developed [1]. One cryptosystem which may have potential to withstand attack by quantum computers is the McEliece cryptosystem which is based on the family of Goppa codes [1]. It is conjectured that this family of Goppa codes is near to random codes and a categorization has so far eluded researchers [2]. There have been many attempts to count the number of Goppa codes for fixed parameters and the author of this paper produced in 2004 a computer program which gives the best upper bound available today for the number of such codes [3]. Recent research has clearly shown that many Goppa codes become equivalent when extended by a parity check [4] and so the question of categorizing Goppa codes through their extended versions is now being proposed. As a first step, we investigate the possibility of counting extended Goppa codes using the tools which were developed for counting the nonextended versions. We begin by defining a degree irreducible Goppa code over of length in terms of a single field element of degree over . We then define the extended code . We give the well-known sufficient conditions on two elements of degree and , for the corresponding extended irreducible Goppa codes (the extended Goppa codes defined by and ) to be equivalent. Denoting the set of all elements of degree as , counting the cardinality of the set , and using the well-known conditions for equivalence we produce an upper bound on the number of inequivalent extended irreducible Goppa codes over of degree and length .

2. Background

Let be a power of a prime number; let be the field of order and its extension of order . In this paper all codes will be over . The family of Goppa codes was first introduced by Goppa in 1971 [5]. For our purposes we focus on irreducible Goppa, codes, and define irreducible Goppa codes as follows.

Definition 1. Let be irreducible of degree and let . Then the irreducible Goppa code is defined as the set of all vectors with components in which satisfy the condition

The polynomial is called the Goppa Polynomial. The set is called the Defining Set. Since is irreducible over the code is called an irreducible Goppa code. Since is of degree the code is called a Goppa code of degree . In this paper is always irreducible of degree over .

Remark 2. The definition we have given is specific for “irreducible Goppa codes.” In the literature, in general, a Goppa code is defined with Defining Set such that no element of is a root of the Goppa polynomial . Since, in this paper, is irreducible we take as large as possible; that is, . Note further that in fixing an order on the elements in we are implicitly putting an order on the coordinates of the Goppa code as the ordered elements in label the component positions in the codewords. Thus the length of the Goppa code is .

Next we define extended irreducible Goppa codes.

Definition 3. Let be a Goppa code of length over . Then the extended code is defined by

Remark 4. The extended code is often described as the code obtained from by adding a parity check to each codeword of .

It is shown in [6] that if is any root of the Goppa polynomial then is completely described by any root of and a parity check matrix is given by where . We may denote this code by .

Remark 5. denotes the same code as , where .

Remark 6. Note that in using this parity check matrix to define we are implicitly fixing an order on and, consequently, an order on the components of the codewords in the code .

Considering that any irreducible Goppa code can be defined by an element of degree over and, conversely, any such element of degree defines an irreducible Goppa code, we make the following definition.

Definition 7. The set is the set of all elements in of degree over .

Finally, as background material, we recall a sufficient condition which is well known for two extended irreducible Goppa codes to be equivalent.

Consider the maps defined on by for fixed , and   where , and .

For simplicity, where there is no confusion, we write for .

It is well known that if then is equivalent to (see [7]).

Remark 8. Note that in the definition of the scalars and are defined up to scalar multiplication. Hence we may assume that or if .

Remark 9. Note that the map can be broken up into the composition of two maps, namely, (1)the map defined on by and(2)the map , where denotes the Frobenius automorphism of leaving fixed.

We immediately justify the statement that is a map on .

Lemma 10. is a map defined on .

Proof. Suppose where is an element of degree strictly less that over (note that and so cannot have degree greater than over ) But this is impossible since and so the right hand side is an element of contradicting the fact that is an element of degree over .

In the light of the foregoing, we make two more definitions

Definition 11. Let denote the set of all maps .

Definition 12. Let denotes the set of all maps .

Lemma 13. together with the operation of composition of maps is a group.

Proof. Let . First we show that is closed under the operation of : We need to show . The Left Hand Side (LHS) is equal to and the Right Hand side (RHS) is equal to . Observe that the first and last terms of the LHS are the same as the first and last terms of the RHS and so our task now is to show that or equivalently show that ; that is .
But this is immediate from the fact that and .
Secondly, associativity follows from the associativity of mappings. Thirdly, observe that is the identity. Finally, given to find such that is a matter of solving the equation . See above. This is a matter of solving the four equations:(a)(b)(c)(d)in the four unknowns and . We know from linear algebra that this is always possible.

Lemma 14. together with the operation of composition of maps is a cyclic group of order .

Proof. Observe where is the Frobenius automorphism of leaving fixed. Since is the identity on the result follows.

3. Strategy to Count All Extended Irreducible Goppa Codes for Fixed ,  , and

We apply the following method to count the number of extended Goppa codes. Observe that each element defines an extended irreducible Goppa code over of degree of length and conversely each such extended Goppa code is defined by an element . We count the number of orbits in under the action of the group and this gives us an upper bound on the number of irreducible extended Goppa codes.

We first confirm the details that acts on .

Lemma 15. The group acts on .

Proof. We have already seen that . Clearly . That is merely the definition of .

The orbit containing is the set and we denote this set by . We denote the set of all orbits in under the action of by ; that is, . It follows from Group Theory and Lemma 15 that the set of all orbits in under the group action of partition the set and that partitions the set .

Theorem 16. For any , :

First remember that the elements are defined up to scalar multiplication so we may assume that, if , then (see Remark 8).(1)If , then w.l.o.g. and there are possibilities.(2)If , then we need to exclude the cases when .(a)Consider , and then exclude(i)the cases when and ,(ii)the cases when and .(b)Consider  . There are such cases. In each such case, for each (and there are of them) there is a unique solution for . Hence there are possibilities when .

So the total number of possibilities under item is .

Adding the possibilities under and we get .

Theorem 17. The number of inequivalent extended irreducible Goppa codes over of degree and length is less than or equal to .

Proof. Any extended irreducible Goppa code is defined by an element of . The elements of contained in the orbit define codes equivalent to . Since partitions and by Theorem 16 every set in has elements, we conclude that . This gives an upper bound on the number of inequivalent extended irreducible Goppa codes.

Remark 18. Note that this bound can be improved upon by further action of the group of Frobenius automorphisms. It is possible to show that acts on and then the number of orbits in under gives an improved upper bound on the number of inequivalent extended irreducible Goppa codes. This research is in progress.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author wishes to acknowledge part funding towards this research provided by MASI (MASAMU Advanced Study Institute) supported by the National Science Foundation (NSF) of the USA.