Enumeration of irreducible and extended irreducible Goppa codes

03/26/2019 ∙ by Kondwani Magamba, et al. ∙ 0

We obtain upper bounds on the number of irreducible and extended irreducible Goppa codes over GF(p) of length q and q+1, respectively defined by polynomials of degree r, where q=p^t and r≥ 3 is a positive integer.

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1 Introduction

It is well known that Goppa codes have few invariants and that the number of inequivalent Goppa codes grows exponentially with the length and dimension of the code [13]. These facts led to the exploitation of Goppa codes in the McEliece cryptosystem. There has been research work on the enumeration of extended Goppa codes but most of it has been confined to particular cases. The reference [11] gives an upper bound on the number of inequivalent extended irreducible binary quartic Goppa codes. Recently, an upper bound on the number of inequivalent extended irreducible Goppa codes of degree and length , where with the restriction that and be primes was found in [7]. Also, in 1978, Chen [3] gave upper bounds on the number of inequivalent irreducible and extended irreducible Goppa codes of length which are not tight. In this paper we derive upper bounds on the number of irreducible and extended irreducible Goppa codes which are tighter than the bounds found in [3]. Our approach takes advantage of recent work by various researchers on the action of and on the set of irreducible polynomials in , see [10] and [4]. This work sheds more light on the structure of Goppa codes and the strength of the McEliece cryptosytem.

2 Preliminaries

2.1 Irreducible and extended irreducible Goppa Codes

We begin by defining an irreducible Goppa code.

Definition 2.1.

Let be a positive integer, be a power of a prime number and be irreducible of degree . Let . Then an irreducible Goppa code

is defined as the set of all vectors

with components in which satisfy the condition

(1)

The polynomial is called the Goppa polynomial. Since is irreducible and of degree over , does not have any root in and the code is called an irreducible Goppa code of degree . In this paper is always irreducible of degree over .

It can be shown, see [3], that if is any root of the Goppa polynomial then is completely described and a parity check matrix is given by

(2)

where .

Next we give the definition of an irreducible Goppa code extended with an overall parity check.

Definition 2.2.

Let be an irreducible Goppa code of length . Then the extended code is defined by .

In this paper we take . That is, we consider irreducible and extended irreducible Goppa codes of length and , respectively.

2.2 Matrices of a given order in

Let be of order . We obtain a characterization of the elements of based on minimal polynomials, conjugacy classes and the order of each matrix. We focus our attention on elements of which fit our purpose. Elements of of a given order turn out to be useful in the enumeration of irreducible and extended irreducible Goppa codes.

It is well known that if the order of a matrix is then or and that the minimal polynomial of divides . Combining these facts, Proposition 4.2.2 in [1] and Lemma 2.1 in [7] we obtain the following theorem.

Theorem 2.1.

Let be of order and . Denote the minimal polynomial of by . Then

  1. If , then . Thus .

  2. If then and is conjugate with a matrix of the form where .

  3. If , and for some where

    is not a multiple of the identity matrix, then

    is conjugate with a matrix of the form where .

  4. If and where and where is irreducible over , then is conjugate with a matrix of the form .

2.3 Equivalence Classes

2.3.1 The set

An irreducible Goppa code can be defined by any root of its Goppa polynomial. As such the set of all roots of such polynomials is important and we make the following definition.

Definition 2.3.

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

2.3.2 Maps on

We define the following maps on .

Definition 2.4.

Let . Mappings of of Types 1, 2 and 3 are defined as follows:

  • where denotes the Frobenius automorphism of leaving fixed and ;

  • where .

  • , where .

It has been shown in [2] that the composition of Type 1 and Type 2 sends irreducible Goppa codes into equivalent irreducible Goppa codes and the composition of Type 1 and Type 3 maps sends extended irreducible Goppa codes into equivalent extended irreducible Goppa codes. Note that the “action” of Type 1 and Type 2 on was used in [3], Theorem 1, to find bounds on the number of equivalence classes of irreducible Goppa codes.

2.3.3 Groups arising from Type 1, Type 2 and Type 3 maps

In this section we define groups which arise from Type 1, Type 2 and Type 3 maps. The action of these groups on will help in counting irreducible and extended irreducible Goppa codes.

Definition 2.5.

Let denote the set of all maps . forms a group under the composition of mappings. It is the group of Frobenius automorphisms. It is shown in [12] that acts on .

Definition 2.6.

Let denote the set of all maps .

forms a group under the composition of mappings and is isomorphic to the group of affine linear transformations.

Observe that there is an action of the projective linear group on via the map where and , see [7].

2.3.4 Actions of , and

We first consider the action of the affine group on . For each , the action of on induces orbits denoted where , and called the affine set containing . We denote the set of all affine sets, , by . Since then . It can be shown that acts on the set , see [14]. We will then consider the action of on to obtain orbits in of . The number of orbits in under will give us an upper bound on the number of irreducible Goppa codes.

Next we consider the action of on . The action of on induces orbits denoted by where . We will refer to as a projective linear set. By Theorem 2.3 in [7], .

We denote the set of all projective linear sets in under the action of by . That is, . Observe that partitions the set and that acts on the set [12].

It is shown in [12] that each projective linear set in can be partitioned into affine sets. See the theorem below.

Theorem 2.2.

For where .

Observe that the sets and are different. and are both partitions of but .

We will use the actions of and on to find an upper bound on the number of extended irreducible Goppa codes. Firstly, we will apply the action of on to obtain projective linear sets . Then we will consider the action of on . The number of orbits in under the action of will give an upper bound on the number of extended irreducible Goppa codes. To find the number of orbits we will use the Cauchy-Frobenius counting theorem, see [6].

The group is cyclic of order . In analysing the action of we will make use of the fact that subgroups of are of the form where . Clearly, .

Next we note that if then the number of projective linear sets in is . That is, there is just one projective linear set containing all elements of . We put the result in a theorem.

Theorem 2.3.

When the set consists of just one projective linear set, that is, for any .

Corollary 2.1.

All extended irreducible Goppa codes of length with are equivalent.

The result in Corollary 2.1 is well known, for example see [3] and [9].

Now, suppose that where . We see that if is the smallest positive integer such that , then since . Observe that implies that . Thus the order of is .

3 Enumeration of irreducible Goppa codes

We count irreducible Goppa codes by using the tools developed in [14] where an upper bound on the number of irreducible Goppa codes of degree and length is given. The upper bound is found by counting the number of affine sets in fixed under the action of subgroups and then applying the Cauchy-Frobenius Theorem. As opposed to [14], where the upper bound is given in the form of an algorithm, we obtain analytic formulas for the upper bound.

Suppose is fixed by . Then . So we have for some . Thus where . Now, so is a matrix of order . We divide our analysis according to whether , and .

3.1 Affine sets fixed under when

If then and and , see Theorem 2.1. Now if and only if contains elements of . We know that this is true if and only if . It is easy to see that every affine set is fixed under . By Corollary 3.5 in [13] the number of affine sets fixed under is

Example 3.1.

Let and . There are 17 affine sets in and all of them are fixed under .

3.2 Affine sets fixed under when

Now suppose that is fixed under where . Then we have that and by Theorem 2.1, we may take of order . Thus and as such satisfies an equation of the form

(3)

where is of order .

Next we note that the factorization of was considered in [4]. Using our notation and Theorem 4 in [4] we obtain the following result.

Theorem 3.1.

Let and suppose and that . Let and be the set of all roots of irreducible factors of degree in the factorization of , where is of order . If then . Otherwise

Observe that since is of order there are conjugacy classes in in this case and a polynomial arising from a representative of each conjugacy class contributes roots to . Thus there are roots which lie in . Note that this closed formula is a partial answer to Remark 4.5 in [13].

Example 3.2.

Let and . There are 672 irreducible factors of degree 6 in the factorization of where is a primitive element of . Hence there are 8,064 roots of polynomials of the form which lie in where is of order .

Next we find the number of affine sets fixed under . We know that if where and then there are roots of the equations of the form which lie in . By [13, Theorem 4.4], each polynomial has roots in exactly one . Thus we have proved the following theorem.

Theorem 3.2.

Suppose where and . Then there are affine sets fixed by .

3.3 Affine sets fixed under when

Suppose that is fixed under where . Then we have that and by Theorem 2.1, we may take where . Thus and as such satisfies an equation of the form

(4)

Observe that if hence then we will take since we can show, by direct computation, that the order of is . As such, we have and we may assume that satisfies an equation of type

(5)

Next we note that the factorization of polynomials of the form was considered in [4]. Using our notation and [4, Theorem 2] we obtain the following result.

Theorem 3.3.

Suppose where and . Let , and be the set of roots of irreducible factors of degree in the factorization of . If , then . Otherwise

It is easy to see that if satisfies Equation 5 then all the elements of the set also satisfy 5 while the remaining elements in do not. Hence if satisfies equation 5 then contains precisely roots of Equation 5. We have proved the following theorem.

Theorem 3.4.

If then there are affine sets fixed by where .

Corollary 3.1.

If and then and there is one affine set fixed by .

Example 3.3.

Let and . Then and there are affine sets fixed by .

Now suppose that . We consider . Thus, we have . So and so the order of divides . We know that matrices of order do not exist. So we consider matrices of order . We obtain an equation of the form and all roots of this equation lie in and not in . We have the following theorem.

Theorem 3.5.

If then and there is no polynomial of degree in the factorization of where .

Next suppose that where is some other divisor of . Then and . Then . If we take of order then we have then satisfies Equation 3. Also if we take of order then we have then satisfies Equation 5 and this is not possible. So there is no irreducible polynomial of degree in the factorization of in this case. We have the following result.

Theorem 3.6.

There is no affine set fixed under if and or where is some other divisor of .

Example 3.4.

Let and . The subgroups and where and respectively do not fix any affine set.

Putting the results together, we have proved the following:

Theorem 3.7.

With the notation we have established:

  1. There are affine sets fixed by .

  2. There are affine sets fixed by if .

  3. If then

    1. there are affine sets fixed by if .

    2. there is affine set fixed by if .

Remark 3.1.

This result agrees with [13, Theorem 4.13] for . Our main contribution here is that we have found closed formulas for and .

4 Counting extended irreducible Goppa codes

4.1 Strategy for counting extended irreducible Goppa codes

We will use the actions of and on to find the maximum number of extended irreducible Goppa codes. Firstly, we will apply the action of the group on to obtain projective linear sets . Then we will consider the action of on . The number of orbits in under the action of will give us an upper bound on the number of extended Goppa codes.

Recall that a projective linear set can be decomposed as where . Observe that if a projective linear set is fixed under then acts on and partitions this set of affine sets. We see that some projective linear sets fixed under contain fixed affine sets and there is also a possibility of having a fixed projective linear set that does not contain fixed affine sets. We will consider the following possibilities: ; and ; ; and where . We will discuss each of the four cases in separate sections.

4.2 Projective linear sets fixed when

If then and , see Theorem 2.1. Now if and only if contains elements of . That is, . By Section 3.1 we know that the number of affine sets fixed under where is .

By an argument similar to the one in [7, Section 4.3.2], we find that the number of projective linear sets fixed under is .

4.3 Projective linear sets fixed when and

Suppose and that . Theorem 3.2 gives the number of affine sets fixed by in this case.

Next we find the number of projective linear sets fixed under . We will do this by finding how many affine sets fixed under lie in each fixed projective linear set.

Suppose is fixed under . Then acts on , a set of affine sets. partitions this set of affine sets. The possible lengths of an orbit are and factors of . Now, since then . We claim that each fixed under contains 2 affine sets which are fixed under . Observe that if is prime then we are done. So we will suppose that is composite. That is, we can find non-negative integers such that where , , and . Note that we can always choose a factor , such that is of prime order and fixes . Now, an fixed under contains two fixed affine sets so it follows that an fixed under also contains two fixed affine sets. Thus if then each fixed under contains 2 fixed affine sets.

We have the following theorem.

Theorem 4.1.

Let and . The number of projective linear sets fixed under is where is defined in Theorem 3.2.

4.4 Projective linear sets fixed when

In this section we obtain the number of projective linear sets fixed when . By Theorem 3.4, there are affine sets fixed by when . We will do this by finding how many affine sets fixed under lie in a projective linear set fixed under .

We claim that each of the in fixed under contains precisely one affine set which is fixed under . It suffices to show that cannot contain two affine sets which are fixed under . Without loss of generality, suppose is fixed under . Recall that . We show that none of the affine sets after in the above decomposition of is fixed under . This is done by showing that no element in any of these affine sets satisfies Equation 5. It is sufficient to show that no element in satisfies . A typical element of has the form and substituting this into we get , since is an element of degree over . We conclude that is not fixed under and in fact is the only affine set in fixed under . It follows that the number of projective linear sets in which are fixed under where is . Thus we have proved the following.

Theorem 4.2.

If and , then the number of projective linear sets fixed under is the same as the number of affine sets fixed under .

4.5 Projective linear sets fixed by where and

Suppose that is fixed by where , and . Then we have that , where and . Then, by Theorem 2.1, is conjugate with a matrix of the form where the minimal polynomial of , , is an irreducible quadratic polynomial over . Without loss of generality, we will take where, as above, is an irreducible quadratic polynomial over .

Now, implies that satisfies an equation of the form

(6)

It is clear from the foregoing discussion that in order to find the number of projective linear sets fixed under we need to find roots of Equation 6 which lie in . Observe that in this case there is no affine set fixed in the decomposition of .

Note that there are polynomials of the form each of which corresponds to a representative of a conjugacy class of matrices of order where the minimal polynomials of such matrices are irreducible quadratic polynomials over , see Theorem 2.2 in [7].

We now consider the factorization of . We begin by considering the factorization of where , and is even. Note that the assumption that where is even implies that the characteristic of

is odd.

Suppose that is a root of where is even. Then where is an integer. Thus we have . So , where . Now, since , without loss of generality we can take since the only elements such that are and . Thus satisfies an equation of the form . By the argument in Section 4.2 there are no irreducible polynomials of degree in the factorization of