Algorithmic aspects of elliptic bases in finite field discrete logarithm algorithms

TL;DR

This paper explores the practical use of elliptic bases in finite field discrete logarithm algorithms, proposing a new model that could lead to efficient heuristic methods in small characteristic fields.

Contribution

It introduces a new elliptic curve model for elliptic bases, enabling adaptation of existing Frobenius techniques for heuristic discrete logarithm algorithms.

Findings

Preliminary experiments suggest comparable performance to current methods.

The new model simplifies adaptation of Frobenius-based techniques.

Potential for practical heuristic algorithms in small characteristic fields.

Abstract

Elliptic bases, introduced by Couveignes and Lercier in 2009, give an elegant way of representing finite field extensions. A natural question which seems to have been considered independently by several groups is to use this representation as a starting point for small characteristic finite field discrete logarithm algorithms. This idea has been recently proposed by two groups working on it, in order to achieve provable quasi-polynomial time for discrete logarithms in small characteristic finite fields. In this paper, we don't try to achieve a provable algorithm but, instead, investigate the practicality of heuristic algorithms based on elliptic bases. Our key idea, is to use a different model of the elliptic curve used for the elliptic basis that allows for a relatively simple adaptation of the techniques used with former Frobenius representation algorithms. We haven't performed any record computation with this new method but our experiments with the field F 3 1345 indicate that switching to elliptic representations might be possible with performances comparable to the current best practical methods.

Tables

Table 1. Table 1: Usual functions of 𝔽 q ​ [ 𝒞 ] subscript 𝔽 𝑞 delimited-[] 𝒞 {\mathbb{F}}_{q}[{\mathscr{C}}] and their corresponding height in ℰ ℰ {\mathscr{E}} .

Table 2. Table 2: Functions appearing on both side of the diagram, and their corresponding heights in ℰ ℰ {\mathscr{E}} . A 𝐴 A and B 𝐵 B are linear combinations of monomials from ℳ t subscript ℳ 𝑡 \mathcal{M}_{t} .

Equations

$$g^x=h.$$

$${\theta }^q=\frac{h_0(\theta )}{h_1(\theta )},$$

$$\prod _{f\in \mathcal{F}}{f}^{e_f}=1.$$

$$\sum _{f\in \mathcal{F}}{e}_f.{\log }_{g}f=0.$$

$$h\prod _{f\in \mathcal{F}}{f}^{e_f}=1.$$

$$\#(\mathcal{E}/{\mathbb{F}}_q)=\mu k,$$

$$\pi (F)=F+P_1,$$

$$\begin{array}[]{rcc}\Phi:{\mathscr{E}}&\mapsto&\overline{{\mathbb{F}}}_{q}^{3}\\ Q&\mapsto&(x_{Q-P_{1}},x_{Q},x_{Q+P_{1}})\end{array}$$

$$A^{q}B-A{B}^q=\prod _{\alpha \in {\mathbb{P}}_1({\mathbb{F}}_q)}(A-\alpha B)$$

$$A(V,W)B(U,V)-A(U,V)B(V,W)$$

$$D=\sum _{\mathfrak{p}\in {\Sigma }_{\mathbb{K}(\mathcal{C})}}{n}_{\mathfrak{p}}(\mathfrak{p}),$$

$$D=\sum _{P\in \mathcal{C}/{\overline{\mathbb{F}}}_q}{n}_P(P),$$

$$\text{mbox}Deg(D)=\sum _{\mathfrak{p}\in {\Sigma }_{\mathbb{K}(\mathcal{C})}}{n}_{\mathfrak{p}}\text{mbox}Deg(\mathfrak{p})=\sum _{P\in \mathcal{C}/{\overline{\mathbb{F}}}_q}{n}_P.$$

$$\sum _{i=0}^{d-1}{\pi }^i(Q)-d(\mathcal{O}).$$

$$D_{0}=\sum_{\begin{array}[]{c}\mathfrak{p}\in\Sigma_{{\mathbb{K}}({\mathscr{C}})}\\ n_{\mathfrak{p}}\geq 0\end{array}}n_{\mathfrak{p}}(\mathfrak{p})\quad\mbox{and}\quad D_{\infty}=\sum_{\begin{array}[]{c}\mathfrak{p}\in\Sigma_{{\mathbb{K}}({\mathscr{C}})}\\ n_{\mathfrak{p}}<0\end{array}}-n_{\mathfrak{p}}(\mathfrak{p}).$$

$$\begin{array}[]{cccc}\Xi:&{\mathbb{K}}({\mathscr{C}})^{*}&\mapsto&\mbox{Div}({\mathscr{C}})\\ &f&\mapsto&\displaystyle\Xi(f)=\sum_{\mathfrak{p}\in\Sigma_{{\mathbb{K}}({\mathscr{C}})}}\mbox{\textscriptv}_{\mathfrak{p}}(f)\,\mathfrak{p}.\end{array}$$

$$\Xi (f)=\Xi (f{)}_0-\Xi (f{)}_{\infty }.$$

$$\begin{array}[]{cccc}\Psi:&\mbox{Princ}({\mathscr{C}})&\mapsto&{\mathbb{L}}/{\mathbb{K}}^{*}\\ &D&\mapsto&\Psi(D)=f(F),\end{array}$$

$$\Psi (D)=\Psi (hD{)}^{1/h}.$$

$$\pi (F)=F+P_1,$$

$$S_3(x_{Q_1},x_{Q_2},x_{Q_3})=0\iff \exists (e_1,e_2,e_3)\in \{-1,1{\}}^3,e_1{Q}_1+e_2{Q}_2+e_3{Q}_3=\mathcal{O}.$$

$$\begin{array}[]{rcc}\Phi:{\mathscr{E}}&\mapsto&\overline{{\mathbb{F}}}_{q}^{3}\\ Q&\mapsto&(x_{Q-P_{1}},x_{Q},x_{Q+P_{1}})\end{array}$$

$$S_3(U,V,x_1)=0,\frac{S_3(U,V,x_1)-S_3(V,W,x_1)}{U-W}=0,S_3(U,W,x_2)=0.$$

$$\pi (F)=F+P_1.$$

$${\mathcal{D}}_e=\sum _{i=1}^d{\pi }^i(Q)-d(\mathcal{O}),$$

$$\Psi ({\mathcal{D}}_e)={\left(f_{{\mathcal{D}}_e}\left(F-\mathcal{O}\right)\right)}^{1/N_D}.$$

$$U^q=Vand{V}^q=W$$

$$\Phi (F)=[{\theta }^{q^{k-1}},\theta ,{\theta }^q]$$

$$\Psi \circ \Xi \circ {\Phi }^{\ast }(U^q)=({\theta }^{q^{k-1}}{)}^q=\theta =\Psi \circ \Xi \circ {\Phi }^{\ast }(V)\mathrm{mod}{\mathbb{F}}_q^{\ast }$$

$$\Psi \circ \Xi \circ {\Phi }^{\ast }(V^q)={\theta }^q=\Psi \circ \Xi \circ {\Phi }^{\ast }(W)\mathrm{mod}{\mathbb{F}}_q^{\ast }.$$

Full text

This idea has been recently proposed by two groups working on it, in order to achieve provable quasi-polynomial time for discrete logarithms in small characteristic finite fields.

In this paper, we don’t try to achieve a provable algorithm but, instead, investigate the practicality of heuristic algorithms based on elliptic bases. Our key idea, is to use a different model of the elliptic curve used for the elliptic basis that allows for a relatively simple adaptation of the techniques used with former Frobenius representation algorithms.

We haven’t performed any record computation with this new method but our experiments with the field ${\mathbb{F}}_{3^{1345}}$ indicate that switching to elliptic representations might be possible with performances comparable to the current best practical methods.

1 Introduction

The discrete logarithm problem (DLP) is a fundamental problem underlying the security of many cryptographic systems. Given $G$ a finite cyclic group denoted multiplicatively and $g$ a generator of the group, solving the discrete logarithm problem in $G$ means being able, for any arbitrary element $h\in G$, to find an integer $x$ such that:

[TABLE]

The integer $x$ is defined modulo the order of $G$ and is called the discrete logarithm of $h$.

Among the groups considered for cryptographic use, we find the multiplicative group of finite fields. There is a long history of algorithms to address this problem that we do not recall here. In the case of “small” characteristic fields, tremendous progress was made in 2013 and the years after. They are surveyed in [JP16]. This led to extreme computational improvements and to two flavors of heuristic quasi-polynomial time algorithms. One of the fundamental tools used to achieved this result is a special representation of a finite field extension above ${\mathbb{F}}_{q}$, called the Frobenius representation which requires an element $\theta$ satisfying a relation of the form:

[TABLE]

where $h_{0}$ and $h_{1}$ are co-prime polynomials with very low degree.

A widely believed heuristic assumption is that any finite field extension can be represented in that way, unless one of the known obstructions applies. These known obstructions are that $h_{0}$ and $h_{1}$ can both have degree $\leq 1$. Furthermore, it is clearly not possible to represent an extension of degree higher than $q+\deg(h_{1}).$ In practice, finding such a representation via exhaustive search among suitable polynomials is a trivial matter. However, proving this assumption seems to be a difficult task.

As a consequence, it is natural to turn to different field representations which can provably be constructed and try to adapt the discrete logarithm algorithms to work with them. Elliptic bases, also called elliptic periods [CL09], form a natural candidate for this purpose. Their use for discrete logarithms was independently considered by several groups. We are aware of two attempts which have been made public. In 2016, in his master’s thesis [Lid16], Lido proposed a discrete logarithm based on elliptic representations, using a descent method made of two halves. His presentation states a theorem concerning one half of the descent and a conjecture for the other half. On June 26th, 2019, Kleinjung and Wesolowski released a preprint [KW19] on the eprint archive announcing a fully provable quasi-polynomial time algorithm based on elliptic representation. The next day, Schoof gave a talk at the conference NutMiC 2019 presenting the work of Lido. He also sent us a not yet publicly available document [Lid19] that extends [Lid16] and contains a theorem announcing an algorithm to compute logarithms in a finite field ${\mathbb{F}}$ in provable time $(\log|{\mathbb{F}}|)^{O(\log\log|{\mathbb{F}}|)})$.

The result announced in [KW19], has a different form. It states that discrete logarithms in ${\mathbb{F}}_{p^{n}}$ can be computed in provable time $(pn)^{2\log_{2}{n}+O(1)}.$ Both forms affect a very large range of characteristic. Indeed, until now, the best provable discrete logarithm algorithms for finite fields had complexity $L(1/2)$. They are thus outperformed as soon as $p<L_{p^{n}}(1/2-\epsilon)$, for an arbitrary small $\epsilon>0$.

In this paper, we present the work we independently performed on a similar idea. However, we do not consider the provable aspects. Instead, we focus more on the algorithmic aspects of heuristic variants of elliptic representation discrete logarithm methods. Our formulation differs in many details. As such, it might shed a different light on the topic and help the reader to study the theoretical breakthrough on provable algorithms. As of now, our proposal remains slightly inferior to the method in [JP14], the fastest currently known (heuristic) method to compute discrete logarithms in small characteristic. However, it gets very close to it, while leaving room for improvement in the analysis.

Quick overview of Frobenius representation algorithms.

The common strategy of all heuristic algorithms in the Frobenius representation family is the following.

• •

A preliminary step, often called the representation phase, we construct a representation of an extension of degree $k$ of ${\mathbb{F}}_{q}$, assuming that:

Heuristic**.**

There exists a small divisor $n$ of $k$ (possibly one), two coprime polynomials $h_{0}$ and $h_{1}$ in ${\mathbb{F}}_{q^{n}}[X]$ of low degree (often of degree at most two) and an irreducible factor $I$ of $h_{1}X^{q}-h_{0}$ having degree $k/n$.

This gives the representation of the target field ${\mathbb{F}}_{q^{k}}$ as ${\mathbb{F}}_{q^{n}}[X]/(I)$.

• •

Relation generation. Set a small set ${\mathcal{F}}$ of particular elements, small in some sense, such that ${\mathcal{F}}\subset{\mathbb{F}}_{q^{k}}$. Collect vectors $(e_{f})_{f\in{\mathcal{F}}}$ such that:

[TABLE]

These vectors create linear relations between the discrete logarithm of factor base elements since:

[TABLE]

• •

Linear algebra. Find a solution of the above system. It gives the discrete logarithms of elements of ${\mathcal{F}}$ (maybe not all). Note that the number of non zero coefficients $e_{f}$ per relation is very low, let’s say a constant $\lambda$. Then it’s possible to compute this phase with sparse linear algebra: if $|{\mathcal{F}}|$ is an upper bound on the number of unknowns, then algorithms such as Wiedmann algorithm find solutions in $O(\lambda|{\mathcal{F}}|^{2})$ operations.

• •

Individual discrete logarithm. Also known as the descent phase, here the aim is to find a relation only involving the target and factor base elements:

[TABLE]

Since the discrete logarithms of ${\mathcal{F}}$ are known, we can reconstruct $\log_{g}h$ by computing $\log_{g}h=-\sum_{f\in{\mathcal{F}}}{e_{f}}.\log_{g}f$.

Because of potential dependencies, it’s not easy to see when we have enough relations, but in practice having a few more relations than factor base elements is usually sufficient.

To turn this into a provable algorithm, it is not only necessary to prove that the finite field representation can be constructed, it is also required to change the relation generation and linear algebra. Essentially, one follows the approach of [Pom87], which consists in decomposing plenty of elements of the form $g^{a}h^{b}$ over the factor base. Linear algebra can then be used to eliminate the contribution of the factor base by combining the equations which leads to a random identity of the form $g^{A}h^{B}=1$ from which the discrete logarithm can be recovered. Unfortunately, this is much less practical. In particular, there no longer is an individual logarithm phase in this approach and the full computation has to be restarted from scratch for every discrete logarithm computation in the same field.

Assuming that the field representation exists, this makes the computation of discrete logarithms provable. For Frobenius representation algorithms, this type of provable approach (assuming that the field representation is given) has been studied in [GKZ18, KW18, GJ18].

Overview of the goal with elliptic representations.

Let’s ${\mathbb{F}}_{p^{k}}$ be the target finite field in which we want to compute discrete logarithms. To simplify exposition, we assume $p\geq 5$, since the equation of the curves needs to be chosen differently in characteristic $2$ or $3$. We summarize our construction as follows:

Representation. Create an elliptic curve ${\mathscr{E}}$ over ${\mathbb{F}}_{q}$ ($q$ being a power of $p$) such that:

[TABLE]

with $\mu$ a natural integer. If $k$ is square, there necessarily exists a $k$-torsion point $P_{1}\in{\mathscr{E}}$. Otherwise, for every prime $\ell$ dividing $k$, let $\ell^{e_{\ell}}$ be the largest power of $\ell$ dividing $k$. For each $\ell$ with $e_{\ell}>1$, it might be necessary to change ${\mathscr{E}}$ to an isogenous curve by applying a sequence of $\ell$-isogenies in order to guarantee that a point of order $\ell^{e_{\ell}}$ exists. These changes to ${\mathscr{E}}$ suffice to guarantee the existence of $P_{1}$.

Finally, find a point $F\in{\mathscr{E}}$ such that:

[TABLE]

where $\pi$ is the Frobenius action in the field ${\mathbb{F}}_{q}$. In particular, if we note $F=(\theta,\tau)$ then we can write ${\mathbb{F}}_{q^{k}}$ as ${\mathbb{F}}_{q}[\theta,\tau]$.

This almost gives the desired representation of the target field. Indeed, ${\mathbb{F}}_{q}[\theta]$ is either ${\mathbb{F}}_{q^{k}}$ or ${\mathbb{F}}_{q^{k/2}}$. In the sequel, we assume that ${\mathbb{F}}_{q}[\theta]={\mathbb{F}}_{q^{k}}$, multiplying $k$ by $2$ if necessary.

Then, since ${\mathbb{F}}_{p^{k}}$ is a subfield of both ${\mathbb{F}}_{q^{k}}$ or ${\mathbb{F}}_{q^{2k}}$, we see that computing discrete logarithms in ${\mathbb{F}}_{q}[\theta]$ is sufficient to achieve the desired goal. 2. 2.

Commutative Diagram. We now define the full representation we want to use from a curve ${\mathscr{C}}$ in 3 dimensions obtained as the image of the following rational map:

[TABLE]

At first, this might seem to be a strange model of an elliptic curve. However, the intuition is that, with this model, the image of $F$ is a point with a really useful property: taking the Frobenius of one of its coordinate leads to the following one. In other words, if $\Phi(F)$ is seen as a point of ${\mathscr{C}}$ in the affine space ${\mathbb{F}}_{q}[U,V,W]$ then it lies on the intersection of the surfaces defined by the two equations $U^{q}=V$ and $V^{q}=W$. This property is at the core of our method for creating relations. Starting from $A$ and $B$ two polynomials in ${\mathbb{F}}_{q}[U,V]$, we construct two big polynomials:

[TABLE]

in one hand, and:

[TABLE]

in the other hand. Each polynomial can be considered as an element of the function field ${\mathbb{F}}_{q}({\mathscr{C}})$. Writing the divisor associated to each side, we can write down an equality between the image of each divisor in ${\mathbb{F}}_{q^{k}}$. For polynomial themselves, the image is simply obtained by evaluation at $\Phi(F)$. This can be extended to divisors as explained in Section 2.2. The equality of the two sides comes from the Frobenius relations between the coordinates of $\Phi(F)$. 3. 3.

Relation collection. We sieve on pairs of polynomials $(A,B)$ such that $A=g_{1}+\alpha g_{3}$ and $B=g_{1}+\beta g_{2}+\gamma g_{3}$ where $\alpha,\beta,\gamma\in{\mathbb{F}}_{q}$ and $g_{1},g_{2}$ and $g_{3}$ are given polynomials constructed by linear combination of the monomials $U,V,UV$ and $1$. Set the factor base ${\mathcal{F}}$ as all the divisors of ${\mathscr{E}}$ with height at most $3$. On one side $\prod_{\alpha\in\mathbb{P}_{1}({\mathbb{F}}_{q})}(A-\alpha B)$ will always lead to divisors that can be written as sum of divisors of ${\mathcal{F}}$, and on the other side $A(V,W)B(U,V)-A(U,V)B(V,W)$ have a low enough height so that the probability that the related divisor $\mathcal{D}$ splits in factor base elements is high enough to get as many relations we want.

Since we only need three degrees of freedom from the four monomials $U,V,UV$ and $1$, we choose $g_{1},g_{2}$ and $g_{3}$ all going through a common point. This nicely reduces the degree of the divisors appearing in the decomposition of the terms $A-\alpha B$. This is essential in making the probability of success during the relation collection phase good enough. 4. 4.

**Linear algebra and individual logarithm. ** Thanks to the action of Frobenius, we can reduce the size of factor base by a factor $k$. In other words, this reduces the effective size of the factor base ${\mathcal{F}}$ to $O(q^{3}/k)$. As a consequence, the cost of the sparse linear algebra, with $O(q)$ entries per equation, is of $O(q^{7}/k^{2})$ arithmetic operations. Note that, when $k$ is chosen close to $q$, this matches the $O(q^{5})$ asymptotic complexity obtained for this step in [JP14]. 5. 5.

Descent Phase. Finally, we need a descent phase to conclude. We give the necessary tools to adapt existing methods in this context.

Outline.

Section 2 gives algebraic preliminaries for this work.

In this paper, we focus on the algorithmic aspects and describe our practical elliptic Frobenius algorithm, that can be helpful to fully understand the general idea. Being aimed at practicality, this algorithm is heuristic. From a performance analysis point of view, our heuristic approach almost achieves the same efficiency as the best pre-existing practical algorithm for DLP in small characteristic finite fields. Almost, because there is a glitch in the analysis of the fast computation of the extended factor base. However, despite this glitch, we were able to implement and use the elliptic representation approach to compute logarithms of an extended factor base for the finite field ${\mathbb{F}}_{3^{1345}}.$

We highlight the heuristics we use as far as we can, in order to clarify the difference with the provable algorithm of [KW19] or [Lid19].

In Section 3 we give our variation on the representation of the target finite field while Section 4 details how to get relations. Finally Section 5 deals with factor base extension and with the individual logarithms phase.

2 A Refresher on the Function Field Sieve Machinery

Many concepts used here originate from the Function Field Sieve (FFS) algorithm [AH99]. The aim of this first section is not to describe FFS itself, but to describes these concepts in a slightly more general form than the original description of Adleman and Huang article.

2.1 Algebraic preliminaries

Let ${\mathbb{K}}={\mathbb{F}}_{q}$ denote a finite field. Let ${\mathscr{C}}$ be a non-singular curve in the $n$-dimensional projective space ${\mathbb{P}}_{n}(\overline{{\mathbb{F}}}_{q})$ defined over ${\mathbb{K}}$ and $\pi$ denote the Frobenius map on ${\mathbb{P}}_{n}(\overline{{\mathbb{F}}}_{q})$. We let ${\mathbb{K}}({\mathscr{C}})$ denote the function field of ${\mathscr{C}}$ over ${\mathbb{K}}$. More details can be found in [CFA*+*05] if needed. A discrete valuation on ${\mathbb{K}}({\mathscr{C}})$ is a map \textscriptv from ${\mathbb{K}}({\mathscr{C}})$ to ${\mathbb{Z}}$ such that for all $x,y\in{\mathbb{K}}({\mathscr{C}})$ we have:

$\mbox{\textscriptv}(xy)=\mbox{\textscriptv}(x)\,\mbox{\textscriptv}(y);$ 2. 2.

$\mbox{\textscriptv}(x+y)\geq\min(\mbox{\textscriptv}(x),\mbox{\textscriptv}(y));$ 3. 3.

$\mbox{\textscriptv}(x+y)=\min(\mbox{\textscriptv}(x),\mbox{\textscriptv}(y))$ when $\mbox{\textscriptv}(x)\neq\mbox{\textscriptv}(y)$.

We define an equivalence relation between valuations by saying that two valuations \textscriptv and $\mbox{\textscriptv}^{\prime}$ are equivalent whenever there exists a non zero rational constant $\alpha$ such that for all $x\in{\mathbb{K}}({\mathscr{C}})$, we have $\mbox{\textscriptv}^{\prime}(x)=\alpha\,\mbox{\textscriptv}(x).$ We recall that a place of ${\mathbb{K}}({\mathscr{C}})$ is an equivalence class of discrete valuations of ${\mathbb{K}}({\mathscr{C}})$ which are trivial on ${\mathbb{K}}$. The set of places of ${\mathbb{K}}({\mathscr{C}})$ is denoted by $\Sigma_{{\mathbb{K}}({\mathscr{C}})}$. In every place $\mathfrak{p}$, there exists a unique valuation whose value group is ${\mathbb{Z}}$, it is called the normalized valuation of $\mathfrak{p}$ and denoted $\mbox{\textscriptv}_{\mathfrak{p}}.$

We recall that, for a non-singular curve ${\mathscr{C}}$, there is a one-to-one correspondance between places of ${\mathbb{K}}({\mathscr{C}})$ and Galois orbit of points on ${\mathscr{C}}$. The degree of a place $\mathfrak{p}$ is the number of points in the corresponding orbit, we denote it by $\mbox{Deg}(\mathfrak{p})$.

The divisor group $\mbox{Div}({\mathscr{C}})$ of ${\mathscr{C}}$ (over ${\mathbb{K}}$) is defined as the free abelian group over $\Sigma_{{\mathbb{K}}({\mathscr{C}})}$. An element $D$ of $\mbox{Div}({\mathscr{C}})$ is expressed as:

[TABLE]

where each $n_{\mathfrak{p}}\in{\mathbb{Z}}$ and $n_{\mathfrak{p}}=0$ for all but finitely many places $\mathfrak{p}$. Since each place corresponds to a Galois orbit of points, a divisor $D$ can also be given in the alternative form:

[TABLE]

where each $n_{P}\in{\mathbb{Z}}$, $n_{P}=0$ for all but finitely many points and $n_{P}=n_{Q}$ if $P$ and $Q$ belong to the same Galois orbit of points. A divisor $D$ is said to be prime when $D=(\mathfrak{p})$ for a place $\mathfrak{p}\in\Sigma_{{\mathbb{K}}({\mathscr{C}})}$.

The degree of a divisor $D$ is defined as:

[TABLE]

In this paper, a degree-[math] divisor that is the difference between a prime divisor and the right number of times the point at infinity $\mathcal{O}$ is defined as an elementary divisor. In particular, any elementary divisor associated to a point $Q\in{\mathscr{C}}/{\mathbb{F}}_{q^{d}}$ is a divisor of the form:

[TABLE]

A divisor $D$ is called effective when $n_{\mathfrak{p}}\geq 0$ for all $\mathfrak{p}$. Any divisor $D$ can be uniquely written as a difference of two effective divisors in the form $D=D_{0}-D_{\infty}$, where:

[TABLE]

The degree map from $\mbox{Div}({\mathscr{C}})$ to ${\mathbb{Z}}$ is a group morphism. Its kernel is denoted $\mbox{Div}_{0}({\mathscr{C}})$ and called the group of degree-[math] divisors of ${\mathscr{C}}$, it is a subgroup of $\mbox{Div}({\mathscr{C}})$.

We define the map111Other authors often use the notation div but we do not want the map to be mistaken with the group. $\Xi$ that sends an element $f\in{\mathbb{K}}({\mathscr{C}})^{*}$ to a divisor in the following way:

[TABLE]

A divisor associated to a function in the above way is called a principal divisor. The image of $\Xi$, i.e. the set of all principal divisors, is denoted $\mbox{Princ}({\mathscr{C}})$. All principal divisors have degree [math] and $\mbox{Princ}({\mathscr{C}})$ is a subgroup of $\mbox{Div}_{0}({\mathscr{C}})$. Every principal divisor can also be written as a difference of effective divisors as:

[TABLE]

The places (or points) that occur in $\Xi(f)_{0}$ or $\Xi(f)_{\infty}$ are respectively called the zeroes or poles of $f$. Note that for two functions $f$ and $g$ of ${\mathbb{K}}({\mathscr{C}})^{*},$ we have $\Xi(f)=\Xi(g)$ if and only if there exists an element $\alpha\in{\mathbb{K}}^{*}$ such that $g=\alpha\,f$.

Since $\mbox{Princ}({\mathscr{C}})$ is a subgroup of $\mbox{Div}_{0}({\mathscr{C}})$, we can form the quotient group, which is called the Picard group (or divisor class group) of ${\mathscr{C}}$ and denoted $\mbox{Pic}_{0}({\mathscr{C}})$. Two divisors have the same representative in the Picard group if and only if their difference is principal.

2.2 A tool from FFS: sending a degree-0 divisor into a finite field

With these algebraic objects in hand, we can now introduce the main tool that we need to import from the Function Field Sieve. Let ${\mathbb{K}}={\mathbb{F}}_{q}$ be a finite field, ${\mathbb{L}}={\mathbb{F}}_{q^{k}}$ be a finite extension of ${\mathbb{K}}$ and ${\mathscr{C}}$ be as before a non-singular curve defined over ${\mathbb{K}}$. Let $F$ be a point of ${\mathscr{C}}/{\mathbb{L}}$ such that the coordinates of $F$ generate ${\mathbb{L}}$ over ${\mathbb{K}}$. Given an arbitrary element $f\in{\mathbb{K}}({\mathscr{C}})^{*}$ which does not have $F$ as a pole, we can evaluate $f$ at $F$ and obtain a value in ${\mathbb{L}}$. As explained in [AH99], this process can be generalized from functions to a large subset of divisors of ${\mathscr{C}}$. Clearly, since $\alpha\,f$ and $f$ have the same divisor for any $\alpha\in{\mathbb{K}}$, we need to proceed with care.

First, we define a map $\Psi$ from $\mbox{Princ}({\mathscr{C}})$ to ${\mathbb{L}}/{\mathbb{K}}^{*}$ defined as follows:

[TABLE]

where $f$ is an arbitrary function such that $\Xi(f)=D$. Since the result is only considered up to multiplication by an arbitrary constant in ${\mathbb{K}}^{*}$, it is independent of the particular choice of $f$.

To generalize to more divisors, we now consider a degree-[math] divisor $D$ together with an integer $h\in{\mathbb{N}}^{*}$ such that $hD$ is principal and $h$ is coprime to the order of ${\mathbb{L}}^{*}/{\mathbb{K}}^{*}=(q^{k}-1)/(q-1)$. For such a divisor, we extend the definition by letting:

[TABLE]

In the terminology of [AH99], we evaluate at $F$ the “surrogate” function that we have associated to $D$ thanks to the multiplication by $h$. Note that replacing $h$ by any other $h^{\prime}$ satisfying the conditions does not change the value of $\Psi(D)$. Furthermore, remark that if $\Psi$ is defined on $D$ and $D^{\prime}$, it is also defined on $D+D^{\prime}$ and $\Psi(D+D^{\prime})=\Psi(D)\cdot\Psi(D^{\prime})$. Similarly, if $\Psi(D)$ is defined and non-zero, then $\Psi(-D)=1/\Psi(D)$.

From a computational point of view, if $D$ has a small support, then $\Psi(D)$ can be efficiently computed by using Miller’s algorithm to compute the evaluation at $F$ of the function corresponding to $hD$. See Section 3.3 for more details.

In the Function Field Sieve, this tool is used as part of the commutative diagram that underlies the construction of multiplicative relations. However, it is not used directly in the algorithm, only in its correctness proof. Similar, in our elliptic representation algorithm, the map $\Psi$ is not really necessary to perform computations. However, having it at our disposal gives a very useful tool for checking the correctness of relations, thus helping to remove undesirable implementation bugs.

3 Representation of the Target Finite Field

Let $q$ and $k$ respectively be a prime power and the extension degree of the field. Our aim is to compute discrete logarithms in it. Let’s write $p$ its characteristic. Since we want to define the extension as ${\mathbb{F}}_{q}[x_{P}]$, with $x_{P}$ the abcissa of some point on an elliptic curve, while the construction naturally writes it as ${\mathbb{F}}_{q}[x_{P},y_{P}]$, there is a small risk of producing a subextension (missing a last quadratic extension) when $k$ is even. If this happens, it suffices to replace $k$ by $2k$ during the initial choice of the elliptic basis. As a consequence, we can safely ignore this point in the sequel.

3.1 Choosing the elliptic basis

The representation step of our algorithm starts by forming an elliptic curve over a small extension of ${\mathbb{F}}_{q}$ with cardinality a multiple of $k$. The following result explicits bounds with respect to $q$ and $k$ for both the extension degree and the multiplying factor.

Theorem 1**.**

*If $q$ and $k$ be a prime power and a positive integer, then there exist $\mu$ and $\nu$ two integers and an elliptic curve over ${\mathbb{F}}_{q^{\mu}}$ with cardinality $\nu k$ such that $\mu\leq\lceil\log(k^{2}/4)/\log q\rceil+1$. *

Proof.

Let $p$ denote the characteristic of ${\mathbb{F}}_{q}$. We can perform following case by case analysis:

First, let us assume that $k<2\sqrt{q}$. Thus there exist at least two multiples $\nu k$ and $(\nu+1)k$ of $k$ in the Hasse interval $\left]q+1-2\sqrt{q},q+1+2\sqrt{q}\right[$. Furthermore, we may assume that $\nu k<q+1.$

If $p$ divides $k$ then $p$ cannot divide the trace $t=q+1-\nu k$. Let us recall now the following result:

Corollary 1** (of Waterhouse’s theorem [Wat69]).**

For each value of the characteristic $p$, for any extension degree $n$ and for every integer $t$ in $\left]-2\sqrt{p^{n}},2\sqrt{p^{n}}\right[$ such that $t\not\equiv 0\mod p$, there exists an elliptic curve over ${\mathbb{F}}_{p^{n}}$ whose number of rational point is exactly ${p^{n}}+1-t$.

The reader can for example find a proof in [Ugh83]. Note that we can run into some cases where the characteristic does divide the trace and yet such that there exists such a curve. Theorem $4$ in [Ugh83] gives the exhaustive list of these special cases.

Back to our discussion of the case where $p$ divides $k$, we see that, Waterhouse’s theorem yields the existence of an elliptic curve over ${\mathbb{F}}_{q}$ with cardinality $\lambda k$, for all values $\lambda k$ in the Hasse interval. In particular, there exist a curve of cardinality $\nu k$.

If $p$ does not divide $k$ then two sub-cases occur. Either $q+1-\nu k$ is not a multiple of $p$, and Waterhouse’s theorem permits to conclude again that there exists an elliptic curve over ${\mathbb{F}}_{q}$ with cardinality $\nu k$, or it is. In the second case, $p$ doesn’t divide $(\nu+1)k$ and we obtain a curve with that cardinality.

Either way, when $k<2\sqrt{q}$ we always find a curve over the base field ${\mathbb{F}}_{q}$. 2. 2.

If $k\geq 2\sqrt{q}$ then there is no guarantee of the existence of two multiples of $k$ in the Hasse interval. Thus, unless we are lucky, we need to increase the size of the finite field to get a larger interval. Let $\mu$ be the smaller integer such that $k<2\sqrt{q^{\mu}}$. Applying the previous case on this extended field, we see that there always exists an elliptic curve over ${\mathbb{F}}_{{q^{\mu}}}$ with cardinality $\nu k$ such that $\nu k<{q^{\mu}}+1+2\sqrt{p^{\mu}}$. To conclude, notice that the additional extension degree $\mu$ that we need is (at most) equal to the ceiling of $\log(k^{2}/4)/\log q$.

∎

Once we have found ${\mathscr{E}}$, it allows us to define the finite field ${\mathbb{F}}_{q^{\mu k}}$, where $\mu$ is the extra extension degree needed to find ${\mathscr{E}}$. To lighten notations, we assume without loss of generality that $\mu=1$. Indeed, it suffices to redefine a new value for $q$ equal to the previous value $q^{\mu}$. Thanks to the upper bound on the extension degree, we see that it does not affect the quasi-polynomial time complexity of the algorithm.

We further assume that ${\mathscr{E}}$ contains a point in ${\mathbb{F}}_{q}$ of order $k$. If necessary, apply low-degree isogenies to the initial curve ${\mathscr{E}}$ until a suitable one is obtained.

Then construct a point $F$ whose coordinates in the algebraic closure satisfy:

[TABLE]

where $\pi$ is the $q$-th power Frobenius action. Write the coordinates of $F$ as $(\theta,\tau).$ From [CL09], we know that ${\mathbb{F}}_{q^{k}}={\mathbb{F}}_{q}[\theta,\tau]$. Furthermore, from our assumption on $k$, we have ${\mathbb{F}}_{q^{k}}={\mathbb{F}}_{q}[\theta]$.

Note that, when focusing on the practical variation of the algorithm, it is important to have $k$ as large as possible compared to $q$. The above proof only guarantees that $q=O(k^{2})$, however, in the best cases we can have $q=O(k)$.

3.2 Representing the curve with a different model

We introduce here a model that represents elliptic curves in the three-dimensional affine space ${\mathbb{F}}_{q}[U,V,W]$. To construct this new model, we start from a curve ${\mathscr{E}}$ together with a $k$-torsion point $P_{1}$ of ${\mathscr{E}}/{\mathbb{F}}_{q}$. We emphasize that $P_{1}$ has all its coordinates in the field ${\mathbb{F}}_{q}$. Let $(x_{1},y_{1})$ denote the coordinates of $P_{1}$ and $(x_{\ell},y_{\ell})$ the coordinates of $P_{\ell}=\ell\,P_{1}$ for $\ell\in[2,\cdots,k-1]$.

Adding some structure.

The idea is to create a new model of ${\mathscr{E}}$ in which we artificially inject extra desirable properties. Namely, for any point $Q\in{\mathscr{E}}/\bar{{\mathbb{F}}}_{q}$, we represent it by the triple of abcissae of the points $Q-P_{1}$, $Q$ and $Q+P_{1}$ , on the one hand and $-Q,P_{1}$, $Q-P_{1}$ on the other hand (see Figure 1).

In this model, there is an easy way to add the $\pi(F)=F+P_{1}$ constraint of the Couveignes and Lercier [CL09] construction of elliptic bases. Graphically, this is shown on Figure 2.

Indeed, for point $F$ we see that the triple of coordinates is $(x_{\pi^{-1}(F)},x_{F},x_{\pi(F)})$. Furthermore, for $\pi(F)$ the triple is $(x_{F},x_{\pi(F)},x_{\pi^{2}(F)}).$ As a consequence, the first two coordinates can be obtained by a simple shift.

Furthermore, it is possible to recover the missing coordinate of $\pi(F)$ from the first two, in a way similar to Montgomery’s ladder technique.

Formal definition of ${\mathscr{C}}$.

Now that we have captured our intuition, we give the equations of the curve in the new model. First, we recall the definition of the third Semaev polynomial $S_{3}$, it is an irreducible and symmetric polynomial of degree-$2$ in ${\mathbb{F}}_{q}[U,V,W]$. Furthermore, for any triple of points $Q_{1}=(x_{Q_{1}},y_{Q_{1}}),Q_{2}=(x_{Q_{2}},y_{Q_{2}})$, $Q_{3}=(x_{Q_{3}},y_{Q_{3}})\in{\mathscr{E}}(\bar{\mathbb{F}}_{q})\setminus\{\mathcal{O}\}$, we have:

[TABLE]

We use this polynomial to describe the image ${\mathscr{C}}$ of the rational map:

[TABLE]

For every point $Q\in{\mathscr{E}}/\bar{{\mathbb{F}}}_{q}$, we use $S_{3}$ to rewrite the three simple identities $(Q-P_{1})-Q+P_{1}=\mathcal{O}$, $Q-(Q+P_{1})+P_{1}=\mathcal{O}$ and $(Q-P_{1})-(Q+P_{1})+P_{2}=\mathcal{O}$. This shows that the point $(x_{Q-P_{1}},x_{Q},x_{Q+P_{1}})$ is a common root of the polynomials $S_{3}(U,V,x_{1})$, $S_{3}(V,W,x_{1}),$ and $S_{3}(U,W,x_{2})$. Yet, the variety defined by these $3$ equations contains several components. One of the components is irreducible and has dimension $1$ whereas the others correspond to extraneous points. This irreducible component is the curve ${\mathscr{C}}$ given by the equations:

[TABLE]

In fact ${\mathscr{C}}$ is isomorphic to the initial elliptic curve ${\mathscr{E}}$. For more details, see Appendix A.

Point in ${\mathscr{E}}$ with coordinates in the target finite field.

Let $F\in{\mathscr{E}}$ be such that:

[TABLE]

Lemma 1**.**

If $F\in{\mathscr{E}}/\bar{{\mathbb{F}}}_{q}$ is such that $\pi(F)=F+P_{1}$ then $F\in{\mathscr{E}}/{\mathbb{F}}_{q^{k}}$. Furthermore, letting $(\theta,\tau)$ denote the coordinates of $F$, we have ${\mathbb{F}}_{q}[\theta,\tau]={\mathbb{F}}_{q^{k}}$. In particular, there exist at least $k$ rational points verifying the same property.

Proof.

Let us compute $\pi^{k}(F)=\pi^{k-1}(F+P_{1})=\pi^{k-1}(F)+P_{1}=\cdots=F+kP_{1}$. We know that $P_{1}$ has precisely order $k$ hence $\pi^{k}(F)=F$. Besides, we note that any point $\pi^{i}(F)$ for $i=1,\cdots,k-1$ satisfies Equation (1) too. ∎

As already mentioned, having possibly doubled the value of $k$ in construction, we may assume that ${\mathbb{F}}_{q}[\theta]={\mathbb{F}}_{q^{k}}.$ in our model of curve, the point $F$ is determined by the fact that $S_{3}(\theta,\theta^{q},x_{1})=0$. The abscissa $\theta$ of $F$ can thus be determined as a root of this polynomial. Note that the choice of the ordinate $\tau$ gives an orientation on the direction of the Frobenius action. We choose the letter $F$ to name this point as a mnemonic to remind that it represents our target Finite Field and that it has a special relationship with the Frobenius map.

3.3 Commutative diagram

From this, we derive the commutative diagram of Figure 3 which serves as the basis for our elliptic Frobenius representation algorithm. Note that the commutative diagram is above ${\mathbb{F}}_{q^{k}}/{\mathbb{F}}_{q}^{*}$, as a consequence, our algorithm doesn’t compute the part of the discrete logarithm corresponding to ${\mathbb{F}}_{q}^{*}$. However, this field is so small that this missing part can be easily obtained.

Remark 1**.**

This diagram could be simplified by removing references to the function field ${\mathbb{F}}_{q}({\mathscr{E}})$ and computing divisors on ${\mathscr{C}}$ directly. However, when using standard computer algebra tools, it is much simpler to work on divisors with the Weirstrass equation of ${\mathscr{E}}$.

Explicit maps to ${\mathbb{F}}_{q^{k}}$ based on Miller algorithm.

The first two maps of the diagram are explicit and the two following ones are canonical injections. $\Phi^{*}$ is given in Appendix and $\Xi$ is as defined in Section 2. In addition, we let $\Psi$ denote a multiplicative group morphism that sends elements of $\mbox{Princ}({\mathscr{E}})$ to ${\mathbb{F}}_{q^{k}}$. Yet, only defining $\Psi$ for principal divisors is not sufficient, since in the relation collection phase we need first to factor divisors in $\mbox{Princ}({\mathscr{E}})$ into elementary divisors before descending them into the finite field. Keep in mind that elementary divisors have no reason to be principal.

For any divisor in $\mbox{Princ}({\mathscr{E}})$ the first thing to do is to decompose it into elementary divisors in $Div^{0}$. Then, we note that since we wish to construct a group morphism that sends elements of $Div^{0}$ to ${\mathbb{F}}_{q^{k}}/{\mathbb{F}}_{q}^{*}$, it suffices to describe this morphism for any of these elementary divisors. Let us consider:

[TABLE]

where $Q\in{\mathscr{E}}/{\mathbb{F}}_{q^{d}}$ is one of the conjugate points in the degree-$d$ place. Fix a maximum degree $D$ for the places we consider and let $N_{D}$ be the least common multiple of the cardinalities of ${\mathscr{E}}$ over each finite field ${\mathbb{F}}_{q^{d}}$ with $1\leq d\leq D$.

From this, we see that $N_{D}\mathscr{D}_{e}$ is a principal divisor, thus there exists a function $f_{\mathscr{D}_{e}}$ in the variables $X$ and $Y$ unique up to multiplication by a constant in ${\mathbb{F}}_{q}$ such that $\mbox{div}(f_{\mathscr{D}_{e}})=N_{D}\mathscr{D}_{e}$. We want to use the point $F$ with coordinates in the target finite field to define it. Since $\theta$ and $\tau$ are respectively the abscissa and the ordinate of this point, it seems natural to send $X$ to $\theta$ and $Y$ to $\tau$, or, in other words, to evaluate the function on the point $F$. However, since $f_{\mathscr{D}_{e}}$ is only defined modulo a constant in ${\mathbb{F}}_{q}$, the result in the finite field would change depending on the choice of the function. To annihilate this constant, we have to divide the evaluation on $F$ by the evaluation on $\mathcal{O}$. Hence to have a well-defined application $\Psi$ we set

[TABLE]

However, evaluation at $\mathcal{O}$ isn’t really necessary, since we are only interested in values in ${\mathbb{F}}_{q^{k}}/{\mathbb{F}}_{q}^{*}$. As done for bilinear pairings, this can be efficiently computed using Miller’s algorithm [Mil04].

Remark 2**.**

In order to raise to the power $1/N_{D}$ and get a uniquely defined value, we need to check that $N_{D}$ is invertible modulo the order of ${\mathbb{F}}_{q^{k}}^{*}/{\mathbb{F}}_{q}^{*}$. This condition needs to be tested for all the orders of the curve ${\mathscr{E}}$ in the extension fields ${\mathbb{F}}_{q},{\mathbb{F}}_{q^{2}},\cdots,F_{q^{D}}.$

We analyze the condition more precisely in Appendix B and provide a replacement for $\Psi$ in the case where $N_{D}$ cannot be inverted.

Intuition about $\Psi$ and commutativity of the diagram.

This definition matches with the following intuitive one. To unsure the commutativity of the diagram we need to verify that $\Psi(\Xi(\Phi^{*}(i(A^{q}B-AB^{q}))))$ is equal in the finite field to the element $\Psi(\Xi(\Phi^{*}(i(A(V,W)B-AB(V,W)))))$. Our intuition is that requiring in some sense:

[TABLE]

would suffice to prove the commutativity. We point out that one point of the elliptic curve ${\mathscr{C}}$, namely $\Phi(F)$, precisely follows this restriction. Indeed, $\Phi(F)$ has abscissa $x_{F-P_{1}}=x_{\pi^{(k-1)}(\pi(F-P_{1}))}=x_{\pi^{(k-1)}(F)}=\pi^{(k-1)}(\theta)$, ordinate $x_{F}=\theta$ and applicate $x_{F+P_{1}}=x_{\pi(F)}=\pi(\theta)$. In a nutshell:

[TABLE]

Thus, in the function field ${\mathbb{F}}_{q}({\mathscr{C}})$, evaluation the functions at $\Phi(F)$ gives the expected relationship to the Frobenius map. As a consequence, evaluation at $F$ after transporting back to the function field of ${\mathscr{E}}$ using $\Phi^{*}$ also gives the desired behavior.

To conclude about the commutativity of the diagram it suffices to note that:

[TABLE]

and similarly:

[TABLE]

Hopefully, since $\Psi\circ\Xi\circ\Phi^{*}\circ i$ is a morphism the equality in the finite field between images of $A^{q}B-AB^{q}$ and $A(V,W)B-AB(V,W)$ holds too.

Extension of the diagram to bigger fields.

As with classical Frobenius representation algorithm, the commutative diagram can also be used when the coefficients of $A$ and $B$ are taken in another extension ${\mathbb{F}}_{q^{d}}.$ In that case, the commutative diagram ends in the compositum of ${\mathbb{F}}_{q^{d}}$ and ${\mathbb{F}}_{q^{k}}$ (again modulo ${\mathbb{F}}_{q}^{*}$). The only difference is that the identity in the finite field is between the images of $A^{q}B-AB^{q}$ and $A^{\pi}(V,W)B-AB^{\pi}(V,W)$, i.e. the coefficients of $A$ and $B$ need to be acted on by Frobenius.

4 Harvesting Relations

4.1 The usual systematic product

As we can see in the previous diagram, our setting uses a mixture of the classical Function Field Sieve and of the Frobenius representation algorithms. As in the classical Function Field Sieve, our algorithm uses function fields instead of polynomials when writing down multiplicative relations. From Frobenius representation algorithms it inherits the use of the systematic relation:

[TABLE]

where as in [JP14], when $\alpha$ is the point at infinity of ${\mathbb{P}}_{1}({\mathbb{F}}_{q})$, the term $A-\alpha\,B$ is used as a shorthand for $B$. For simplicity, we also use a bracket notation and define:

[TABLE]

We underline that our bracket is ${\mathbb{F}}_{q}$-bilinear and antisymmetric, as in [JP14]. Yet, we warn the reader of the difference between the definition of our bracket and previous ones. Our bracket is equal to the entire fraction whereas the authors of [JP14] only consider the numerator of this rational fraction.

4.2 Choice of $A$ and $B$

In the commutative diagram of Figure 3 and in the above discussion, we indicate that relations are obtained from a choice of two bivariate polynomials $A$ and $B$ in $U$ and $V$. However, we need to specify how these polynomials are chosen and which monomials they should contain.

As a preliminary, let us notice that $(A,B)$ and $(\alpha A,B)$ for $\alpha\in{\mathbb{F}}_{q}$ lead to the same relation. Indeed, $(\alpha A)^{q}\,B-(\alpha A)\,B^{q}=\alpha(A^{q}\,B-A\,B^{q})$ so the two divisors associated to the two corresponding functions are equals. Thus $A$ and $B$ are chosen as some kind of monic polynomials: the coefficient of the higher monomial (in the lexicographic order for instance) must be equal to $1$. Then we note that monomials divisible by $(UV)^{2}$ are not useful in $A$ and $B$. Indeed, when going to ${\mathbb{F}}_{q}[{\mathscr{C}}]$, reduction modulo $S_{3}(U,V,x_{1})$ transforms these monomials into smaller monomials $U^{2}V,UV^{2},U^{2},UV,V^{2},U,V$ and $1$. Moreover, it is natural to consider sets of monomials globally symmetric in $U$ and $V$.

As a consequence for all those items, given a parameter $t\geq 1$ we construct $A$ and $B$ as linear combinations of monomials from:

[TABLE]

Each $\mathcal{M}_{t}$ contains $4t$ distinct monomials222Not $4(t+1)$ since each of $1$, $U$, $V$ and $UV$ are included twice..

4.3 Defining a naive factor base

As shown in Figure 3, we define the factor base as a subset of $Div^{0}({\mathscr{E}})$. We now explain how this subset is chosen.

Definition 1**.**

Let $\mathscr{D}$ be a divisor of $Div(\mathcal{V})$. The height of $\mathscr{D}$ denotes the number of positive points counted with multiplicity. We write it $h(\mathscr{D})$.

It matches with the following explicit definition: if the divisor $\mathscr{D}$ is written as $\sum_{Q_{i}\in\mathcal{V}}e_{i}Q_{i}$ we have $h(\mathscr{D})=\sum_{e_{i}>0}e_{i}$. Note that, for any prime or elementary divisor, the height is equal to the degree of the corresponding place.

Proposition 1**.**

If $f_{1}$ and $f_{2}$ are two functions of the function field associated to $\mathcal{V}$ then the following inequalities are verified:

$h(\Xi(f_{1}f_{2}))\leq h(\Xi(f_{1}))+h(\Xi(f_{2}))$. 2. 2.

$h(\Xi(f_{1}+f_{2}))\leq\max(h(\Xi(f_{1}))+\deg g_{2}\,,\,h(\Xi(f_{2}))+\deg g_{1})$* *

where $f_{1}$ (resp. $f_{2}$) has $g_{1}$ (resp. $g_{2}$) as denominator. 3. 3.

h(\Xi(f_{1}))\leq h(\Xi(f_{2}))\quad\hbox{if f_{1}$and$f_{2}$are polynomials such that$f_{1}$divides$f_{2}.}**

Proof.

Let $f_{i}=e_{i}/g_{i}$ for $i=1,2$ be the two functions written such that $e_{i}$ and $g_{i}$ are two polynomials of the function field with no commun factor. Clearly, the function $f_{1}f_{2}$ has $e_{1}e_{2}$ as numerator, thus all the zeros of $f_{1}f_{2}$ are either a zero or $f_{1}$ or $f_{2}$. The inequality of $1.$ is strict if some simplifications appear (for instance if a zero of $f_{1}$ is also a pole of $f_{2}$).

Writing $f_{1}+f_{2}$ as $(e_{1}g_{2}+e_{2}g_{1})/(g_{1}g_{2})$ we see that the number of zero of this sum is upper-bounded by the weighted degree of $e_{1}g_{2}$ or $e_{2}g_{1}$. Note again that some simplifications may appear.

Item $3.$ is straightforward. ∎

Left side.

Following the ideas of all Frobenius representation algorithms, we define the factor base such that the images of $A-\alpha B$ in this set are small. Doing this improve the relation collection phase compared to a classical sieving. Indeed, all elements in the left part of the diagram will belong to the factor base. The relation collection phase produces divisors of the form $\Xi(\Phi^{*}(i(\prod(A-\alpha B))))$, so, thanks to the fact that we consider morphisms, it yields a sum of divisors $\sum(\Xi(\Phi^{*}(i(A-\alpha B))))$. As explained, we require all the divisors noted by:

[TABLE]

to be in the factor base.

Let us find the maximal height they can reach. To do so we set $A$ and $B$ two linear combinations of monomials in $\mathcal{M}_{t}$ and $t$ an integer parameter to define later. All polynomials $A-\alpha B$ are so linear combinations of monomials in $\mathcal{M}_{t}$ too. In other words, we are considering divisors of functions of ${\mathbb{F}}_{q}({\mathscr{E}})$ of the form $\Phi^{*}(\sum_{m\in\mathcal{M}_{t}}a_{m}m)=\sum_{m\in\mathcal{M}_{t}}a_{m}\Phi^{*}(m)$ where $a_{m}$ are constants in the base field. From Proposition 1 we know that the height of the divisors we obtain in the left part of the diagram are dominated by the largest height achieved for any $\Xi(\Phi^{*}(m))$, with $m\in\mathcal{M}_{t}$.

Considering $h(\Xi(\Phi^{*}(U^{t}V^{t^{\prime}})))\leq t\,h(\Xi(\Phi^{*}(U)))+t^{\prime}\,h(\Xi(\Phi^{*}(V)))$, we see that it suffices to determine both the height of the divisors associated to the images of $U$ and $V$ in the function field of ${\mathscr{E}}$. The most significant monomials will be $U^{t}V$ and $UV^{t}$.

On the one hand we have $\Xi(\Phi^{*}(V))=\Xi(X)=([0,\sqrt{b},1])+([0,-\sqrt{b},1])-2(\mathcal{O})$ where $\sqrt{b}$ is the element333Be careful, here we assume that the characteristic differs from $2$. If not, we just consider the corresponding degree-$2$ place. in $\bar{\mathbb{F}}_{q}$ such that its square is equal to $b$. Hence:

[TABLE]

On the other hand, $\Xi(\Phi^{*}(U))=\Xi((Y+y_{1})^{2}-(X-x_{1})^{3})-\Xi((X-x_{1})^{2})$. From $\Xi((Y+y_{1})^{2}-(X-x_{1})^{3})=2(-P_{1})+(Q_{1})+(Q_{2})-4(\mathcal{O})$, where $Q_{1}$ and $Q_{2}$ are two conjugated points of a degree-2 place, and $\Xi((X-x_{1})^{2})=2(P_{1})+2(-P_{1})-4(\mathcal{O})$, it comes $\Xi(\Phi^{*}(U))=(Q_{1})+(Q_{2})-2(P_{1})$. We obtain:

[TABLE]

Putting Equations (3) and (4) together with the upper-bound, we conclude that the most significant monomials $U^{t}V$ and $UV^{t}$ have both height $2t+2$. Yet $\Phi^{*}(U^{t}V)$ and $\Phi^{*}(UV^{t})$ do not share the same denominator so to count their respective contribution in the height of divisors $\sum_{m\in\mathcal{M}_{t}}a_{m}\Phi^{*}(m)$ we need to add the contribution of the residual denominator. Namely, since $\Phi^{*}(U^{t})$ brings the largest denominator, the height of the divisor of $\Phi^{*}(U^{t}V)$ does not change, but for the one of $\Phi^{*}(UV^{t})$ we need to add the number of zeros corresponding to the denominator of $\Phi^{*}(U^{t}/U)$. We note that there is $2(t-1)$ such points. To put it in a nutshell, the most significant monomial is $UV^{t}$ and all divisors on the left are sum of divisors with a height upper-bounded by $\textbf{4t}=(2t+2)+2(t-1)$.

To conclude, starting the relation collection phase with $t=1$ it is natural to set the initial factor base as included in the set of divisors of $Div^{0}$ with height equal or lower than $4$. We emphasize that in this case, all the divisors appearing in the left part belong to the factor base.

Right side.

On the right part of the diagram, divisors are given through the extra variable $W$. We can compute the corresponding height exactly as for $U$. Again it gives:

[TABLE]

Let us consider the polynomials of ${\mathbb{F}}_{q}[U,V,W]$ given on this side and write this time $\mathcal{M}_{t}^{VW}=\left\{V^{i},W^{i},V^{i}\,W,V\,W^{i}|i\in[0\cdots t]\right\}$. Sorting the monomials in the lexicographic order, we recall that the leading coefficient for both $A$ and $B$ can be chosen equal to $1$. Thus, keeping the leading monomial $U^{t}V$ apart and calling $a_{m}$ (resp. $b_{m}$) the coefficients in ${\mathbb{F}}_{q}$ of $A$ (resp. $B$) we obtain on the right side the polynomial:

[TABLE]

Since the monomial $U^{t}V^{t+1}W$ vanishes it yields a linear combination of monomials where the three that dominate the height of the associated divisor are $UV^{t+1}W^{t}$, $U^{t}V^{2}W^{t}$ and $UV^{2t}W$. Indeed, each variables $U$, $V$ and $W$ contributes the same way, thus, the most important monomials are those with the highest additive degree. We note then that we have $h(\Xi(\Phi^{*}(UV^{t+1}W^{t})))=h(\Xi(\Phi^{*}(U^{t}V^{2}W^{t})))=h(\Xi(\Phi^{*}(UV^{2t}W)))=4t+4$. Yet, again, we need to carefully add the zeros raised by the residual denominator. The contribution of $\Phi^{*}(U^{t}V^{2}W^{t})$ is left unchanged but we must add the number of poles of $\Phi^{*}(W^{t}/W)$ to the height of the divisor associated to $\Phi^{*}(UV^{t+1}W^{t})$ and the one corresponding to the denominator of $\Phi^{*}((UW)^{t}(UW)^{-1})$ to the height of the divisor associated to $\Phi^{*}(UV^{2t}W)$. From Proposition 1 and since there are respectively $2(t-1)$ and $4(t-1)$ such points, we conclude that all the divisors appearing on the right side are twice as large as factor base elements since they have height equal or lower than $\textbf{8t}=4t+4+4(t-1)$. In particular, when $t=1$, this gives divisors of height $8$ at most.

Complexity of the linear algebra with a naive factor base.

A first and naive choice of factor base is made of all divisors of height $\leq 4$.

However, this factor base is too large to be competitive when compared to the best Frobenius representation algorithms. To show this, let us briefly analyze the number of operations needed to perform linear algebra with this factor base. The number of divisors in this naive factor base is dominated by the number of degree $4$ places on the curve ${\mathscr{E}}$, corresponding to polynomials of degree $4$ with coefficients in ${\mathbb{F}}_{q}$. So the order of the factor base’s size is dominated by $q^{4}$. Considering the Frobenius action of Section 4.4 that permits to divide the size of the factor base by $k\approx q$, we obtain a factor base of size $O(q^{3})$. Since there are $q$ terms in each linear equations, performing a sparse linear algebra step can be done in $O((q^{3})^{2}q)=O(q^{7})$ operations.

As a comparison, we that the first phase of the algorithm in [JP14] only has a $O(q^{6})$ complexity. Thus, we need to improve our initial factor base.

4.4 Action of Frobenius and translation by $P_{1}$

Let $d$ be the largest possible height of an elementary divisor of the factor base. We would like to explicit how the action of Frobenius on elements of the finite field ${\mathbb{F}}_{q^{k}}$ is related to addition of $-P_{1}$ on the elliptic curve ${\mathscr{E}}$. Considering the divisors:

[TABLE]

related to the place given by any point $Q\in{\mathscr{E}}$ and:

[TABLE]

related to the translation of $Q$ by $-P_{1}$. We show that the two discrete logarithms satisfy a simple relation. More precisely, we have the following result:

Lemma 2**.**

Let $Q\in{\mathscr{E}}$ be a point with coordinates in ${\mathbb{F}}_{q^{d}}$. We consider the divisors

[TABLE]

respectively related to the place given by the point $Q$ in ${\mathscr{E}}$ and the one given by the translation of $Q$ by $-P_{1}$. Then:

[TABLE]

where $N_{d}$ is a common multiple of the cardinalities of ${\mathscr{E}}$ over each finite field ${\mathbb{F}}_{q^{i}}\subset{\mathbb{F}}_{q^{d}}$

Proof.

Let us start from $\mathscr{D}_{Q}$ the degree-$d$ divisor. We recall that to have a principal divisor we need to consider $N_{d}\mathscr{D}_{Q}$. Thanks to Miller algorithm we are able to recover a function $f_{Q}$ with coefficients in the base field ${\mathbb{F}}_{q}$ such that $N_{d}\mathscr{D}_{Q}$ is the divisor of this function. By definition we obtain: $\Psi(\mathscr{D}_{Q})=f_{Q}(F-\mathcal{O})^{1/N_{d}}$. To simplify the notation let us write $\alpha=f_{Q}(\mathcal{O})^{1/N_{d}}$ that is an element in ${\mathbb{F}}_{q}$. Hence on the one hand we have:

[TABLE]

To link this expression to the divisor of $\pi(Q)$, i.e. to the evaluation of $f_{Q}$ in the point $\pi(F)=F+P_{1}$ we define the function $g_{Q}$ such that, for all $S$ in ${\mathscr{E}}$, $g_{Q}(S)=f_{Q}(S+P_{1})$. Let us write the divisor of this new function. Since a zero $S$ (resp. a pole) of $g_{Q}$ is such that $S+P_{1}$ is a zero of $f_{Q}$ (resp. a pole), we obtain:

[TABLE]

Hence on the other hand we have:

[TABLE]

∎

We emphasize that the green term is a constant term. Thanks to this action, we are able to reduce the size of the factor base by a factor $k$ throughout the computations. Indeed, if we know the discrete logarithm of $\Psi(\mathscr{D}_{Q})$ then we learn for free the discrete logarithms of $\Psi(\mathscr{D}_{Q-P_{1}}),\Psi(\mathscr{D}_{Q-P_{2}}),\cdots,\Psi(\mathscr{D}_{Q-P_{k-1}})$.

4.5 Getting a smaller factor base

To be able to reduce the size of the initial factor base, and thus to decrease the complexity of the linear algebra phase, we adapt the idea of systematic factors that was presented in [JP14] to the elliptic case. The idea was twofold: first extracting some systematic factors that appear in every equation, second, restrict the search to a sieving space that induce extra common factors. In this article, we choose to call these extra factors compelled factors to underline the difference with previous ones.

Left part of the diagram: making $P_{3}$ a compelled point.

In our case, our aim is to consider a subgroup of the sieving space where $A$ and $B$ are polynomials such that the associated divisors always present a common (compelled) point. Here, choose to use the special point $P_{3}=3\,P_{1}$. As in [JP14] we select three generators $g_{1}$, $g_{2}$, $g_{3}$ in ${\mathbb{F}}_{q}[U,V]$ leading to divisors going through $P_{3}$. We propose to sieve on pairs of polynomials $(A,B)$ such that $A=g_{1}+\alpha g_{3}$ and $B=g_{1}+\beta g_{2}+\gamma g_{3}$ where $\alpha,\beta,\gamma\in{\mathbb{F}}_{q}$. Indeed, if $A$ (resp. $B$) is a linear combination of those three generators and if $P_{3}$ is a zero of $\Phi^{*}(i(g_{j}))$ for $j=1,2$ and $3$ then it is also a zero of the image of $A$ (resp. $B$) in ${\mathscr{E}}$. As a consequence and for the same reason, $P_{3}$ is a zero of the image of $A-\alpha B$ too, where $\alpha$ belongs to the base field.

Lemma 3**.**

Let $j$ be an integer in $[0,k-1]$ and assume that $P_{0}$ is a shorthand for $\mathcal{O}$. Then:

[TABLE]

Proof.

Let $j$ be an integer between [math] and $k-1$. We recall that $x_{j}$ denotes the abscissa of $P_{j}\in{\mathscr{E}}$ and $x_{0}=0$. Then by definition:

[TABLE]

for all possible values of $j$. It means that, over the curve ${\mathscr{C}}$, $\Phi(P_{j})$ is a zero of $U-x_{j-1}$, $V-x_{j}$ and $W-x_{j+1}$. Going back to the curve ${\mathscr{E}}$ it yields that the point $P_{j}$ is a zero of $\Phi^{*}(U-x_{j-1})$, $\Phi^{*}(V-x_{j})$ and $\Phi^{*}(W-x_{j+1})$. Similarly, from:

[TABLE]

we get that $-P_{j}$ is a zero of $\Phi^{*}(U-x_{j+1})$, $\Phi^{*}(V-x_{j})$ and $\Phi^{*}(W-x_{j-1})$. Besides, writing $\Phi^{*}(U-x_{j})$ as $\left((Y+y_{1})/(X-x_{1})\right)^{2}-X-x_{1}-x_{j}$ we see that $P_{1}$ is a pole of $\Phi^{*}(U-x_{j})$ with multiplicity $2$, and similarly $-P_{1}$ is a pole of $W-x_{j}$ with multiplicity $2$. From $\Phi^{*}(V-x_{j})=X-x_{j}$ we conclude that $\mathcal{O}$ is twice a pole too. ∎

Hence as generators we select:

[TABLE]

From Lemma 3 we have $\Xi(\Phi^{*}(g_{1})))=(P_{3})+(-P_{1})-2(P_{1})$ and $\Xi(\Phi^{*}(g_{2})))=(P_{3})+(-P_{3})-2(\mathcal{O})$. Thus $\Xi(\Phi^{*}(g_{3})))=2(P_{3})+(-P_{1})+(-P_{3})-2(\mathcal{O})-2(P_{1})$. Clearly the point $P_{3}$ is a positive point of each divisor.

Thus, if we start to sieve with a parameter $t$ equal to $1$, we obtain divisors on the left part of the diagram that have a height lower or equal to $4$. Since $P_{3}$ is a positive point for all these divisors, we are left with divisors that have a height lower or equal to $3$. We conclude with the definition of the reduced factor base:

[TABLE]

Remark 3**.**

The factor base is the same on both sides of the diagram.

As previously, we can upper-bound the cardinality of the factor base by the number of divisors with height lower than $3$, so the number of monic degree-$3$ polynomials in ${\mathbb{F}}_{q}$, which is $q^{3}$. Thanks to the Frobenius action, the base is reduces by a factor of $k$, and the reduced factor base has size $O(q^{3}/k)$. At the end, assuming that we get enough equations (we discuss this issue in Section 4.6), linear algebra recovers the discrete logarithms of the initial factor base elements in $O((q^{3}/k)^{2}q)=O(q^{7}/k^{2})$ operations. When $k$ is close to $q$, this matches the result of [JP14].

Remark 4**.**

Note that by contrast with polynomials, ideals of degree $4t$ are determined from $4t$ monomials, instead of $4t+1$. This is unfortunate because it forces us to increase the degrees to get enough degrees of freedom when generating relations. However, this drawback is counter-balanced by the reduction of the factor base size obtained by using the action of Frobenius.

4.6 How to get enough relations with the reduced factor base

Thank to $P_{3}$ which is a compelled point we know that, on the left part of the diagram, we directly have divisors that split into factor base elements only. Now the questions are whereas we manage to easily obtain small height divisors on the right part or not, and how many relation we expect to write. We recall that, to be able to perform linear algebra, we need as many relations as unknowns.

We have around $q^{3}/3$ unknowns and we sieve on $q^{3}$ pairs of polynomials $(A,B)$ consisting in linear combinations of $g_{1},g_{2}$ and $g_{3}$ as given in (6). It means that we need a probability higher than $1/3$ to get a relation. Since a degree-$d$ divisor is clearly linked with an irreducible polynomial of the same degree as seen in Section 2, this probability is assumed to be the same that a random polynomial of degree $d$ to factor into terms of degree at most $3$.

For degree $d=8$, the probability is easy to compute. A polynomial fail to factor into terms of degree at most $3$ when one factor as degree $4$ or more. Since there can only be a single factor when the degree is $5$, $6$, $7$ or $8$ and at most two of degree $4$. Thus, for large fields, the probability of success approaches $1-(1/8+1/7+1/6+1/5+3/16+1/32)\approx 0.147.$ Unfortunately, this is much smaller than $1/3$.

Right part of the diagram: the two compelled points $P_{2}$ and $P_{3}$.

Thus we need to look at the right part of the diagram more carefully. Going back to the analysis made in Section 4.3, Right part we see that choosing $A$ and $B$ as monic (in some sense) does not reduce the height of the associated divisor. Hence, for $(A,B)$ a pair of linear combinations of $g_{1},g_{2}$ and $g_{3}$ as in (6), the divisor:

[TABLE]

has still a height of $8$. Because $P_{3}$ is a zero of $\Phi^{*}(A(U,V))$ and $\Phi^{*}(B(U,V))$, we note that $P_{3}$ is a zero of $\Phi^{*}(\left[{A},{B}\right])$ too. Yet it is not enough and we need to extract another compelled positive point of the associated divisor to the image of $A(V,W)$ (resp. $B(V,W)$) over ${\mathscr{E}}$. We start by underlining that $g_{1},g_{2}$ and $g_{3}$ respectively becomes $V-x_{2},W-x_{3}$ and $(V-x_{2})(W-x_{3})$, when sending $V$ to $W$ and $U$ to $V$. Thus according to Lemma 3, the point $P_{2}=2\,P_{2}$ is a zero of all the generators, and so a zero of $\left[{A},{B}\right]$ as $P_{3}$. We conclude that we are left with a divisor of height at most $6$. The probability that it splits into a sum a divisors with height at most $3$ is so roughly equals to:

[TABLE]

as $q$ grows. As a consequence, we heuristically expect to get a linear system with enough equations to get the discrete logarithms of all elements of ${\mathcal{F}}$ in $O(q^{5})$ operations.

Heuristic**.**

The heuristic in this linear algebra step and in all the following ones comes from the fact that we have no argument to prove we really get enough equations. We can count them and expect that when their number slightly exceeds the number of unknows, we are able to find a solution. Yet, nothing provably indicates whether the kernel of our matrix of relations has dimension $1$ or not.

5 Extended Factor Base and Individual Discrete Logarithm

We only sketch here the last two main steps of our practical algorithm, the computation of an Extended Factor Base and the Individual Discrete Logarithm step. Indeed, they are an adaptation of the techniques that already exist for Frobenius representation algorithms to our setting.

5.1 From divisors of height $3$ to divisors of height $4$

As done in Frobenius representation algorithms, we extend the initial factor base and now include all elementary divisors up to height $4$. Thanks to the Frobenius action, there are approximately $q^{4}/4k\approx q^{3}/4$ unknowns. The naive approach we showed earlier gives the desired logarithms at a cost of $O(q^{7})$ arithmetic operations (or $O(q^{9}/k^{2})$ when $k$ is away from $q$).

Practical speed up with regrouping

To speed up the computation of height $4$ divisors, it is possible to decompose the height $4$ factor base into small groups, in a way similar to [JP14], in order to perform several linear algebra steps on these small groups, instead of a single big linear algebra step. In Appendix C, we give details on how to produce relation in these groups. One technicality is the interaction of the groupings with the reduction of the factor base size given the action of Frobenius.

Once the height $4$ divisors are obtained, it is a simple matter to continue extending the factor base to height $5$ divisors. For that final step, no additional linear algebra is needed. It suffices to keep relations where a single height $5$ divisors appears, the rest being of lower degree. See Appendix C for a detailed explanation.

However, for the height $4$ extension, the expected number of produced relations seems to be slightly too low asymptotically to guarantee its success. Nevertheless, we tested the method on a practical example to check its viability. Namely, for the target finite field ${\mathbb{F}}_{3^{1345}}$, we were able to compute logarithms for an extended factor base comprising divisors of height up to $5$. This was done by choosing a curve of (prime) cardinality $269$ over ${\mathbb{F}}_{243}$. Studying the exact behavior of the height $4$ extension to understand this gap is thus a matter of further research.

5.2 Computing Individual Discrete Logarithms

To solve the discrete logarithm problem in our target finite field, we need not only to know the logarithms of extended factor base elements but to be able to compute the discrete logarithm of any arbitrary element. This is the aim of this paragraph. Various descent phases were previously proposed by various authors, the idea is to show how to adapt to our context. In practice, one can use the bilinear descent, the classical descent and the zig-zag descent or a mix of them. Indeed the quasi-polynomial descent of [BGJT14] is unlikely to be practical for currently accessible computations.

For the classical descent which simply consist in writing the target finite field element whose discrete logarithm is wanted as a product or quotient of relatively low-degree polynomials in $\theta$, no adaptation is needed. We only need to check that any polynomial in $f(\theta)$ can be injected in the commutative diagram. This is simply done by written the divisor of $f(V)$ since $\Psi(V)=\theta$. When $f$ is irreducible, the corresponding divisor is either the sum of two elementary divisors of height $\deg(f)$ or a single elementary divisor of height $2\deg(f)$.

We illustrate the adaptation with the bilinear and zigzag descents:

Bilinear descent for our setting.

The bilinear descent step is easy to adapt from [JP14]. Remark that we usually need to unbalance the degrees of freedom in $A$ and $B$, thus choosing different sets of generating polynomials. Instead of constructing the polynomials just from $1$, $U$, $V$ and $UV$ we built them from higher degree polynomials in $\mathcal{M}_{t_{a}}$ and $\mathcal{M}_{t_{b}}$ respectively. We assume that $t_{a}\geq t_{b}$. Let us first analyze the case where we use all these monomials, remembering that there are $4t_{a}$ and $4t_{b}$ of them. As usual, we force $A$ and $B$ to be monic and remove the head monomial of $B$ from $A$. All of the other coefficients are replaced by a corresponding formal unknown. Thus, the polynomial $A$ contains $4t_{a}-2$ unknowns. If $t_{a}\neq t_{b}$, $B$ contains $4t_{b}-1$ monomials. If $t_{a}=t_{b}$, we can remove an extra unknown from $B$. Furthermore, we know that the height of factors of the form $A-\alpha B$ is upper bounded by $4t_{a}$. We also know from Table 2 that the height of the bracket is at most $8t_{a}$

If we want to adjust the values modulo $4$ of the number of degrees of freedom, it is necessary to use compelled points. More precisely, we can force $A$ and $B$ to go through one, two or three forced points. This reduces the degrees of freedom by the same amount on both sides. It also reduces height on the left by the same value and heights on the right by its double.

As in Frobenius representation algorithms, the coefficients of each monomials in $\left[{A},{B}\right]$ are bilinear (or linear or constant) in the $A$ and $B$ unknowns. To force an elementary divisor of degree $d$ to appear in $\left[{A},{B}\right]$, it suffices to require that the bracket vanishes when evaluated at each of the $d$ conjugate points corresponding to the associated prime divisor. This yields a bilinear system of $d$ equations in the $A$ and $B$ unknowns. This equation can be solved using Gröbner basis techniques exactly as in the case of Frobenius representation.

The only extra (and minor) restriction here is the relation between the number of $A$ variables and $B$ variables modulo $4$.

5.3 Zig-zag descent.

The zig-zag descent seems to be the best option to achieve provable quasi-polynomial complexity. In particular, it is used both in  [KW19] and [Lid19]. As a consequence, we also show how to adapt it to our setting. As it is more lenghty to describe than the bilinear descent, we assign a separate section to it.

Short recap on the zig-zag descent.

First the main idea is to adapt the zig-zag descent presented in [GKZ14] to our setting. Let us give an insight of this descent in the classical settings. We call $z$ our target, which is an irreducible polynomial in ${\mathbb{F}}_{q}[X]$ of degree 444Indeed, one can use Wan’s theorem [Wan97, Theo 5.1] to ensure that any field element is equivalent to an irreducible polynomial of degree a power of $2$ only slightly larger than the extension degree $k$. $2d=2^{t+1}$. One crucial point of this method is that for any relation in ${\mathbb{F}}_{q^{2}}[X]$ implying degree-$d$ polynomials, one can find a relation in the subfield ${\mathbb{F}}_{q}[X]$ at the price of having polynomials of degree twice as large. Thus, in order to make $z$ appear in a polynomial relation of ${\mathbb{F}}_{q}[X]$, we write it as a product of two degree-$d$ conjugated polynomials $\tilde{z}$ and ${\tilde{z}}^{*}$ over the extended field ${\mathbb{F}}_{q^{2}}[X]$ and we try to get one relation (in the extended field) involving one of this degree-$d$ polynomials. Multiplying by the same conjugated relation we would obtain a relation (in the subfield) where $z$ appears.

Recursively manipulating this trick on a tower of extensions as presented in Figure 4, we write in fact $z$ as a product of conjugated degree-$2$ polynomials over ${\mathbb{F}}_{q^{2^{t}}}[X]$. Indeed, this descent method rests upon the existence of an extended field in which any degree-$2$ polynomials evaluated in $\theta$ can be written as product of linear evaluations in $\theta$. Thus at the end, we get a relation of the form $z(\theta)=\prod_{i}L_{i}(\theta)$ where $L_{i}$ are linear polynomials.

To adapt this descent to our settings, a idealized method would be to exhibit a sufficiently large extension of the curve ${\mathscr{C}}$ in which any height-$2$ divisor can be written not as a degree-$2$ place but as a sum of points on this exact extension (and not the larger following one). This precisely would have translated the requirement that all degree-$2$ polynomials split in linear polynomials when the extension degree of the field is sufficiently large. Unfortunately, to the best of our knowledge, this ideal adaptation isn’t possible.

On the technical side, we see that the method is much easier to describe when computing logarithm in ${\mathbb{F}}_{q^{k}}$ for an odd extension degree $k$. Indeed, in that case, the compositum of ${\mathbb{F}}_{q^{k}}$ and any extension ${\mathbb{F}}_{q^{2^{i}}}$ is simply ${\mathbb{F}}_{q^{k2^{i}}}$. Making this assumption is very convenient to describe the adaptation to the elliptic representation.

5.3.1 Elliptic zig-zag descent.

We now go back to the elliptic representation setting, with the additional restriction that the extension degree $k$ is odd.

Points and divisors over extensions.

As mentioned in the last paragraph of Section 3, the commutative diagram in Figure 3 can be used not only over ${\mathbb{F}}_{q}$ but also over extensions. We now give more details for ${\mathbb{F}}_{q^{d}}$, assuming that $d$ and $k$ are coprime.

This we now use polynomials $A$ and $B$ with coefficients in the larger field ${\mathbb{F}}_{q^{d}}$. Everything remains almost identical, except the definition and properties of the bracket. With a larger field, we use:

[TABLE]

where $A^{\pi}$ denotes the polynomial derived from $A$ by raising each coefficient of $A$ to the power $q$ (while keeping the same monomials). This new bracket $\left[{\cdot},{\cdot}\right]_{*}$ is ${\mathbb{F}}_{q}$-bilinear (but not ${\mathbb{F}}_{q^{d}}$-bilinear).

Bootstrapping the descent.

Let $z\in{\mathbb{F}}_{q^{k}}$ be our target arbitrary element for which we want to find a discrete logarithm. Thanks to the diagram of Figure 3, we know that there exists a polynomial Pol in ${\mathbb{F}}_{q}[U,V]$ such that:

[TABLE]

In fact, there are many such polynomials. We choose $\ell$ such that $2^{\ell}>k$ and search a representation by a polynomial Pol in ${\mathbb{F}}_{q}[U,V]$ such that:

$z=\Psi(\Xi(\Phi^{*}(\mbox{Pol}(U,V)))).$ 2. 2.

$h(\Xi(\Phi^{*}(\mbox{Pol}(U,V))))=2^{\ell}.$ 3. 3.

$\Xi(\Phi^{*}(\mbox{Pol}(U,V)))$ exactly corresponds to a place of degree $2^{\ell}.$

Let us call $p_{z}$ such a place in $\Sigma_{{\mathbb{F}}_{q}({\mathscr{C}})}$. We could lift it to $\Sigma_{{\mathbb{F}}_{q^{2^{\ell}}}({\mathscr{C}})}$ so that it corresponds to $2^{\ell}$ points. However, for the rest of the method, it suffices to decompose it into degree-$8$ places. Theses places appear in $\Sigma_{{\mathbb{F}}_{q^{2^{\ell-3}}}({\mathscr{C}})}$.

Descending degree-$8$ places.

Using a series of relations based on the bracket $\left[{A},{B}\right]_{*}$, there is a way to express the logarithm of the divisor corresponding to a degree-$8$ place as a sum of logarithms of divisors of degree at most $4$. Once this is done, we can pair conjugates divisors and go one step down in the tower of quadratic extension. This at most doubles the height of divisors. Iterating the process, we now encounter places of degree $6$ and $8$ whose divisors need to be expressed as combination of divisors of degree at most $4$. Finally, at the bottom of the tower, everything can be expressed using divisors of height at most $4$, this in turn permit to compute the logarithm of $z$.

Keeping this strategy in mind we now describe the transformation of logarithm of divisors into sums of divisors of lower height. More precisely, we first transform degree $8$ places as sums using divisors of height at most $6$. Places of degree $6$ can be expressed using divisors of height up to $5$. Finally place of degree $5$ are transformed using divisors of height up to $4$.

The exact degrees appearing in the descent strategy depend on the relative position in the tower of extension. Except at the lower levels, it is possible to descent directly from degree $8$ to degree $5$ and from degree $6$ to degree $4$. Except at the lower levels, it even possible to descend from degree $4$ to degree $3$.

Thus, from a practical point of view,there are two essentially equivalent options for the descent. Either one starts from a degree-$8$ place and encounters descent steps from $8$ to $5$ then $4$ and descent steps from $6$ to $4$, except in the lower levels where longer chains from $8$ to $6$ then $5$ and finally $4$ appear. Or one starts from a degree $4$ place and encounters descent steps from $6$ to $4$ then $3$ and steps from $4$ to $3$. At the lower levels, this approach gets stuck.

In our presentation, we choose the approach that starts from a degree-$8$ place. Note than in the context of provable algorithms, using degree $8$ possibly leads to a more difficult proof.

Degree-$5$ places.

We start with degree $5$ places since it is slightly simpler and illustrates the general idea. We let $d$ be the power of $2$ corresponding to our current position in the tower of extensions.

Again, we create somehow relations from:

[TABLE]

where $(A,B)$ is a pair of polynomials with coefficients in ${\mathbb{F}}_{q^{d}}$. For degree $5$, the polynomials are built from monomials in $\mathcal{M}_{1}$, i.e. from $1$, $U$, $V$ and $UV$. To check whether there are enough degrees of freedom to force a place too appear, we need to consider how many (non equivalent) pairs of candidates relations we can try.

Since we use the new bracket instead of  $\left[{\cdot},{\cdot}\right]$, the counting changes slightly. Before considering the property of the bracket, there is a total of $8$ coefficients in ${\mathbb{F}}_{q^{d}}$, four in each of $A$ and $B$. Remark that, for any $\Lambda\in{\mathbb{F}}_{q^{d}}$, we have $\left[{\Lambda A},{\Lambda B}\right]_{*}=\Lambda^{q+1}\left[{A},{B}\right]_{*}$. Simultaneously, the left side corresponding to $(\Lambda A,\Lambda B)$ is $\Lambda^{q+1}\prod_{\alpha\in{\mathbb{P}}_{1}({\mathbb{F}}_{q})}(A-\alpha B)$. Since $\Lambda^{q+1}$ appears on both sides, we see that $(A,B)$ and $(\Lambda A,\Lambda B)$ generate the same equation. Thus, we can set the leading coefficient of $A$ to $1$. This removes one of the coefficients.

In addition, because of the ${\mathbb{F}}_{q}$-linearity of the bracket, for any $\lambda\in{\mathbb{F}}_{q}$, we have $\left[{A},{B-\lambda A}\right]_{*}=\left[{A},{B}\right]_{*}.$ Thus we can fix one component of the leading coefficient of $B$ to [math]. Then, using $\left[{A},{\lambda B}\right]_{*}=\lambda\left[{A},{B}\right]_{*}$ we can fix one component of another coefficient of $B$ to $1$. Finally, thanks to the relation $\left[{A-\lambda B},{B}\right]_{*}=\left[{A},{B}\right]_{*},$ we can set the corresponding component in $A$ to [math].

This decreases the numbers of degrees of freedom to $7-3/d>5,$ when $d>1$. Thus, we have enough degrees of freedom available. In the case $d=1$, we are in the base field where the logarithms of the degree-5 places have already been precomputed as part of the extended factor base.

Let $p_{5}$ be a place of degree $5$ in $\Sigma_{{\mathbb{F}}_{q^{d}}({\mathscr{C}})}$. Using a variation on bilinear descent and solving a bilinear system of equations in $6$ unknowns over the extension field ${\mathbb{F}}_{q^{d}}$, we can obtained a relation involving $p_{5}$. Since the number of variables is a small constant, this is a very efficient computation.

The relation can be written in the form:

[TABLE]

where $D_{3}$ is a divisor of height $3$. This shows that we can descend any divisors of height $5$ as a sum of divisors of height at most $4$. Note that $D_{3}$ and the divisors on the left may not be elementary, however, in that case they decompose into elementary divisors of lower degrees.

Note that we do not prove here that such a decomposition always exists. Instead out counting of the degrees of freedom gives heuristic support to this fact. It might be possible to adapt the proofs of [Lid19] or [KW19] to our specific setting.

Degree-$6$ places.

For degree $6$, there are two options depending of the extension degree $d$.

When $d\geq 4$, we can again build relations using only the monomials $1$, $U$, $V$ and $UV$. In this case, it gives $7-3/d>6$ degrees of freedom. Thus, we can directly descend to a sum of divisors of height at most $4$.

For the remaining cases, $d=2$ or $d=1$, we need to use monomials from $\mathcal{M}_{2}$ to provide more degrees of freedom. However, if we use them all, the height of the left-hand factor become $8$ and the height of the bracket is $16$. To control this explosion, it suffices to fix three (essentially arbitrary) compelled points and keep a basis of all functions going through these $3$ points. This basis contains $5$ polynomials, say $g_{1}$, …, $g_{5}$. Forming $A$ and $B$ as linear combinations of the $g_{i}$s induces a systematic factor of total height $3$ in every term $A-\alpha B$ (corresponding to the compelled points). Furthermore, this systematic factor also appears in the decomposition of the bracket together with an extra systematic factor also of height $3$. This extra factor corresponds to the compelled points translated by $-P_{1}$. Thanks to the systematic factors, the height of the left becomes $5$ while the height of the right becomes $10$. There is a total of $10$ coefficients in $A$ and $B$, which corresponds to $9-3/d\geq 6$ degrees of freedom when removing identical relations as in the previous case. More precisely, we can fix the coefficient of $g_{1}$ in $A$ to $1$, one component of the coefficient of $g_{1}$ in $B$ to [math], and one component of $g_{2}$ to $1$ in $B$ and [math] in $A$.

Solving a bilinear system, we can find coefficients that lead to an equation:

[TABLE]

where $D_{4}$ is a divisor of height $4$. This expresses the logarithm $p_{6}$ as a sum of logarithms of divisors of height at most $5$.

Degree-$8$ places.

For degree $8$ places, we proceed as in the second method for degree $6$. We use monomials from $\mathcal{M}_{2}$. With three compelled points as in degree $6$, we have $9-3/d$ degree of freedom. This is more than $8$ as soon as $d\geq 4$. In this case, we can write the logarithm $p_{8}$ as a sum of logarithms of divisors of height at most $5$.

When $d$ is $1$ or $2$, we use only two compelled points. We thus have a basis of $6$ polynomials and $9-3/d\geq 8$ degrees of freedom. The height after removing the systematic factors become $6$ for the left factors and $12$ for the bracket. Thus, in the lower levels of the tower of extension, we can write the logarithm $p_{8}$ as a sum of logarithms of divisors of height at most $6$.

Practical (un)efficiency of the approach.

In the Frobenius representation zig-zag, every step down the tower was based on the creation of one relation. As a consequence, at every level, the total number of elements under consideration was multiplied by $O(q)$.

By contrast, here, we need two levels of relations for each of the middle levels of the tower. As a consequence, the total number at each level is multiplied by $O(q^{2})$, which makes this approach much less appealing in practice.

Acknowledgments

This work has been supported by the European Union’s H2020 Programme under grant agreement number ERC-669891.

Appendix A Details on the curve model

In order to understand our model of ${\mathscr{C}}$, we analyze how points of ${\mathscr{E}}$ are mapped by $\Phi$ to the model ${\mathscr{C}}$ on the three variables $U$, $V$ and $W$. We assume for simplicity that ${\mathscr{E}}$ is given by a reduced Weirstrass equation, but this can be generalized to include characteristic $2$ and $3$.

Equations of ${\mathscr{C}}$.

We assume that ${\mathscr{E}}$ is given by a reduced Weierstrass equation:

[TABLE]

In this case, the third summation polynomial is given by:

[TABLE]

where the $\sigma_{i}$ are the symmetric polynomials:

[TABLE]

Let us first consider the variety given by the equations $S_{3}(U,V,x_{1})=0,$ $S_{3}(V,W,x_{1})=0$ and $S_{3}(U,W,x_{2})=0.$ To determine its components, let us consider its intersection with the hyperplane $U=W$. This intersection is described by $S_{3}(U,V,x_{1})=0$, $U=W$ and $S_{3}(U,U,x_{2})=0.$ From the third equation that is a degree-$4$ polynomial in $U$, we know that $U$ has finitely many values. Thus we want to remove the extraneous points lying in this hyperplane.

We look so at the components in the complement of this hyperplane and assume that $U\neq W$. In this case, since $S_{3}(U,V,x_{1})-S_{3}(V,W,x_{1})$ is divisible by $U-W$ we obtain a lower degree polynomial, namely:

[TABLE]

The variety defined thanks to the equations $S_{3}(U,V,x_{1})=0$, $S_{\delta}=0$ and $S_{3}(U,W,x_{2})=0$ is now irreducible. We call it ${\mathscr{C}}$ and prove that it is a genus $1$ curve isomorphic to ${\mathscr{E}}$. To see that, let us give rational maps between ${\mathscr{E}}$ and ${\mathscr{C}}$.

Mappings between ${\mathscr{E}}$ and ${\mathscr{C}}$.

In the forward direction, let us consider the rational map:

[TABLE]

Every point $P$ in ${\mathscr{E}}$ is such that $\Phi(P)\in{\mathscr{C}}$.

Besides, the images of the three points $\mathcal{O}$, $P_{1}$ and $-P_{1}$ are at infinity on ${\mathscr{C}}$ and that, by homogenization, we may check that there are exactly three points at infinity on ${\mathscr{C}}$.

As usual, $\Phi$ induces a map $\Phi^{*}$ from the function field ${\mathbb{F}}_{q}({\mathscr{C}})$ to ${\mathbb{F}}_{q}({\mathscr{E}})$ (expressed with the two variables $X$ and $Y$) using the following replacement:

[TABLE]

where $y_{1}$ is the ordinate of the point $P_{1}$ in ${\mathscr{E}}$. Developing and using the curve equation, the images of $U$ and $W$ can be respectively simplified to:

[TABLE]

In this form, it is clear that the map can be easily inverted when $X\neq x_{1}$. Moreover, given a pair $(U,V)$ values we can compute the value of $W$ and similarly, from $(V,W)$ we can compute $U$.

When $X=x_{1}$, we have two possibilities $\Phi(P_{1})=(\infty,x_{P_{1}},x_{P_{2}})$ and $\Phi(-P_{1})=(x_{P_{2}},x_{P_{1}},\infty)$. These are distinct (unless $P_{1}$ has order $2$), which means that $\Phi$ is a bijection and thus an isomorphism.

Appendix B Analysis of the invertibility of $N_{D}$

In this appendix, we analyze the condition that appears in Section 3.3 when explicitly writing down the definition of the morphism $\Psi$. Indeed, as previously explained, we need $N_{D}$ to be invertible in the group where we want to compute discrete logarithms.

This analysis requires us to follow standard practice and first decompose our target group ${\mathbb{F}}_{q^{k}}^{*}$, in order to apply Pohlig-Hellman algorithm [PH78]. Thanks to this method it suffices to compute discrete logarithms in all prime order subgroups of ${\mathbb{F}}_{q^{k}}^{*}$.

An important technicality is that we would need to first factor $q^{k}-1$. Unfortunately, this would completely dominate the cost of computation. However, to study the invertibility of $N_{D},$ we do not need to factor $q^{k}-1$ fully, the existence of the factorization suffices.

Let $\gamma$ be a prime factor dividing $q^{k}-1$, it suffices to check that $N_{D}\neq 0\mod\gamma$, for $\Psi$ to be well-defined in subgroups of order a power of $\gamma$ in ${\mathbb{F}}_{q^{k}}^{*}$. Since $N_{D}$ is defined as the least common multiple of the cardinalities of ${\mathscr{E}}$ over each of the finite fields ${\mathbb{F}}_{q^{d}}$ with $1\leq d\leq D$, where $D\leq k$ is the maximum degree of the places we want to consider, this gives us an extra condition on ${\mathscr{E}}$. Namely, it should satisfy the following property:

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For any $i=1,\cdots,D$, $|{\mathscr{E}}/{\mathbb{F}}_{q^{i}}|\neq 0\mod\gamma$.

Let us study this condition. We denote by $t$ the trace of ${\mathscr{E}}$ over ${\mathbb{F}}_{q}$ and factor the characteristic polynomial of the Frobenius of ${\mathscr{E}}$:

[TABLE]

with $r$ and $s$ in ${\mathbb{F}}_{\gamma^{2}}$. We know that the number of points of ${\mathscr{E}}/{\mathbb{F}}_{q^{i}}$ is equal to $(1-r^{i})(1-s^{i})\mod\gamma$. Thus, to ensure that the cardinalities of ${\mathscr{E}}$ over the field extensions ${\mathbb{F}}_{q^{i}}$, with $i$ in $[1,D]$, all differ from [math] modulo $\gamma$ we just need to verify that both $r^{i}\neq 1\mod\gamma$ and $s^{i}\neq 1\mod\gamma$. In order to do that, let us first study the order of the product $rs=q\mod\gamma$.

By definition of $\gamma$, we have $q^{k}=1\mod\gamma$. Furthermore, the order of $q$ is strictly smaller than $k$ modulo $\gamma$ if and only if $\gamma$ already divides the order of the multiplicative group of a subfield of ${\mathbb{F}}_{q^{k}}$. In that case, we compute this part of the logarithm by applying our method to the smallest such subfield.

We now assume that the order of $q$ is precisely $k\mod\gamma$. Thus, for any $i$ not a multiple of $k$, at most one of $r^{i}$ or $s^{i}$ can be equal to $1\mod\gamma$. Exchanging $r$ and $s$ if necessary, we now study the case $r^{i}=1\mod\gamma.$ In that case, we have $s^{i}=q^{i}\neq 1\mod\gamma.$ This implies that ${\mathscr{E}}({\mathbb{F}}_{q^{i}})$ contains a $\gamma$-torsion point $Q_{\gamma}$ but not the full $\gamma$-torsion ${\mathscr{E}}[\gamma]$. Thus, the Tate pairing provides a non-degenerate bilinear map to the $\gamma$ roots of unity:

[TABLE]

Fixing an arbitrary non-zero element from ${\mathscr{E}}({\mathbb{F}}_{q^{i}})/\gamma{\mathscr{E}}({\mathbb{F}}_{q^{i}})$, we obtain a linear map $\tilde{\Psi_{i}}$ from the subgroup generated by $Q_{\gamma}$ to the $\gamma$-th roots of unity.

Possibly after renormalization, $\tilde{\Psi_{i}}$ gives a compatible replacement for $\Psi$ that can be applied to the $\gamma$-torsion point. The renormalization consists in replacing $\tilde{\Psi_{i}}$ by $\tilde{\Psi_{i}}^{\beta_{i}}$, where $\beta_{i}$ is the renormalization constant. As a consequence, it is mathematically possible to extend $\Psi$ to all divisors. One computational caveat is that determining the value of $\beta_{i}$ can be expressed as a discrete logarihm problem in the group of order $\gamma$. It does not affect the efficiency of the overall algorithm but prevents independent check on relations containing divisors of degree $i$ not compatible with the definition of $\Psi$ during the precomputation phase.

To see how $\beta_{i}$ can be determined, let us take a place $p_{i}$ of degree $i$ and compute $\tilde{\Psi_{i}}(p_{i})$ (as usual this it the product of the value for all the conjugate points in $p_{i}$). Then apply one step of the descent algorithm to relate $p_{i}$ to places of degrees $\neq i$ which are all compatible with the computation of $\Psi$. Multiplying these contributions gives the renormalized value $\tilde{\Psi_{i}}(p_{i})^{\beta_{i}}.$ Thus, if we wish to do so, we can compute $\beta_{i}$ from the individual logarithms of these two values.

Appendix C Relations for factor base extension

Let us describe our decomposition in groups to extend the factor base to all elementary divisors of height equal or lower than $4$. The idea is to write a partition of $q$ groups with $q^{2}$ elements in each and to be able to decrease the height of the divisor associated to the bracket on the right again. To illustrate the process, we define a first group with the monomials:

[TABLE]

Defining then $G$ as all the linear combinations of these three monomials with coefficients in ${\mathbb{F}}_{q}$ permits to set our first (special) group as:

[TABLE]

All the divisors in the special group have height lower than $4$. We now sieve on pairs of polynomials $(A,B)$ such that $A=g_{1}+\alpha g_{2}$ and $B=g_{1}+\beta g_{3}$ where $\alpha,\beta\in{\mathbb{F}}_{q}$. On the left side it is clear that all polynomials raised in the product belong to $G$. So all the divisors in the corresponding sum on the left side have height lower than $4$ (see Table 1) and belong to $\mathcal{G}$. On the right side, we are left with a bracket $\left[{A},{B}\right]$ leading to a height lower than $8$. Again, the probability that it splits into divisors with a height lower than $3$ is too low. Yet, computing the brackets:

[TABLE]

and thanks to bilinearity we obtain that $W-U$ is a common factor of $\left[{A},{B}\right]$. Besides we note that $h(\Xi(\Phi^{*}(W-U)))=4$.

Removing this constant contribution, we are left with a residual height of $4$ on the right side. If it decomposes into lower height divisors, this gives us a linear equation involving the logarithms a subset of the divisors in $\mathcal{G}$. With enough such equations, we again use linear algebra to compute the logarithms of the elements of $\mathcal{G}$.

Note that the probability to find a good relation is $3/4$ when $q$ grows. We thus expect $3q^{2}/4$ equations in $q^{2}/4$.

Note that the pairs $(A,B)$ that fail to give an equation are nonetheless useful! Indeed, a pair $(A,B)$ of sieving polynomials fails if $\left[{A},{B}\right]$ leads to a divisor with height precisely $4$. It means that after obtain the logarithm of elements of $\mathcal{G}$ we can derive the logarithm of these extra divisors for free.

Construction of groups with one compelled point.

Following the idea of the special group $\mathcal{G}$, we would like to construct small other groups of divisors that have two properties. First, for each group, we need to be able to create relations involving only heigh-$4$ divisors from this group on the left, possibly with divisors of lower height. Second, we need to control the splitting probability of the bracket on the right of the equation. We proceed using compelled points.

How to choose our generators $g_{1},g_{2}$ and $g_{3}$ in this case ? We recall that the naive height-$4$ sieving is based on the monomials $1,U,V,UV$. Since there is no reason to favor nor $U$ neither $V$, we propose to preserve symmetry between the two variables, writing:

[TABLE]

where $k_{1}$ and $k_{2}$ are in the base field ${\mathbb{F}}_{q}$. Defining again groups:

[TABLE]

with $q^{2}$ divisors each, we sieve on pairs of polynomials $(A,B)$ such that $A=g_{1}+\alpha g_{2}$ and $B=g_{1}+\beta g_{3}$ where $\alpha,\beta\in{\mathbb{F}}_{q}$. On the left side all divisors have height lower than $4$ and belong to $\mathcal{G}_{k_{1},k_{2}}$. On the right side, we are left with a bracket $\left[{A},{B}\right]$ leading to a height lower than $8$. To decrease this height we consider the brackets:

[TABLE]

Note that $\left[{A},{B}\right]$ is a linear combination of these brackets and that the last two ones are associated to divisors of height lower than $4$. Thus, removing a point in $\Xi(\Phi^{*}(\left[{g_{1}},{g_{2}}\right]))$ will suffice. Let us look at $k_{1}\left[{U},{UV}\right]+k_{2}\left[{UV},{V}\right]+k_{1}k_{2}\left[{U},{V}\right]$ in details. Calling $c_{f}$ the coefficient in ${\mathbb{F}}_{q}$ of the leading monomial555Considering the weighted degree in $X$ and $Y$ of each monomial. of $\left[{g_{1}},{g_{2}}\right]$ for the denominator of any fraction $f$, we see that we can force $k_{1}c_{\left[{U},{UV}\right]}+k_{2}c_{\left[{UV},{V}\right]}+k_{1}k_{2}c_{\left[{U},{V}\right]}=0$ in ${\mathbb{F}}_{q}$. We underline that for any fixed constant $k_{1}\neq-c_{\left[{UV},{V}\right]}c_{\left[{U},{V}\right]}^{-1}$ there exists a unique $k_{2}$ such that the previous equality is verified. It means that we create $q-1$ such groups. Besides, this annihilates the leading monomial so decreases the weighted degree of $\left[{g_{1}},{g_{2}}\right]$ and leads to remove a point in the associated divisor of $\left[{A},{B}\right]$. Hence, we are left with a residual height of $7$. We want the corresponding divisor to be written as a sum of divisors of height $3$ at most. The heuristic probability to get a good relation is so equal to:

[TABLE]

as $q$ tends to infinity.

This is slightly too low for the purpose. As a consequence, we need either to improve the group construction or to make good use of the equations with a single height-$4$ divisor in the bracket. Nevertheless, since $0.24$ is close to $1/4$, it is conceivable that we can find enough relations in practice. We decided to test it and we computed all the discrete logarithms up to height-$5$ for the target finite field ${\mathbb{F}}_{3^{1345}}={\mathbb{F}}_{243^{269}}$ with this method.

Remark 5**.**

It is useful to know that:

[TABLE]

Interaction with the action of Frobenius.

Looking at our groupings, we see that we have built a total of $q$ different ones (including the special group $\mathcal{G}$. Since each grouping contains about $q^{2}/4$, the computations (if successful) gives us about $q^{3}/4$ logarithms of height $4$. This is much less than the total expected number which is close to $q^{4}/4$. However, the action of the Frobenius potentially multiply these logarithms by a factor of $k$. For practical, we heuristically assume that this is the case. The fact that we were able to compute the logarithms of all height-$4$ divisors for ${\mathbb{F}}_{3^{1345}}$ supports this assumption.

Going to height $5$.

We continue the extension to height $5$ in similar fashion. Since, this requires more degree of freedom, we no longer need to use compelled points. Instead, we sieve on more general polynomials of the forms $A=UV+a_{U}U+a_{1}$ and $B=UV+b_{V}V+b_{1}$. On the left-hand side, all factors of the form $A-\alpha B$ have height $4$. Thus they decompose in divisors of height at most $4$ and their logarithms can be directly obtained. On the right-hand side, the bracket has height at most $8$. We expect that it contains an elementary divisor of height $5$ with probability close to $1/5$. As a consequence, we obtain about $q^{4}/5$ divisors of height $5$ without performing any linear algebra.

Again, thanks to the action of Frobenius, we expect to recover an overwhelming fraction of divisors of height $5$. This turn out to work in practice for our example ${\mathbb{F}}_{3^{1345}}$.