On the first fall degree of summation polynomials

TL;DR

This paper improves the bound on the first fall degree of polynomial systems derived from Weil descent and Semaev's summation polynomials, which are used in solving the Elliptic Curve Discrete Logarithm Problem with Gr"obner basis methods.

Contribution

It provides a tighter bound on the first fall degree for these polynomial systems, enhancing understanding of their complexity in cryptographic applications.

Findings

Improved the first fall degree bound for specific polynomial systems.

Enhanced analysis of the complexity of solving elliptic curve discrete logs.

Contributed to the theoretical understanding of polynomial system behavior in cryptography.

Abstract

We improve on the first fall degree bound of polynomial systems that arise from a Weil descent along Semaev's summation polynomials relevant to the solution of the Elliptic Curve Discrete Logarithm Problem via Gr\"obner basis algorithms.

Tables

Table 1. Table 1. Empirical data for the Weil descent along the summation polynomial S m + 1 subscript 𝑆 𝑚 1 S_{m+1} over 𝔽 2 n subscript 𝔽 superscript 2 𝑛 \mathbb{F}_{2^{n}} with n ′ superscript 𝑛 ′ n^{\prime} -dimensional factor basis. Displayed are the observed first fall degree D f ​ f subscript 𝐷 𝑓 𝑓 D_{ff} , degree of regularity D r ​ e ​ g subscript 𝐷 𝑟 𝑒 𝑔 D_{reg} , the time in seconds s 𝑠 s and space requirement in gigabyte GB. All values are averaged over 10 10 10 repetitions. For the case m = 2 𝑚 2 m=2 see also § 3 .

$m$	$n$	$n^{\prime }$	$m\left(m-1\right)+1$	$D_{ff}$	$D_{reg}$	$s$	GB
$2$	$34$	$17$	$3$	$2$	$4$	$188$	$1.2$
$2$	$35$	$18$	$3$	$2$	$4$	$1237$	$16.1$
$2$	$36$	$18$	$3$	$2$	$4$	$1342$	$16.4$
$2$	$37$	$19$	$3$	$2$	$5$	$2542$	$29.2$
$2$	$38$	$19$	$3$	$2$	$5$	$2815$	$25.2$
$2$	$39$	$20$	$3$	$2$	$5$	$4785$	$45.6$
$2$	$40$	$20$	$3$	$2$	$5$	$4858$	$46.3$
$2$	$41$	$21$	$3$	$2$	$5$	$7930$	$65.3$
$2$	$42$	$21$	$3$	$2$	$5$	$8901$	$66.7$
$2$	$43$	$22$	$3$	$2$	$5$	$16816$	$95.5$
$2$	$44$	$22$	$3$	$2$	$5$	$15690$	$96.8$
$2$	$45$	$23$	$3$	$2$	$5$	$38352$	$140.0$
$2$	$46$	$23$	$3$	$2$	$5$	$31735$	$140.7$
$2$	$47$	$24$	$3$	$2$	$5$	$103200$	$207.7$
$2$	$48$	$24$	$3$	$2$	$5$	$86636$	$208.2$
$3$	$13$	$5$	$7$	$7$	$7$	$14$	$0.6$
$3$	$14$	$5$	$7$	$7$	$7$	$14$	$0.7$
$3$	$15$	$5$	$7$	$7$	$7$	$14$	$0.7$
$3$	$16$	$6$	$7$	$7$	$7$	$597$	$13.5$
$3$	$17$	$6$	$7$	$7$	$7$	$656$	$13.3$
$3$	$18$	$6$	$7$	$7$	$7$	$729$	$34.1$
$3$	$19$	$7$	$7$	$7$	$7$	$16571$	$92.2$
$3$	$20$	$7$	$7$	$7$	$7$	$17684$	$101.2$
$3$	$21$	$7$	$7$	$7$	$7$	$17681$	$90.2$
$4$	$13$	$4$	$13$	$13$	$13$	$467$	$25.0$
$4$	$14$	$4$	$13$	$13$	$13$	$487$	$25.8$
$4$	$15$	$4$	$13$	$13$	$13$	$592$	$26.3$
$4$	$16$	$4$	$13$	$13$	$13$	$755$	$27.6$

Equations

$$S={\mathbb{F}}_{2^n}[X_0,\dots ,X_{N-1}]/(X_0^2,\dots ,X_{N-1}^2)$$

$$S=S_0\oplus S_1\oplus \cdots \oplus S_N.$$

$$\begin{array}[]{cccl}\phi:&S^{r}&\longrightarrow&I\\ &(g_{1},\ldots,g_{r})&\mapsto&g_{1}h_{1}+\cdots+g_{r}h_{r}.\end{array}$$

$$\displaystyle h_{j}e_{i}+h_{i}e_{j}\mbox{ and }h_{k}e_{k}\mbox{, where $i,j,k=1,\ldots,r$,}$$

$$\begin{array}[]{cccl}\phi_{j-d}:&S^{r}_{j-d}&\longrightarrow&I\cap S_{j}\end{array}$$

$${\bar{\varphi }}_{j-d}:S_{j-d}^r/U_{j-d}\to I\cap S_j.$$

$$A_{{\mathbb{F}}_{2^n}}={\mathbb{F}}_{2^n}[X_0,\dots ,X_{N-1}]/(X_0^2-X_0,\dots ,X_{N-1}^2-X_{N-1})$$

$$\mathrm{Gr}(A_{{\mathbb{F}}_{2^n}})={\mathbb{F}}_{2^n}[X_0,\dots ,X_{N-1}]/(X_0^2,\dots ,X_{N-1}^2)$$

$$[\mathrm{Gr}(A_{\mathbb{F}_{2^{n}}})]_{d}=[A_{\mathbb{F}_{2^{n}}}]_{d}/[A_{\mathbb{F}_{2^{n}}}]_{d-1}\mbox{, for $d\in\mathbb{N}$},$$

$$D_{ff}\left(V\right)=\begin{cases}d&\mbox{, $\dim_{\mathbb{F}_{2^{n}}}\bar{V}<\dim_{\mathbb{F}_{2^{n}}}V$,}\\ D_{ff}(\bar{V})&\mbox{, else,}\end{cases}$$

$$S_2(x_1,x_2)$$

$$S_3(x_1,x_2,x_3)$$

$$S_{m+1}(x_1\dots ,x_m,x_{m+1})$$

$$x_i=\sum _{l=1}^{n^{\prime }}{y}_{il}{\nu }_l,$$

$$S_{m+1}(x_1,\dots ,x_m,c)$$

$$=f_0(y_{ij})+z{f}_1(y_{ij})+\cdots +z^{n-1}{f}_{n-1}(y_{ij})$$

$$s_k\equiv f_k\mathrm{mod}(y_{11}^2-y_{11},\dots ,y_{m{n}^{\prime }}^2-y_{m{n}^{\prime }})\text{mbox},wherek=0,\dots ,n-1.$$

$$(x_1\cdots x_m{)}^{2^{m-1}-1}\cdot x_{m+1}$$

$$S_3(x_1,x_2,x_3)$$

$$S_{m+1}(x_1\dots ,x_m,x_{m+1})$$

$${\mathrm{Res}}_X(f,g)$$

$$=\det \left(\begin{array}{cccccccc}{f}_k& & \cdots & & f_0& & & \\ & f_k& & \cdots & & f_0& & \\ & & \ddots & & & & \ddots & \\ & & & f_k& & \cdots & & f_0\\ g_l& & \cdots & & g_0& & & \\ & g_l& & \cdots & & g_0& & \\ & & \ddots & & & & \ddots & \\ & & & g_l& & \cdots & & g_0\end{array}\right)$$

$$S_3(x_m,x_{m+1},X)$$

$$S_m(x_1,\dots ,x_{m-1},X)$$

$$\displaystyle S_{m+1}(x_{1}\ldots,x_{m},x_{m+1})=\det\big{(}\operatorname{Syl}(S_{m},S_{3})\big{)}.$$

$$\left(\begin{array}{cccccc}{c}_{2^{m-2},m}& c_{2^{m-2}-1,m}& \cdots & c_{0,m}& 0& \\ 0& c_{2^{m-2},m}& \cdots & c_{1,m}& c_{0,m}& \\ x_m^2+x_{m+1}^2& x_m{x}_{m+1}& x_m^2{x}_{m+1}^2+t& & & \\ & \ddots & & \ddots & & \\ & & x_m^2+x_{m+1}^2& x_m{x}_{m+1}& x_m^2{x}_{m+1}^2+t& \end{array}\right)$$

$$\displaystyle\det\big{(}\operatorname{Syl}(S_{m},S_{3})\big{)}=\sum_{\pi}\mathrm{sgn}(\pi)\prod_{i=1}^{2^{m-2}+2}\operatorname{Syl}(S_{m},S_{3})_{i,\pi_{i}}$$

$$\sigma$$

$$=\left(2^{m-2}+1,2^{m-2}+2,1,2,\dots ,2^{m-2}\right)$$

$$S_{m+1}(x_1\dots ,x_m,x_{m+1})$$

$$=c_{0,m}{c}_{0,m}\prod _{i=1}^{2^{m-2}}(x_m^2+x_{m+1}^2)+\dots$$

$$={\left((x_1\cdots x_{m-1}{)}^{2^{m-2}}\right)}^2\cdot x_m^{2^{m-1}}+\dots$$

$$=(x_1\cdots x_{m-1}{x}_m{)}^{2^{m-1}}+\dots$$

$$\tau$$

$$=\left(2^{m-2},2^{m-2}+2,1,\dots ,2^{m-2}-1,2^{m-2}+1\right)$$

Full text

KEYWORDS. Polynomial systems, Gröbner bases, Discrete logarithm problem, Elliptic curve cryptosystem.

2010 MATHEMATICS SUBJECT CLASSIFICATION. 13P15 Solving polynomial systems, 13P10 Gröbner bases, 14H52 Elliptic curves.

1. Introduction

Finding solutions to algebraic equations is a fundamental task. A common approach is a Gröbner basis computation via an algorithm such as Faugère’s $F4$ and $F5$ (see [4, 5]). In recent applications Gröbner basis techniques have become relevant to the solution of the Elliptic Curve Discrete Logarithm Problem (ECDLP). Here one seeks solutions to polynomial equations arising from a Weil descent along Semaev’s summation polynomials [13] which represents a crucial step in an index calculus method for the ECDLP, see e.g. [12, 14]. The efficiency of Gröbner basis algorithms is governed by a so-called degree of regularity, that is the highest degree occurring along the subsequent computation of algebraic relations. It is widely believed that this often intractable complexity parameter is closely approximated by the degree of the first non-trivial algebraic relation, the first fall degree. In particular, the algorithms for the ECDLP of Petit and Quisquater [12] are sub-exponential under the assumption that this approximation is in o$(1)$.

In the present paper, we will improve Petit’s and Quisquater’s [12] first fall degree bound $m^{2}+1$ for the system arising from the Weil descent along Semaev’s $(m+1)$-th summation polynomial. That is, we prove that a degree fall occurs at degree $m^{2}-m+1$ by exhibiting the highest degree homogeneous part of that polynomial system. In fact, this degree is $m^{2}-m$, so that we expect the bound to be sharp except for the somewhat pathological case $m=2$ that has been discussed by Kosters and Yeo [10]. This allows us to sharpen the asymptotic run time of the index calculus algorithm for the ECDLP as exhibited in the complexity analysis of Petit and Quisquater [12].

2. The first fall degree

The notion of the first fall has been described by Faugère and Joux [6, Section 5.1], Granboulan, Joux and Stern [7, Section 3], Dubois and Gama [3, Section 2.2], and Ding and Hodges [2, Section 3]. Although the concept of the first fall degree has been called minimal degree [6] and degree of regularity [2, 3, 7], we actually adopt the terminology and definition of Hodges, Petit and Schlather [8]. For readability reasons we include a brief and tailored account of the first fall degree and refer the reader to [8, Section 2] for details and greater generality.

Our considerations take place over a degree $n$ extension $\mathbb{F}_{2^{n}}$ of the binary field $\mathbb{F}_{2}$. Consider the decomposition of the graded ring

[TABLE]

into its homogeneous components

[TABLE]

Each $S_{j}$ is the $\mathbb{F}_{2^{n}}$-vector space generated by the monomials of degree $j$. Let $I$ be an ideal in $S$ generated by homogeneous polynomials $h_{1},\ldots,h_{r}\in S_{d}$ all of the same degree $d$. Then we have a surjective map

[TABLE]

Without loss of generality we furthermore assume $0<r=\dim_{\mathbb{F}_{2^{n}}}\sum_{j=1}^{r}\mathbb{F}_{2^{n}}h_{j}$. Let $e_{i}$ denote the canonical $i$-th basis element of the free $S$-module $S^{r}$. The $S$-module $U$ generated by the elements

[TABLE]

is a subset of $\ker(\phi)$. If we restrict $\phi$ to the $\mathbb{F}_{2^{n}}$-subvector space $S^{r}_{j-d}\subset S^{r}$ we obtain a surjective map

[TABLE]

whose kernel contains the $\mathbb{F}_{2^{n}}$-subvector space $U_{j-d}=U\cap S^{r}_{j-d}$ and hence factors through

[TABLE]

Definition \thedfn (Cf. [8, Definition 2.1]).

The first fall degree of a homogeneous system $h_{1},\ldots,h_{r}\in S_{d}$ and its linear span $\sum_{j=1}^{r}\mathbb{F}_{2^{n}}h_{j}$, respectively, is the smallest $j$ such that the induced $\mathbb{F}_{2^{n}}$-linear map $\bar{\phi}_{j-d}$ is not injective, that is the smallest $j$ such that $\dim_{\mathbb{F}_{2^{n}}}(I\cap S_{j})<\dim_{\mathbb{F}_{2^{n}}}(S^{r}_{j-d}/U_{j-d})$. It is denoted by $D_{ff}(\sum_{j=1}^{r}\mathbb{F}_{2^{n}}h_{j})$.

Following [8] we now consider the ring of functions

[TABLE]

as a finite-dimensional filtered algebra whose filtration components $[A_{\mathbb{F}_{2^{n}}}]_{d}$, $d\in\mathbb{N}$, are given by the polynomials up to degree $d$. The associated graded ring of $A_{\mathbb{F}_{2^{n}}}$ is

[TABLE]

whose graded components

[TABLE]

are given by the homogeneous polynomials of degree $d$. Any linear subspace $V\subset[A_{\mathbb{F}_{2^{n}}}]_{d}$ induces a homogeneous linear subspace $\bar{V}\subset[\mathrm{Gr}(A_{\mathbb{F}_{2^{n}}})]_{d}$ via the canonical projection $\pi_{d}:[A_{\mathbb{F}_{2^{n}}}]_{d}\rightarrow[\mathrm{Gr}(A_{\mathbb{F}_{2^{n}}})]_{d}$.

Definition \thedfn (Cf. [8, Definition 2.2]).

Consider a polynomial system $p_{1},\ldots,p_{r}\in[A_{\mathbb{F}_{2^{n}}}]_{d}$ and its linear span $V=\sum_{j=1}^{r}\mathbb{F}_{2^{n}}p_{j}\subset[A_{\mathbb{F}_{2^{n}}}]_{d}$, respectively. We assume without loss of generality that $\dim_{\mathbb{F}_{2^{n}}}V=r>0$. The first fall degree of $V$ is

[TABLE]

where $D_{ff}(\bar{V}=\sum_{j=1}^{r}\mathbb{F}_{2^{n}}\pi_{d}(p_{j}))$ is given in § 2.

3. Weil descent along summation polynomials

We prove that the first fall degree of the polynomial system that arises from a Weil descent along Semaev’s summation polynomial $S_{m+1}$ is bounded from above by $m^{2}-m+1$. This is an improvement over $m^{2}+1$ that results from [8, Theorem 5.2] and [12, Section 4]. Let us briefly introduce the summation polynomials and describe the Weil descent.

Semaev [13] introduced the $m$-th summation polynomial $S_{m}(x_{1},\ldots,x_{m})\in\mathbb{K}[x_{1},\ldots,x_{m}]$ on an elliptic curve $E:y^{2}=x^{3}+a_{4}x+a_{6}$ over a finite field $\mathbb{K}$ with $\mathrm{char}(\mathbb{K})\neq 2,3$ by the following defining property: for elements $x_{1},\ldots,x_{m}$ in the algebraic closure $\bar{\mathbb{K}}$ one has $S_{m}(x_{1},\ldots,x_{m})=0$ if and only if there exist $y_{1},\ldots,y_{m}\in\bar{\mathbb{K}}$ such that $(x_{1},y_{1}),\ldots,(x_{m},y_{m})\in E(\bar{K})$ and $(x_{1},y_{1})+\ldots+(x_{m},y_{m})=0$ on $E$. Semaev gave a recursive formula based on resultants to compute those polynomials and described some properties [13, Theorem 1]. The summation polynomials can also be given in characteristic $2$. We consider $\mathbb{K}=\mathbb{F}_{2^{n}}$, an ordinary, i.e. non-singular, elliptic curve $E:y^{2}+xy=x^{3}+a_{2}x^{2}+a_{6}$, and the projection to the $x$-coordinate $x(P_{i})=x(x_{i},y_{i})=x_{i}$ of $P_{i}\in E$ . Then, still

[TABLE]

and from Diem’s general description [1, Lemma 3.4, Lemma 3.5] one can deduce

[TABLE]

and the degree of $S_{m+1}$ in each variable $x_{i}$ is $2^{m-1}$. Note that these formulas have also been outlined by Petit and Quisquater [12, Section 5] who also refer to Diem [1].

To describe the Weil descent along those summation polynomials (see e.g. [12, Section 4]) we fix a basis $1,z,\ldots,z^{n-1}$ of $\mathbb{F}_{2^{n}}$ over $\mathbb{F}_{2}$ and let $W$ be a subvector space in $\mathbb{F}_{2^{n}}$ of dimension $n^{\prime}$ and basis $\nu_{1},\ldots,\nu_{n^{\prime}}$ over $\mathbb{F}_{2}$. We introduce $mn^{\prime}$ variables $y_{ij}$ that model the linear constraints

[TABLE]

set $x_{m+1}$ to an arbitrary element $c\in\mathbb{F}_{2^{n}}$, and obtain the equation system

[TABLE]

The first fall degree of interest is that of the reduced polynomial system

[TABLE]

Note that $s_{0},\ldots,s_{n-1}\in\mathbb{F}_{2}[y_{11},\ldots,y_{mn^{\prime}}]/(y_{11}^{2}-y_{11},\ldots,y_{mn^{\prime}}^{2}-y_{mn^{\prime}})$.

By the definition of the first fall degree we are interested in the highest degree homogeneous part of $s_{0},\ldots,s_{n-1}$ whose degree can be determined as follows.

Lemma \thelem.

Let $m\geq 3$. The highest degree homogeneous part of the polynomial system $s_{0},\ldots,s_{n-1}\in\mathbb{F}_{2}[y_{11},\ldots,y_{mn^{\prime}}]/(y_{11}^{2}-y_{11},\ldots,y_{mn^{\prime}}^{2}-y_{mn^{\prime}})$ from Equation 3.1 is induced by the monomial

[TABLE]

in the summation polynomial $S_{m+1}(x_{1},\ldots,x_{m},x_{m+1})$, and hence its degree is less than or equal to $m^{2}-m$.

Proof.

First, we show the existence of the monomial $(x_{1}\cdots x_{m})^{2^{m-1}-1}\cdot x_{m+1}$ in $S_{m+1}(x_{1},\ldots,x_{m},x_{m+1})$. We have

[TABLE]

and the degree of $S_{m+1}$ in each variable $x_{i}$ is $2^{m-1}$. The resultant of $f,g\in\mathbb{F}_{2^{n}}[X]$ of degree $k$ and $l$ is the determinant of the Sylvester matrix

[TABLE]

That is, with

[TABLE]

where each $c_{i,m}\in\mathbb{F}_{2^{n}}[x_{1},\ldots,x_{m-1}]$, we have

[TABLE]

To be concrete, $\operatorname{Syl}(S_{m},S_{3})$ is the matrix

[TABLE]

with a total of $2^{m-2}+2$ rows and columns. In order to prove our claim we have to identify specific summands in the Leibniz formula of the determinant. That is, we consider

[TABLE]

and argue that for the relevant summands no cancellation over $\mathbb{F}_{2^{n}}$ occurs. Note that the sign of a permutation is $1\in\mathbb{F}_{2^{n}}$.

Step $1$: Prove by induction (start with $x_{1}^{2}x_{2}^{2}$ in $S_{3}$) that $S_{m+1}$ contains the monomial $(x_{1}\cdots x_{m})^{2^{m-1}}$ in its term $c_{0,m+1}$. For that we consider the permutation

[TABLE]

and obtain

[TABLE]

Note that specifying $\sigma_{1}=2^{m-2}+1$ and $\sigma_{2}=2^{m-2}+2$ determines $\sigma$ since the remaining entries in $\operatorname{Syl}(S_{m},S_{3})$ form an upper triangular matrix with $x_{m}^{2}+x_{m+1}^{2}$ on the diagonal.

Step $2$: Prove by induction (start with $x_{1}x_{2}x_{3}$ in $S_{3}$) that $S_{m+1}$ contains the monomial $(x_{1}\cdots x_{m})^{2^{m-1}-1}\cdot x_{m+1}$, i.e. $(x_{1}\cdots x_{m})^{2^{m-1}-1}$ in its term $c_{1,m+1}$. For that we consider the permutation

[TABLE]

and obtain

[TABLE]

Note that specifying $\tau_{1}=2^{m-2}$ and $\tau_{2}=2^{m-2}+2$ determines $\tau$ since the remaining entries in $\operatorname{Syl}(S_{m},S_{3})$ form an upper triangular matrix with $x_{m}^{2}+x_{m+1}^{2},\ldots,x_{m}^{2}+x_{m+1}^{2},x_{m}x_{m+1}$ on the diagonal.

Second, in order to exclude potential cancellations we have to show that the permutations $\sigma$ in (3.3) and $\tau$ in (3.4) are the only possible choices to produce the monomials $(x_{1}\cdots x_{m})^{2^{m-1}}$ and $(x_{1}\cdots x_{m})^{2^{m-1}-1}\cdot x_{m+1}$ in $S_{m+1}$, respectively. For that, we prove by induction (start with $x_{1}x_{2}$ in $S_{3}$) that the only multiples of $(x_{1}\cdots x_{m})^{2^{m-1}-1}$ in the coefficients of $S_{m+1}$ are $(x_{1}\cdots x_{m})^{2^{m-1}}$ in $c_{0,m+1}$ and $(x_{1}\cdots x_{m})^{2^{m-1}-1}$ in $c_{1,m+1}$. Indeed, the factor $(x_{1}\cdots x_{m-1})^{2^{m-1}-1}$ in the variables $x_{1},\ldots,x_{m-1}$ can only be produced by products $c_{i,m}\cdot c_{j,m}$ of entries taken from the first two rows of the Sylvester matrix $\operatorname{Syl}(S_{m},S_{3})$. Since the degree of $S_{m}$ in each variable $x_{1},\ldots,x_{m-1}$ is $2^{m-2}$, each of the entries $c_{0,m},\ldots,c_{2^{m-2},m}$ is a sum of monomials in the variables $x_{1},\ldots,x_{m-1}$ where each monomial is either

1. (i)

no multiple of $(x_{1}\cdots x_{m-1})^{2^{m-2}-1}$ or 2. (ii)

a multiple $(x_{1}\cdots x_{m-1})^{2^{m-2}-1}\cdot x_{1}^{\delta_{1}}\cdots x_{m-1}^{\delta_{m-1}}$, with $\delta_{i}\in\{0,1\}$.

Therefore, the monomials in the products $c_{i,m}\cdot c_{j,m}$ that contribute to the determinant (3.2) occur in the following forms

[TABLE]

where $\mu$ and $\mu^{\prime}$ denote elements that are no multiples of $(x_{1}\cdots x_{m-1})^{2^{m-2}-1}$. Consequently, a monomial in the product $c_{i,m}\cdot c_{j,m}$ that is now a multiple of $(x_{1}\cdots x_{m-1})^{2^{m-1}-1}$ can only arise in case (3.5) if for each $k=1,\ldots,m-1$ the following condition holds

[TABLE]

Due to the degree restriction of $S_{m}$ a product $c_{i,m}\cdot c_{j,m}$ where the monomials in $c_{i,m}$ and $c_{j,m}$ are all of the form (3.6) or (3.7) cannot produce a multiple of $(x_{1}\cdots x_{m-1})^{2^{m-1}-1}$. Therefore, we are left with products of the terms $c_{0,m}$ and $c_{1,m}$ by the induction hypothesis. Since $c_{1,m}\cdot c_{1,m}$ only produces $(x_{1}\cdots x_{m-1})^{2^{m-1}-2}$, the permutations $\pi=(\pi_{1},\pi_{2},\ldots,\pi_{2^{m-2}+2})$ in the Leibniz formula (3.2) that produce multiples of the monomial $(x_{1}\cdots x_{m-1})^{2^{m-1}-1}$ must have either $(\pi_{1},\pi_{2})=(\sigma_{1},\sigma_{2})$ or $(\pi_{1},\pi_{2})=(\tau_{1},\tau_{2})$ as given in (3.3) and (3.4), respectively. This determines our permutations $\sigma$ and $\tau$ completely.

To finish the proof, our degree claim in § 3 is argued as follows. The variables $y_{ij}$ of the $s_{k}$ are over $\mathbb{F}_{2}$ where taking squares is a linear operation. Therefore the degrees of the homogeneous parts of the system $s_{0},\ldots,s_{n-1}$ depend only on the Hamming weight $\operatorname{wt}(x_{1}^{\alpha_{1}}\cdots x_{m}^{\alpha_{m}})=\sum\operatorname{wt}(\alpha_{i})$ of a monomial in $S_{m+1}$. Since the degree of $S_{m+1}$ in each variable $x_{i}$ is $2^{m-1}$ the monomial $(x_{1}\cdots x_{m})^{2^{m-1}-1}\cdot x_{m+1}$, when $x_{m+1}$ is set to an element $c\in\mathbb{F}_{2^{n}}$, produces the highest Hamming weight $\sum_{i=1}^{m}\operatorname{wt}(2^{m-1}-1)=m(m-1)$. To be precise, we consider

[TABLE]

and obtain

[TABLE]

which is of degree less than or equal to $m(m-1)$ in the variables $y_{ij}$. ∎

We are ready to prove the main result.

Theorem 3.1**.**

Let $n^{\prime}\geq m\geq 3$ and $c\in\mathbb{F}_{2^{n}}\setminus\{0\}$, and consider the polynomial system $s_{0},\ldots,s_{n-1}\in\mathbb{F}_{2}[y_{11},\ldots,y_{mn^{\prime}}]/(y_{11}^{2}-y_{11},\ldots,y_{mn^{\prime}}^{2}-y_{mn^{\prime}})$ from Equation 3.1, that results from the Weil descent along the summation polynomial $S_{m+1}(x_{1},\ldots,x_{m},c)$. The first fall degree of $s_{0},\ldots,s_{n-1}$ is less than or equal to $m^{2}-m+1$.

Proof.

Consider the finite-dimensional filtered algebra

[TABLE]

The linear span

[TABLE]

is inside the degree $d=m^{2}-m$ subspace of the filtered algebra $A_{\mathbb{F}_{2}}$ due to § 3. By [8, Corollary 2.4] an extension of the base field, i.e.

[TABLE]

does not affect the first fall degree. That is,

[TABLE]

By [8, Definition 2.2], the first fall degree of the subspace $\sum_{j=0}^{n-1}\mathbb{F}_{2^{n}}s_{j}$ of $A_{\mathbb{F}_{2^{n}}}$ is

[TABLE]

where $\bar{V}$ denotes the induced homogeneous subspace of $\sum_{j=0}^{n-1}\mathbb{F}_{2^{n}}s_{j}$ in the associated graded ring

[TABLE]

If $\dim_{\mathbb{F}_{2^{n}}}\bar{V}<\dim_{\mathbb{F}_{2^{n}}}V$, our claim follows. Otherwise we consider the polynomial

[TABLE]

which is an element of the homogeneous subspace $\bar{V}$ by § 3, and in particular Equation 3.8. Now, for any

[TABLE]

we have a non-trivial relation

[TABLE]

of degree $d+1=m^{2}-m+1$ unless $P_{0}=0\in\mathrm{Gr}(A_{\mathbb{F}_{2^{n}}})$. Therefore it remains to show that $P_{0}\neq 0$. For that purpose, we recall that $c\in\mathbb{F}_{2^{n}}\setminus\{0\}$, $v_{1},\ldots,v_{n^{\prime}}$ are linearly independent, and $n^{\prime}\geq m$. Consider the linear change of variables

[TABLE]

This is induced by the $m\times n^{\prime}$ matrix

[TABLE]

that can be completed to an invertible linear transform by [11, Lemma 3.51], since we have assumed $v_{1},\ldots,v_{n^{\prime}}$ to be linearly independent and $n^{\prime}\geq m$. By using such an invertible linear transform on any block of variables

[TABLE]

we get new variables

[TABLE]

Under this change of variables $P_{0}$ is mapped to the non-zero element

[TABLE]

Remark \therem.

Our Theorem 3.1 remains true also in the case $m=2$ with first fall degree less than or equal to $2\cdot 1+1=3$. This bound is not sharp though, in fact the first fall degree in the case $m=2$ equals $2$ [10, Corollary 4.11 and Remark 4.12].

4. Experiments and Conclusion

In the light of the first fall degree bound given in Theorem 3.1 we computed a Gröbner basis for the ideal resulting from the Weil descent along the summation polynomial $S_{m+1}(x_{1},\ldots,x_{m},x_{m+1})$ for $m=2,3,4$ on an AMD Opteron CPU with Magma’s GroebnerBasis() function. Again, we set the verbose level to $1$ and extracted the empirical first fall degree $D_{ff}$ as the step degree of the first step where new lower degree (i.e. less than step degree) polynomials are added. The empirical degree of regularity $D_{reg}$ is the highest step degree that appears during the Gröbner basis computation. In each experiment we chose a random non-singular elliptic curve over $\mathbb{F}_{2^{n}}$, a random subvector space of dimension $n^{\prime}=\lceil n/m\rceil$ as the factor basis, and set $x_{m+1}$ to the $x$-coordinate of a random point on the curve. The experimental results that extend the ones present in the literature by Petit and Quisquater [12] and Kosters and Yeo [10] are displayed in Table 1.

Like Kosters and Yeo [10, Section 5] we observed a raise in the regularity degree for $m=2$ in our experiments and were able to verify their observation that with the low degree polynomials $W=\mbox{span}\{1,z,\ldots,z^{n^{\prime}}\}$ chosen as the factor basis (Cf. [14, Section 4.5]) the raise in the regularity degree was produced for slightly greater $n=45$. It would be very interesting to observe a raise in the degree of regularity for higher Semaev polynomials, but time and memory amounts become a serious issue for $m\geq 3$. However, such observations might neither falsify [12, Assumption 2] that $D_{reg}=D_{ff}+\mbox{o}(1)$ nor lead to further evidence that the gap between the degree of regularity and the first fall degree depends on $n$ as discussed in [9, Section 5.2].

However, we believe our first fall degree bound $m^{2}-m+1$ for Semaev polynomials to be sharp for $m\geq 3$, and rephrase [12, Assumption 2] as the following question:

[TABLE]

Note that our upper bound on the first fall degree of summation polynomials is a first step towards answering this question. The first fall degree generically bounds the degree of regularity from below. Hence, any further lower bound on the degree of regularity associated to the specific case of a Weil descent along summation polynomials can potentially answer (4.1).

Assuming an affirmative answer to (4.1), we can furthermore sharpen the asymptotic complexity of the index calculus algorithm for the ECDLP as presented by Petit and Quisquater [12, Section 5]. In the paragraph A new complexity analysis of [12, Section 5] it is argued that the complexity of the index calculus approach via summation polynomials is dominated by the Gröbner basis computation. Under the assumption that the degree of regularity is approximated closely by the first fall degree [12, Assumption 2], Petit and Quisquater derive [12, Proposition 4], i.e. that the discrete logarithm can asymptotically be solved in sub-exponential time

[TABLE]

where $c=2\omega/3$, $\omega$ is the linear algebra constant ($\omega=\log(7)/\log(2)$ is used in the following estimates), and $n^{2/3}+1$ is an upper bound for the first fall degree of the $m$-th summation polynomial when $m=n^{1/3}$ [12, Proposition 1]. They state that, by following this analysis the index calculus approach beats generic algorithms with run time $\mathcal{O}(2^{n/2})$ for any $n\geq N$ where $N$ is an integer approximately equal to $2000$. Now, based on Theorem 3.1 we assume $D_{reg}\approx m^{2}-m+1=n^{2/3}-n^{1/3}+1$ and sharpen (4.2) to

[TABLE]

Hence, the turning point to solve the ECDLP faster than a generic algorithm is an integer approximately equal to $1250$. Note that this is still far from cryptographically relevant sizes of $n$ up to $521$.