# Recent algebraic attacks on the McEliece cryptosystem

Jean-Pierre Tillich

► To cite this version:

Jean-Pierre Tillich. Recent algebraic attacks on the McEliece cryptosystem. International Conference on Finite Fields and Their Applications 2023 (Fq15), Sihem Mesnager, Jun 2023, Aubervilliers, France. hal-04276638

HAL Id: hal-04276638

https://inria.hal.science/hal-04276638v1

Submitted on 9 Nov 2023

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
Recent algebraic attacks on the McEliece
cryptosystem

J.-P. Tillich (Inria de Paris)

June 21, 2023
Joint work

► “On the dimension and structure of the square of the dual of a Goppa code”, R. Mora, J.-P. Tillich, *Designs, Codes and Cryptography*, 2022

► “Polynomial time key-recovery attack on high rate random alternant codes”, M. Bardet, R. Mora, J.-P. Tillich, https://arxiv.org/abs/2304.14757

► “A new approach based on quadratic forms to attack the McEliece cryptosystem”, A. Couvreur, R. Mora, J.-P. Tillich, https://eprint.iacr.org/2023/950
# 1. The McEliece cryptosystem

* **1978 McEliece cryptosystem based on Goppa codes.**

* **Secret Key:** The algebraic structure of an $[n, k]_q$ Goppa code $C$ which has an efficient decoding algorithm for decoding $t$ errors

* **Public Key:** an arbitrary generator matrix $G$ of $C$.

* **Encryption:** $m \in \mathbb{F}_q^k \mapsto y \stackrel{\text{def}}{=} mG + e$ with $|e| = t$.

* **Decryption:** $y \mapsto m = \text{Decode}(y)$.
# Advantages/drawbacks

## Advantages

* Post Quantum, 4th-round finalist of the NIST competition;

* 55 years of research have not changed much the picture

> Compl. of key attacks ≫ Compl. of message attacks ≈ 2$^{αn}$
> with $α ≈$ Constant

* Efficient encryption and decryption

## Drawbacks

* Huge size of the keys, Classic McEliece ≈ 260 Kbytes.
Key Attacks/Message Attacks (pre 2011)

<figure><img src="image_2.png" /></figure>

► **Key attack:** (pre 2011) Enumerating all Goppa polynomial + solving the code-equivalence problem

► **Message attack:** Decoding a **generic** linear code
# Key Attacks/Message Attacks (2011)

* Distinguishing Attack
  [Faugère-Gauthier-Otmani-Perret-Tillich-2011]: distinguishing a
  Goppa/alternant code from a generic linear code

* $n-k = \Theta(n)$

<figure><img src="image_3.png" /></figure>

* $n-k = O(n^{\frac{1}{2}})$

<figure><img src="image_4.png" /></figure>
# Key Attacks/Message Attacks [BMT23]

## Assumptions:

1. the code should be distinguishable (⇒ $n-k = O(n^{\frac{1}{2}})$) with the method of [FGOPT11]

2. should be a generic alternant code rather than a Goppa code

3. $q \in \{2, 3\}$

<figure><img src="image_5.png" /></figure>
# A new approach and a new distinguisher [CMT23]

* Associating a space of quadratic forms to an alternant/Goppa code and looking for low rank quadratic forms in it by Gröbner basis techniques

⇒ a new distinguisher in characteristic 2

$$s \stackrel{\text{def}}{=} n-k$$

$$n = s^{\alpha}$$

$$\alpha \in (1,2)$$

<figure><img src="image_6.png" /></figure>
# A new approach and a new attack [CMT23]

## Assumption:

* Generic alternant and distinguishable [FGOPT11] codes
or

* distinguishable [FGOPT11] Goppa codes with
  $\deg(\text{Goppa polynomial}) < q - 1$

<figure><img src="image_7.png" /></figure>
2.Alternant and Goppa Codes
Generalized Reed-Solomon codes

**Definition 1.** [**Generalized Reed-Solomon code**] $k$ and $n$ integers with $1 \le k < n \le q$ where $q$ is a power of a prime number. The generalized Reed-Solomon code $\text{GRS}_k(x, y)$ of dimension $k$ associated to a pair $(x, y) \in \mathbb{F}_q^n \times \mathbb{F}_q^n$ where $x$ is an $n$-tuple of distinct elements of $\mathbb{F}_q$ and the entries $y_i$ are arbitrary nonzero elements in $\mathbb{F}_q$ is defined as:

$$ \text{GRS}_k(x, y) \stackrel{\text{def}}{=} \{ (y_1p(x_1), \dots, y_np(x_n)) : p \in \mathbb{F}_q[X], \deg p < k \}. $$

$x$ is the **support** and $y$ the **multiplier**.

**[Sidelnikov-Shestakov1992]**: recover from an arbitrary generator matrix of a GRS code $C$, a tuple $(x, y)$ such that $C = \text{GRS}(x, y)$ (all what is needed to decode $C$ efficiently).
Alternant codes

**Definition 1.** [alternant/Goppa code] $\boldsymbol{x} \in \mathbb{F}_{q^m}^n, \boldsymbol{y} \in \mathbb{F}_{q^m}^n$ as in def. of GRS codes, $\Gamma \in \mathbb{F}_{q^m}[z]$, $\deg \Gamma = r$. The alternant code $\text{Alt}_r(\boldsymbol{x}, \boldsymbol{y})$, resp. Goppa code $\text{Gop}(\boldsymbol{x}, \Gamma)$ of support $\boldsymbol{x}$, degree $r$ and extension degree $m$ are defined by

$$ \text{Alt}_r(\boldsymbol{x}, \boldsymbol{y}) \stackrel{\text{def}}{=} \text{GRS}_r(\boldsymbol{x}, \boldsymbol{y})^{\perp} \cap \mathbb{F}_q^n \\ = \text{GRS}_{n-r}(\boldsymbol{x}, \boldsymbol{y}^{\perp}) \cap \mathbb{F}_q^n \text{ (subfield subcode of GRS)} $$

$$ \text{Gop}(\boldsymbol{x}, \Gamma) = \text{Alt}_r(\boldsymbol{x}, \frac{1}{\Gamma(\boldsymbol{x})}) $$

$$ \text{GRS}_r(\boldsymbol{x}, \boldsymbol{y})^{\perp} = \text{GRS}_{n-r}(\boldsymbol{x}, \boldsymbol{y}^{\perp}) $$

$$ \boldsymbol{y}^{\perp} \stackrel{\text{def}}{=} \left( \frac{1}{y_i \pi_x'(x_1)}, \dots, \frac{1}{y_i \pi_x'(x_1)} \right) $$

$$ \pi_x(\boldsymbol{x}) \stackrel{\text{def}}{=} (x - x_1) \cdots (x - x_n) $$
3. The square code

► Key ingredient for the [FGOPT11]/[MT22] distinguisher and the
[BMT23] attack

► Introduced for cryptanalyzing the Berger-Loidreau scheme by
Wieschebrink in 2010
The square code

**Definition 2. [Componentwise product]** Given two vectors $\mathbf{a} = (a_1, \dots, a_n)$ and $\mathbf{b} = (b_1, \dots, b_n) \in \mathbb{F}_q^n$, we denote by $\mathbf{a} \star \mathbf{b}$ the componentwise product

$$ \mathbf{a} \star \mathbf{b} \stackrel{\text{def}}{=} (a_1 b_1, \dots, a_n b_n) $$

**Definition 3. [Product of codes & square code]** The star product code denoted by $\mathcal{A} \star \mathcal{B}$ of $\mathcal{A}$ and $\mathcal{B}$ is the vector space spanned by all products $\mathbf{a} \star \mathbf{b}$ where $\mathbf{a}$ and $\mathbf{b}$ range over $\mathcal{A}$ and $\mathcal{B}$ respectively. When $\mathcal{B} = \mathcal{A}$, $\mathcal{A} \star \mathcal{A}$ is called the square code of $\mathcal{A}$ and is rather denoted by $\mathcal{A}^2$.
Dimension of the square code

A and B codes with respective bases ($\mathbf{a}_i$) and ($\mathbf{b}_j$).

$$
1. \dim(\mathcal{A} \star \mathcal{B}) \le \dim(\mathcal{A}) \dim(\mathcal{B}) \text{ (generated by the } \mathbf{a}_i \star \mathbf{b}_j \text{'s)}
$$

$$
2. \dim(\mathcal{A}^2) \le \left( \binom{\dim(\mathcal{A})+1}{2} \right) \text{ (generated by the } \mathbf{a}_i \star \mathbf{a}_j \text{'s with } i \le j)
$$
# What is wrong with generalized Reed-Solomon codes ?

When $\mathcal{C}$ is a random code of length $n$, with high probability

$$ \dim(\mathcal{C}^2) = \min \left\{ \binom{\dim(\mathcal{C}) + 1}{2}, n \right\} $$

When $\mathcal{C}$ is a generalized Reed-Solomon code

$$ \dim(\mathcal{C}^2) = \min \{2 \dim(\mathcal{C}) - 1, n\} $$
The explanation

$\mathbf{c} = (y_1p(x_1), \dots, y_np(x_n))$, $\mathbf{c}' = (y_1q(x_1), \dots, y_nq(x_n)) \in \mathbf{GRS}_k(\mathbf{x}, \mathbf{y})$

where $p$ and $q$ are two polynomials of degree at most $k-1$.

$\mathbf{c} \star \mathbf{c}' = (y_1^2p(x_1)q(x_2), \dots, y_n^2p(x_n)q(x_n)) = (y_1^2r(x_1), \dots, y_n^2r(x_n))$

where $r$ is a polynomial of degree $\le 2k-2$.

$$ \implies \mathbf{c} \star \mathbf{c}' \in \mathbf{GRS}_{2k-1}(\mathbf{x}, \mathbf{y}^2) $$
# What happens for alternant/Goppa codes
## [FGOPT11][MT22]?

Alternant/Goppa code $\mathcal{C}$ over $\mathbb{F}_q$ of length $n$, degree $r$, ext. deg. $m$

$$ \dim \mathcal{C}^{\perp} \stackrel{\text{typ.}}{=} rm $$

$$ \dim \left( \left( \mathcal{C}_{\text{Gop}}^{\perp} \right)^2 \right) \le \dim \left( \left( \mathcal{C}_{\text{alt}}^{\perp} \right)^2 \right) $$

$$ \dim \left( \left( \mathcal{C}^{\perp} \right)^2 \right) \le \min \left( n, \underbrace{\binom{mr+1}{2}}_{\dim \mathcal{R}^2} - \Omega(mr^2 \log r) \right) $$

$$ \mathcal{R} \stackrel{\text{def}}{=} \text{random code of length } n, \text{ dim } rm $$
4. [BMT23] : filtration attack + Gröbner bases

<figure><img src="image_8.png" /></figure>

► Transforming the distinguisher into an attack
A theorem

C a code of length $n$, $I, J \subset \{1, \dots, n\}$

$$ \boldsymbol{x}_J = (x_j)_{j \in J} $$

$$ \boldsymbol{x}_{\setminus I} = (x_j)_{j \notin I} $$

$$ \text{Shortened code } \mathbf{Sh}_I(C) = \{ \boldsymbol{c}_I : \boldsymbol{c} \in c_i = 0, \forall i \in I \} $$

**Theorem 1. [BMT23]** Let $\mathbf{Alt}_r(\boldsymbol{x}, \boldsymbol{y})$ be an alternant code on $\mathbb{F}_q$ such that $r \ge q+1$ and let $C \stackrel{\text{def}}{=} \mathbf{Alt}_r(\boldsymbol{x}_{\setminus i}, \boldsymbol{y}_{\setminus i})^{\perp}$, $D \stackrel{\text{def}}{=} (\mathbf{Sh}_i(\mathbf{Alt}_r(\boldsymbol{x}, \boldsymbol{y})^{\perp}))^2$, then

$$ C \star \mathbf{Alt}_{r-1}(\boldsymbol{x}_{\setminus i}, \boldsymbol{y}_{\setminus i}(\boldsymbol{x}_{\setminus i} - \boldsymbol{x}_i))^{\perp} \subseteq D $$
Filtration

► For given subspaces $\mathcal{A}$ and $\mathcal{B}$, computing the largest subspace $\mathcal{X}$ such that

$$ \mathcal{A} \star \mathcal{X} \subseteq \mathcal{B} $$

is just solving a linear system.

**Conjecture 1.** For $r \ge q+1$ and if $(\mathrm{Sh}_i(\mathrm{Alt}_r(\mathbf{x}, \mathbf{y})^{\perp}))^2$ is not the full space. Let $\mathcal{X}$ be largest subspace such that $\mathcal{C} \star \mathcal{X} \subset \mathcal{D}$. Then for a generic alternant code

$$ \mathcal{X} = \mathrm{Alt}_{r-1}(\mathbf{x}_{\setminus i}, \mathbf{y}_{\setminus i}(\mathbf{x}_{\setminus i} - \mathbf{x}_i))^{\perp} $$

For a Goppa code $\mathrm{Alt}_{r-1}(\mathbf{x}_{\setminus i}, \mathbf{y}_{\setminus i}(\mathbf{x}_{\setminus i} - \mathbf{x}_i))^{\perp} \subset \mathcal{X}$
Filtration (ii)+attack

► $\mathbf{Alt}_r(x, y) \to \mathbf{Alt}_{r-1}(x \setminus i, \dots)^\perp \to \dots \to \mathbf{Alt}_q(x \setminus I, \dots)^\perp$

► The support $x$ and multiplier $y$ of an alternant code $\mathbf{Alt}_r(x, y)$
can be recovered **efficiently** by a suitable algebraic modeling +
Gröbner bases techniques when $r = 3 \Rightarrow$

efficient attack when $q \in \{2, 3\}$
GRS codes

$$
\begin{align*}
\mathcal{C} &\stackrel{\text{def}}{=} \mathbf{GRS}_k(\mathbf{x}_{\setminus i}, \mathbf{y}_{\setminus i}) \\
&= \{(y_j P(x_j))_{j \neq i} : \deg P \le k-1\}
\end{align*}
$$

$$
\begin{align*}
\mathcal{D} &\stackrel{\text{def}}{=} (\mathbf{Sh}_i (\mathbf{GRS}_k(\mathbf{x}, \mathbf{y})))^2 \\
&= \{(y_j (x_j - \mathbf{x}_i)^2 P(x_j))_{j \neq i} : \deg P \le 2k - 4\} \text{ since}
\end{align*}
$$

$$
\mathrm{Sh}_i(\mathrm{GRS}_k(\mathbf{x}, \mathbf{y})) = \{(y_j(x_j - \mathbf{x}_i)P(x_j))_{j \neq i} : \deg P \le k-2\}
$$

$$
\begin{align*}
\mathcal{X} &= \mathbf{GRS}_{k-2}(\mathbf{x}_{\setminus i}, \mathbf{y}_{\setminus i}(\mathbf{x}_{\setminus i} - \mathbf{x}_i)^2) \\
&= \{(y_j(x_j - \mathbf{x}_i)^2 P(x_j))_{j \neq i} : \deg P \le k-3\}
\end{align*}
$$

$C * X = D$
5. A new tool : quadratic forms
Going to the big field

$\mathcal{C} \subseteq (\mathbb{F}_q)^n$, $\mathcal{C}_{\mathbb{F}_{q^m}} \stackrel{\text{def}}{=} \text{linear span of } \mathcal{C} \text{ over } \mathbb{F}_{q^m}$.

**Proposition 1.**

$$ (\mathbf{Alt}_r(\mathbf{x}, \mathbf{y})^{\perp})_{\mathbb{F}_{q^m}} = \sum_{j=0}^{m-1} \mathbf{GRS}_r(\mathbf{x}, \mathbf{y})^{(q^j)} = \sum_{j=0}^{m-1} \mathbf{GRS}_r(\mathbf{x}^{q^j}, \mathbf{y}^{q^j}) $$

$$ c_a \stackrel{\text{def}}{=} yx^a \in (\mathbf{Alt}_r(\mathbf{x}, \mathbf{y})^{\perp})_{\mathbb{F}_{q^m}} $$

$$ c_0 * c_2 = c_1^2 $$
Quadratic relations between codewords

C an $[n, k]$-code over $\mathbb{F}$, $\mathcal{V} = \{\mathbf{v}_1, \dots, \mathbf{v}_k\}$ a basis of $\mathbb{C}$

$$ \mathcal{C}_{\text{rel}}(\mathcal{V}) \stackrel{\text{def}}{=} \{ \mathbf{c} = (c_{i,j})_{1 \le i \le j \le k} \mid \sum_{i \le j} c_{i,j} \mathbf{v}_i \star \mathbf{v}_j = \mathbf{0} \} \subseteq \mathbb{F}^{\binom{k+1}{2}} $$

$$ Q_c(x_1, \dots, x_k) = \sum_{i \le j} c_{i,j} x_i x_j $$

$$ \mathbf{x} M_c \mathbf{y}^\top = Q_c(\mathbf{x} + \mathbf{y}) - Q_c(\mathbf{x}) - Q_c(\mathbf{y}) $$

$$ \mathcal{C}_{\text{mat}}(\mathcal{V}) \stackrel{\text{def}}{=} \{ \mathbf{M}_c = (m_{i,j})_{\substack{1 \le i \le k \\ 1 \le j \le k}} \mid \mathbf{c} = (c_{i,j})_{1 \le i \le j \le k} \in \mathcal{C}_{\text{rel}}(\mathcal{V}) \} $$

$$ M_c \stackrel{\text{def}}{=} \begin{cases} m_{i,j} \stackrel{\text{def}}{=} m_{j,i} \stackrel{\text{def}}{=} c_{i,j}, & 1 \le i < j \le k, \\ m_{i,i} \stackrel{\text{def}}{=} 2c_{i,i}, & 1 \le i \le k. \end{cases} $$
Low rank matrices

$$
c_0 * c_2 = c_1^2 \Rightarrow
$$

<figure><img src="image_9.png" /></figure>

basis invariant, since ∃ an invertible matrix P such that

$$
\mathcal{C}_{\text{mat}}(\mathcal{A}) = \mathcal{P} \mathcal{C}_{\text{mat}}(\mathcal{B}) \mathcal{P}^{\top}
$$
MinRank Problem

**Problem 1. [odd characteristic]** Let $M_1, \cdots, M_K$ be $K$ symmetric matrices in $F^{N \times N}$, find an $M \in \langle M_1, \cdots, M_K \rangle_F$ of rank 3.

**Problem 2. [characteristic 2]** Let $M_1, \cdots, M_K$ be $K$ skew-symmetric matrices in $F^{N \times N}$, find an $M \in \langle M_1, \cdots, M_K \rangle_F$ of rank 2.

► Geometric argument on matrix codes $\Rightarrow$ probability that there are rank 2 (char 2) or rank 3 matrix in a random matrix code of the same dimension(s) as $C_{\text{mat}}(\mathcal{V})$ when $\text{rate}(\mathcal{C}) \le 1/3$ is $o(1)$.
Algebraic modeling in characteristic 2

• $\mathcal{C}$ an $[n, k]$ code,

• $\mathcal{C}_{\text{mat}}(\mathcal{V}) \in \mathbb{F}^{k \times k}$ the associated skew-symmetric matrix code
$\dim \mathcal{C}_{\text{mat}}(\mathcal{V}) = \binom{k}{2} - (n-k)$ when $\mathcal{C}$ is not [FGOPT11]-
distinguishable.

• skew-symmetric $M = (m_{ij})_{\substack{1 \le i \le k \\ 1 \le j \le k}} \in \mathbb{F}^{k \times k}$

$$
\begin{align*}
\text{rank}(M) \le 2 &\Rightarrow \text{ all principal minors of } M \text{ of size } 4 \text{ are } 0 \\
&\Rightarrow m_ab m_c d + m_a c m_b d + m_a d m_b c = 0, \forall a, b, c, d
\end{align*}
$$

$$
M \in \mathcal{C}_{\text{mat}}(\mathcal{V}) \rightarrow n-k \text{ linear equations}
$$
Hilbert Series

• Homogeneous ideal $\mathcal{I} \in \mathbb{K}[z]$, $z = (z_1, \dots, z_n)$,

• Hilbert function of the ring $R = \mathbb{K}[z]/\mathcal{I}$ defined as

$$ \mathrm{HF}_R(d) \stackrel{\mathrm{def}}{=} \dim_{\mathbb{K}}(\mathbb{K}[z]_d) - \dim_{\mathbb{K}}(\mathcal{I}_d) $$

$$ \mathbb{K}[z]_d \stackrel{\mathrm{def}}{=} \{f \in \mathbb{K}[z] \mid \mathrm{deg}(f) = d\} $$

$$ \mathcal{I}_d \stackrel{\mathrm{def}}{=} \mathcal{I} \cap \mathbb{K}[z]_d $$

Can be computed in time $\approx n^{\omega d}$
Generic linear code/Goppa or alternant code

► alternant/Goppa case: $HF(d) > 0$ for all $d$.

► generic linear $[n, k]$-code:

$$ 
\begin{aligned}
HF(d) &= \max(0, \sum_{i=0}^{d} (-1)^i \binom{s}{i} HF(d-i)) \\
&= \max(0, \sum_{i=0}^{d} (-1)^i \binom{s}{i} \left( \binom{k+d-i-2}{d-i}^2 - \binom{k+d-i-2}{d-i-1} \right))
\end{aligned}
$$

where $s \stackrel{\text{def}}{=} n-k$.

$$ d_0 \stackrel{\text{def}}{=} \min\{d : HF(d) = 0\} \sim c \frac{k^2}{n-k} $$
Generalization of the distinguisher

► HF(1) = old distinguisher : $HF(1) = \binom{k+1}{2} - \dim \mathcal{C}^2$

<table><caption>Table 1: $HF(2)$ for random, alternant and Goppa codes with $q = 4, m = 4, r = 4$. In bold = distinguishable lengths.</caption><thead><tr><th>HF(2)</th><th>256 ≥ n ≥ 77</th><th>n = 76</th><th>n = 75</th><th>n = 74</th><th>n = 73</th></tr></thead><tbody><tr><td>Random code</td><td>0</td><td>10</td><td>71</td><td>133</td><td>196</td></tr><tr><td>Alternant code</td><td><b>20</b></td><td><b>20</b></td><td>71</td><td>133</td><td>196</td></tr><tr><td>Goppa code</td><td><b>80</b></td><td><b>80</b></td><td><b>80</b></td><td>133</td><td>196</td></tr></tbody></table>
## 6. New attack in the (old) distinguishable regime

**Assumption 1.** The parameters are in the old distinguishable regime and $r < q + 1$ (alternant) or $r < q - 1$ (Goppa).

► Recall that

$$ (\mathbf{Alt}_r(\mathbf{x}, \mathbf{y})^{\perp})_{\mathbb{F}_{q^m}} = \sum_{j=0}^{m-1} \mathbf{GRS}_r(\mathbf{x}, \mathbf{y})^{(q^j)} = \sum_{j=0}^{m-1} \mathbf{GRS}_r(\mathbf{x}^{q^j}, \mathbf{y}^{q^j}) $$

⇒ Basis $\mathcal{A}$ of $(\mathbf{Alt}_r(\mathbf{x}, \mathbf{y})^{\perp})_{\mathbb{F}_{q^m}}$ of the form

$$ \mathcal{A} = (\mathbf{a}_0, \dots, \mathbf{a}_{r-1}, \mathbf{a}_0^q, \dots, \mathbf{a}_{r-1}^q, \dots, \mathbf{a}_0^{q^{m-1}}, \dots, \mathbf{a}_{r-1}^{q^{m-1}}), $$

$$ \mathbf{a}_i \stackrel{\text{def}}{=} \mathbf{y}\mathbf{x}^i $$

► From [MT22] the quadratic relations are generated by

$$ \mathbf{a}_a^{q^l} \star \mathbf{a}_b^{q^l} - \mathbf{a}_c^{q^l} \star \mathbf{a}_d^{q^l} \quad \text{for } a + b = c + d. $$
The Crucial Observation

The elements $\mathbf{A}$ of $\mathcal{C}_{\text{mat}}(\mathcal{A})$ are of the form (blocks of size $r$)

$$ \mathbf{A} = \begin{bmatrix} \mathbf{A}_0 & 0 & \cdots & 0 \\ 0 & \mathbf{A}_2 & & 0 \\ \vdots & & \ddots & 0 \\ 0 & \cdots & 0 & \mathbf{A}_{m-1} \end{bmatrix}, $$

If $\text{rank}(\mathbf{A}) = rm - 1 \Rightarrow \exists j : \text{rank}(\mathbf{A}_j) = r - 1, \text{rank}(\mathbf{A}_i) = r, i \neq j$

if $\mathbf{v} \in \text{ker}(\mathbf{A}) \Rightarrow \mathbf{v} = (\mathbf{0}_r, \dots, \mathbf{0}_r, \mathbf{v}_j, \mathbf{0}_r, \dots, \mathbf{0}_r)$

$\Rightarrow$ identify the vectors generating a single GRS code $\mathbf{GRS}_r(\mathbf{x}^{q^j}, \mathbf{y}^{q^j})$
Conclusion/Open Problems

▶ Finding a distinguisher in odd characteristic

▶ Turning the distinguisher into an attack

▶ Attacking Goppa codes in the whole distinguishable regime

▶ A general methodology for studying the security of the McEliece cryptosystem