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Distinguisher-based attacks on public-key cryptosystems using Reed–Solomon codes

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Abstract

Because of their interesting algebraic properties, several authors promote the use of generalized Reed–Solomon codes in cryptography. Niederreiter was the first to suggest an instantiation of his cryptosystem with them but Sidelnikov and Shestakov showed that this choice is insecure. Wieschebrink proposed a variant of the McEliece cryptosystem which consists in concatenating a few random columns to a generator matrix of a secretly chosen generalized Reed–Solomon code. More recently, new schemes appeared which are the homomorphic encryption scheme proposed by Bogdanov and Lee, and a variation of the McEliece cryptosystem proposed by Baldi et al. which hides the generalized Reed–Solomon code by means of matrices of very low rank. In this work, we show how to mount key-recovery attacks against these public-key encryption schemes. We use the concept of distinguisher which aims at detecting a behavior different from the one that one would expect from a random code. All the distinguishers we have built are based on the notion of component-wise product of codes. It results in a powerful tool that is able to recover the secret structure of codes when they are derived from generalized Reed–Solomon codes. Lastly, we give an alternative to Sidelnikov and Shestakov attack by building a filtration which enables to completely recover the support and the non-zero scalars defining the secret generalized Reed–Solomon code.

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Notes

  1. It means that \(\mathsf{\textit{Prob} }(e_i=0)=1-\eta \) and \(\mathsf{\textit{Prob} }(e_i=x)=\frac{\eta }{q-1}\) for any \(x\) in \(\mathbb {F}_{q}\) different from zero.

  2. See [24] which contains much more examples of codes with this kind of behavior

References

  1. Baldi M., Bianchi M., Chiaraluce F., Rosenthal J., Schipani D.: Enhanced public key security for the McEliece cryptosystem. ArXiv:1108.2462v2 (2011, Submitted).

  2. Baldi M., Bianchi M., Chiaraluce F., Rosenthal J., Schipani D.: Enhanced public key security for the McEliece cryptosystem. ArXiv:1108.2462v3 (2012, Submitted).

  3. Berger T.P., Loidreau P.: How to mask the structure of codes for a cryptographic use. Des. Codes Cryptogr. 35(1), 63–79 (2005).

    Google Scholar 

  4. Bernstein D.J., Lange T., Peters C.: Wild McEliece. In: Selected Areas in Cryptography, pp. 143–158 (2010).

  5. Bogdanov A., Lee C.H.: Homomorphic encryption from codes. ArXiv:1111.4301. This paper was accepted for publication in the proceedings of the 44th ACM Symposium on Theory of Computing (STOC). The authors withdrew their paper after they learned that their scheme was threatened (2011).

  6. Bosma W., Cannon J.J., Playoust Catherine: The Magma algebra system. I: the user language. J. Symb. Comput. 24(3/4), 235–265 (1997).

    Google Scholar 

  7. Brakerski Z.: When homomorphism becomes a liability. In: TCC, pp. 143–161 (2013).

  8. Cascudo I., Chen H., Cramer R., Xing C.: Asymptotically good ideal linear secret sharing with strong multiplication over any fixed finite field. In: Halevi S. (ed.) Advances in Cryptology: CRYPTO 2009. Lecture Notes in Computer Science, vol. 5677, pp. 466–486. Springer, Berlin (2009).

  9. Cascudo I., Cramer R., Xing C.: The torsion-limit for algebraic function fields and its application to arithmetic secret sharing. In: Rogaway P. (ed.) Advances in Cryptology: CRYPTO 2011. Lecture Notes in Computer Science, vol. 6841, pp. 685–705. Springer, Berlin (2011).

  10. Chizhov I.V., Bordodin M.A.: The failure of McEliece PKC based on Reed–Muller codes. Cryptology ePrint Archive, Report 2013/287 (2013).

  11. Couvreur A., Otmani A., Tillich J.P.: Polynomial time attack on wild McEliece over quadratic extensions. In: EUROCRYPT (2014) (To appear).

  12. Faugère J.-C., Gauthier V., Otmani A., Perret L., Tillich J.-P.: A distinguisher for high rate McEliece cryptosystems. In: Proceedings of the Information Theory Workshop 2011, ITW 2011, Paraty, Brasil, pp. 282–286 (2011).

  13. Faugère J.-C., Gauthier-Umaña V., Otmani A., Perret L., Tillich J.-P.: A distinguisher for high-rate McEliece cryptosystems. IEEE Trans. Inf. Theory, 59(10), 6830–6844 (2013).

    Google Scholar 

  14. Faure C., Minder L.: Cryptanalysis of the McEliece cryptosystem over hyperelliptic curves. In: Proceedings of the Eleventh International Workshop on Algebraic and Combinatorial Coding Theory, Pamporovo, Bulgaria, pp. 99–107 (2008).

  15. Gauthier V., Otmani A., Tillich J.-P.: A distinguisher-based attack on a variant of McEliece’s cryptosystem based on Reed–Solomon codes. http://arxiv.org/abs/1204.6459 (2012).

  16. Gibson J.: Equivalent Goppa codes and trapdoors to McEliece’s public key cryptosystem. In: Davies D. (ed.) Advances in Cryptology: EUROCRYPT 91. Lecture Notes in Computer Science, vol. 547, pp. 517–521. Springer, Berlin (1991).

  17. Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).

  18. Kötter R.: A unified description of an error locating procedure for linear codes. In: Proceedings of the Algebraic and Combinatorial Coding Theory, Voneshta Voda, pp. 113–117 (1992).

  19. Loidreau P., Sendrier N.: Weak keys in the McEliece public-key cryptosystem. IEEE Trans. Inf. Theory 47(3), 1207–1211 (2001).

    Google Scholar 

  20. Márquez-Corbella I., Martínez-Moro E., Pellikaan R.: Evaluation of public-key cryptosystems based on algebraic geometry codes. In: Borges J., Villanueva M. (eds.) Proceedings of the Third International Castle Meeting on Coding Theory and Applications, Barcelona, pp. 199–204 (2011).

  21. Márquez-Corbella I., Martínez-Moro E., Pellikaan R.: The non-gap sequence of a subcode of a generalized Reed–Solomon code. In: Finiasz M., Sendrier N., Charpin P., Otmani A. (eds.) Proceedings of the 7th International Workshop on Coding and Cryptography WCC 2011, Paris, pp. 183–193 (2011).

  22. Márquez-Corbella I., Martínez-Moro E., Pellikaan R.: The non-gap sequence of a subcode of a generalized Reed–Solomon code. Des. Codes Cryptogr. 66, 1–17 (2012).

    Google Scholar 

  23. Márquez-Corbella, I., Martínez-Moro, E., Pellikaan, R.: On the unique representation of very strong algebraic geometry codes. Des. Codes Cryptogr. 70, 1–16 (2012).

    Google Scholar 

  24. Márquez-Corbella I., Pellikaan R.: Error-correcting pairs for a public-key cryptosystem (2012) (preprint).

  25. MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes, 5th edn. North-Holland, Amsterdam (1986).

  26. McEliece R.J.: A public-key system based on algebraic coding theory, pp. 114–116. Jet Propulsion Lab, DSN Progress, Report 44 (1978).

  27. Minder L., Shokrollahi A.: Cryptanalysis of the sidelnikov cryptosystem. In: EUROCRYPT 2007, Barcelona. Lecture Notes in Computer Science, vol. 4515, pp. 347–360 (2007).

  28. Niederreiter H.: Knapsack-type cryptosystems and algebraic coding theory. Probl. Control Inf. Theory 15(2), 159–166 (1986).

    Google Scholar 

  29. Pellikaan R.: On decoding by error location and dependent sets of error positions. Discret. Math. 106–107, 368–381 (1992).

    Google Scholar 

  30. Sidelnikov V.M.: A public-key cryptosystem based on Reed–Muller codes. Discret. Math. Appl. 4(3), 191–207 (1994).

    Google Scholar 

  31. Sidelnikov V.M., Shestakov S.O.: On the insecurity of cryptosystems based on generalized Reed–Solomon codes. Discret. Math. Appl. 1(4), 439–444 (1992).

    Google Scholar 

  32. Wieschebrink C.: Two NP-complete problems in coding theory with an application in code based cryptography. In: IEEE International Symposium on Information Theory, pp. 1733–1737 (2006).

  33. Wieschebrink C.: Cryptanalysis of the Niederreiter public key scheme based on GRS subcodes. In: Sendrier N. (ed.) Post-Quantum Cryptography, Third International Workshop, PQCrypto 2010. Lecture Notes in Computer Science, vol. 6061, pp. 61–72. Springer, Darmstadt (2010).

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Correspondence to Ayoub Otmani.

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography” .

Appendices

Appendix 1: Proof of Proposition 12

Set \(a \mathop {=}\limits ^{\text {def}}|I|-|J|\) and \(b\mathop {=}\limits ^{\text {def}}|I|\). After a suitable permutation of the support and the indexes of the \(x_j\)’s, the code \(\fancyscript{C}_I\) has a generator matrix of the form

$$\begin{aligned} \left( \begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} x_1 &{} x_2 &{} \cdots &{} x_a &{} x_{a+1} &{} \cdots &{} x_b \\ \vdots &{} \vdots &{} &{} \vdots &{} \vdots &{} &{} \vdots \\ x_1^{\ell } &{} x_2^{\ell } &{} \cdots &{} x_a^{\ell } &{} x_{a+1}^{\ell } &{} \cdots &{} x_b^{\ell }\\ &{} &{} &{} &{} &{} &{} \\ x_1^{\ell +1} &{} x_2^{\ell +1} &{} \cdots &{} x_a^{\ell +1} &{} &{} &{} \\ \vdots &{} \vdots &{} &{} \vdots &{} &{} (0) &{} \\ x_1^k &{} x_2^k &{} \cdots &{} x_a^k &{} &{} &{} \end{array}\right) \end{aligned}$$

We define the maps

$$\begin{aligned}&\Phi _{I} : \left\{ \begin{array}{c@{\quad }c@{\quad }c} \mathbb {F}_{q}[x] &{} \rightarrow &{} \mathbb {F}_{q}^b \\ P &{} \mapsto &{} (P(x_1), \ldots , P(x_b)) \end{array}\right. \quad \text { and }\\&\Phi _{I\setminus J} : \left\{ \begin{array}{c@{\quad }c@{\quad }c} \mathbb {F}_{q}[x] &{} \rightarrow &{} \mathbb {F}_{q}^b \\ P &{} \mapsto &{} (P(x_1), \ldots , P(x_a), 0\ldots , 0) \end{array}\right. . \end{aligned}$$

We have the two following obvious lemmas.

Lemma 22

Both maps \(\Phi _{I}\) and \(\Phi _{I\setminus J}\) are linear. In addition, their restrictions to the vector space \(\langle x^2, \ldots , x^{2k}\rangle \) are injective.

Proof

It is sufficient to prove that the restriction of \(\Phi _{I\setminus J}\) is injective. It is an elementary consequence of polynomial interpolation, since \(a = |I| - |J|\) is assumed to be be larger than \(2k\). \(\square \)

Lemma 23

For all \(P, Q\in \mathbb {F}_{q}[x]\), we have:

$$\begin{aligned} \Phi _{I}\left( P \right) \star \Phi _{I}\left( Q \right)&= \Phi _{I}\left( PQ \right) \end{aligned}$$
(25)
$$\begin{aligned} \Phi _{I\setminus J} \left( P \right) \star \Phi _{I\setminus J} \left( Q \right)&= \Phi _{I\setminus J} \left( PQ \right) \end{aligned}$$
(26)
$$\begin{aligned} \Phi _{I}\left( P \right) \star \Phi _{I\setminus J} \left( Q \right)&= \Phi _{I\setminus J} \left( PQ \right) \end{aligned}$$
(27)

Clearly, we have

$$\begin{aligned} \fancyscript{C}_I = \Phi _{I}\left( \langle x,\ldots , x^{^\ell }\rangle \right) \ \oplus \ \Phi _{I\setminus J} \left( \langle x^{\ell +1},\ldots , x^k\rangle \right) . \end{aligned}$$
(28)

Using (25), (26) and (27), we get

$$\begin{aligned} \fancyscript{C}_I^2&= \Phi _{I}\left( \langle x, \ldots , x^{\ell }\rangle \right) ^2 \ + \ \Phi _{I\setminus J} \left( \langle x^{\ell +1}, \ldots , x^k\rangle \right) ^2\\&\quad + \ \Phi _{I}\left( \langle x, \ldots , x^{\ell }\rangle \right) \star \Phi _{I\setminus J} \left( \langle x^{\ell +1}, \ldots , x^k\rangle \right) \nonumber \\&= \Phi _{I}\left( \langle x^2, \ldots , x^{2\ell }\rangle \right) \ + \ \Phi _{I\setminus J} \left( \langle x^{2\ell +2}, \ldots , x^{2k}\rangle \right) \nonumber \\&\quad +\ \Phi _{I\setminus J} \left( \langle x^{\ell +2 }, \ldots , x^{k+ \ell }\rangle \right) \nonumber \\&= \Phi _{I}\left( \langle x^2, \ldots , x^{2\ell }\rangle \right) \ + \ \Phi _{I\setminus J} \left( \langle x^{2\ell +2}, \ldots , x^{2k}\rangle \ +\ \langle x^{\ell +2 }, \ldots , x^{k+ \ell }\rangle \right) \end{aligned}$$

Since, by assumption, \(\ell <k\), we have

$$\begin{aligned} \langle x^{\ell +2 }, \ldots , x^{k+ \ell }\rangle \ +\ \langle x^{2\ell +2}, \ldots , x^{2k}\rangle \ =\ \langle x^{\ell +2}, \ldots , x^{2k}\rangle \end{aligned}$$

Therefore,

$$\begin{aligned} \fancyscript{C}_I^2 = \Phi _{I}\left( \langle x^2, \ldots , x^{2\ell }\rangle \right) \ +\ \Phi _{I\setminus J} \left( \langle x^{\ell +2}, \ldots , x^{2k}\rangle \right) . \end{aligned}$$
(29)

Lemma 22 entails

$$\begin{aligned} \dim \Phi _{I}\left( \langle x^2, \ldots , x^{2\ell }\rangle \right) = 2\ell -1,\ \quad \text {and}\quad \ \dim \Phi _{I\setminus J} \left( \langle x^{\ell +2}, \ldots , x^{2k}\rangle \right) = 2k-\ell -1. \end{aligned}$$
(30)

To conclude the proof, we need to compute the dimension of the intersection of these spaces. For this purpose, set

$$\begin{aligned} R(x)\mathop {=}\limits ^{\text {def}}\prod _{j=a+1}^b (x-x_j). \end{aligned}$$

An element of \(\Phi _{I}\left( \langle x^2, \ldots , x^{2\ell }\rangle \right) \cap \Phi _{I\setminus J} \left( \langle x^{\ell +2},\ldots , x^{2k}\rangle \right) \) is an element of \(\Phi _{I}\left( \langle x^2 , \ldots , x^{2\ell }\rangle \right) \) which vanishes on the \(|J| = b-a\) last positions: it is an element of \(\Phi _{I}\left( \langle x^2 R(x), \ldots , x^{2\ell -|J|}R(x)\rangle \right) \). Thus,

$$\begin{aligned}&\Phi _{I}\left( \langle x^2, \ldots , x^{2 \ell }\rangle \right) \cap \ \Phi _{I\setminus J} \left( \langle x^{\ell +2}, \ldots , x^{2k}\rangle \right) \nonumber \\&\quad =\Phi _{I}\left( \langle x^2 R,\ldots , x^{2\ell - |J|}R\rangle \right) \cap \Phi _{I\setminus J} \left( \langle x^{\ell +2}, \ldots , x^{2k}\rangle \right) \nonumber \\&\quad = \Phi _{I\setminus J} \left( \langle x^2 R,\ldots , x^{2\ell - |J|}R\rangle \right) \cap \Phi _{I\setminus J} \left( \langle x^{\ell +2}, \ldots , x^{2k}\rangle \right) \nonumber \\&\quad = \Phi _{I\setminus J} \left( \langle x^2R, \ldots , x^{2\ell - |J|}R\rangle \cap \langle x^{\ell +2}, \ldots , x^{2k}\rangle \right) . \end{aligned}$$

The last equality is also a consequence of Lemma 22 since the direct image of an intersection by an injective map is the intersection of the direct images.

Since all the \(x_i\)’s are nonzero, the polynomials \(x^{\ell +2}\) and \(R\) are prime to each other, this yields

$$\begin{aligned} \langle x^2R, \ldots , x^{2\ell - |J|}R\rangle \cap \langle x^{\ell +2}, \ldots , x^{2k}\rangle&= \langle x^{\ell +2}R, \ldots , x^{2\ell - |J|}R\rangle . \end{aligned}$$

Therefore,

$$\begin{aligned} \Phi _{I}\left( \langle x^2, \ldots , x^{2 \ell }\rangle \right) \cap \Phi _{I\setminus J} \left( \langle x^{\ell +2}, \ldots , x^{2k}\rangle \right) = \Phi _{I\setminus J} \left( \langle x^{\ell + 2 } R(x), \ldots , x^{2\ell - |J|}R(x)\rangle \right) \nonumber \\ \end{aligned}$$
(31)

and this last space has dimension \(\ell - |J|- 1\). Finally, combining (29), (30) and (31), we get

$$\begin{aligned} \dim \fancyscript{C}_I^2 = (2k - \ell -1) + (2\ell -1) -(\ell - |J| -1) = 2k + |J| - 1. \end{aligned}$$

Appendix 2: Proof of Lemma 13

Recall that \(\varvec{R}\) has rank \(1\), then so does \(\varvec{R}\varvec{\Pi }^{-1}\) and there exist \(\varvec{a}\) and \(\varvec{b}\) in \(\mathbb {F}_{q}^n\) such that \( \varvec{R}\varvec{\Pi }^{-1} = \varvec{b}^T \varvec{a}\). Set

$$\begin{aligned} \varvec{P}\mathop {=}\limits ^{\text {def}}\mathfrak {I}+ \varvec{R}\varvec{\Pi }^{-1} = \mathfrak {I}+ \varvec{b}^T \varvec{a}. \end{aligned}$$

We first need the following lemmas

Lemma 24

The matrix \(\varvec{Q}\) is invertible if and only if \(\varvec{P}\) is.

Proof

We have \(\varvec{Q}= \varvec{\Pi }+ \varvec{R}= (\mathfrak {I}+ \varvec{R}\varvec{\Pi }^{-1})\varvec{\Pi }= \varvec{P}\varvec{\Pi }\), which yields the proof. \(\square \)

Lemma 25

The matrix \(\varvec{P}\) is invertible if and only if \(\varvec{a}\cdot \varvec{b} \ne -1\). In addition, if it is invertible, then

$$\begin{aligned} \varvec{P}^{-1} = \mathfrak {I}-\frac{1}{1+\varvec{a}\cdot \varvec{b}} \varvec{b}^T \varvec{a}. \end{aligned}$$

Proof

First, assume that \(\varvec{a}\cdot \varvec{b}\ne -1\). Then,

$$\begin{aligned} \varvec{P}\left( \mathfrak {I}-\frac{1}{1+\varvec{a}\cdot \varvec{b}} \varvec{b}^T \varvec{a}\right)&= \left( \mathfrak {I}+ \varvec{b}^T \varvec{a}\right) \left( \mathfrak {I}-\frac{1}{1+\varvec{a}\cdot \varvec{b}} \varvec{b}^T \varvec{a}\right) \\&= \mathfrak {I}+ \left( 1 - \frac{1}{1+\varvec{a}\cdot \varvec{b} }\right) \varvec{b}^T \varvec{a}-\frac{1}{1+\varvec{a}\cdot \varvec{b}} \varvec{b}^T \varvec{a}\varvec{b}^T \varvec{a}\\&=\mathfrak {I}+ \frac{\varvec{a}\cdot \varvec{b}}{1+\varvec{a}\cdot \varvec{b}}\varvec{b}^T \varvec{a}- \frac{\varvec{a}\cdot \varvec{b}}{1+\varvec{a}\cdot \varvec{b}}\varvec{b}^T \varvec{a}\\&= \mathfrak {I}. \end{aligned}$$

To conclude the “only if” part of the proof, there remains to prove that \(\varvec{P}\) is non invertible for \(\varvec{a}\cdot \varvec{b} = -1\). Assume \(\varvec{a}\cdot \varvec{b} = -1\), then

$$\begin{aligned} \varvec{P}^2 = \mathfrak {I}+2\varvec{b}^T \varvec{a}+ \varvec{b}^T \varvec{a}\varvec{b}^T \varvec{a}= \mathfrak {I}+ (2 +\varvec{a}\cdot \varvec{b}) \varvec{b}^T \varvec{a}= \varvec{P}. \end{aligned}$$

Thus, in this situation, \(\varvec{P}\) is a projection distinct from \(\mathfrak {I}\) and hence is non invertible. \(\square \)

Let \(\varvec{c}\) be an element of \(\fancyscript{C}_{\text {pub}}\). Since

$$\begin{aligned} \fancyscript{C}_{\text {sec}}= \fancyscript{C}_{\text {pub}}\varvec{Q}= \fancyscript{C}_{\text {pub}}(\varvec{\Pi }+ \varvec{R})= \fancyscript{C}_{\text {pub}}(\mathfrak {I}+ \varvec{R}\varvec{\Pi }^{-1})\varvec{\Pi }. \end{aligned}$$

We obtain

$$\begin{aligned} \fancyscript{C}=\fancyscript{C}_{\text {sec}}\varvec{\Pi }^{-1}= \fancyscript{C}_{\text {pub}}\varvec{P}\quad \mathrm{where}\quad \varvec{P}\mathop {=}\limits ^{\text {def}}\mathfrak {I}+ \varvec{R}\varvec{\Pi }^{-1}. \end{aligned}$$
(32)

Therefore

$$\begin{aligned} \fancyscript{C}_{\text {pub}}= (\fancyscript{C}_{\text {sec}}\varvec{\Pi }^{-1}) \varvec{P}^{-1} = \fancyscript{C}\varvec{P}^{-1}. \end{aligned}$$

From this, we obtain that there exists \(\varvec{p}\) in \(\fancyscript{C}\) such that \(\varvec{c}= \varvec{p}\varvec{P}^{-1}\). Thus, from Lemma 25 we know that \(\varvec{P}^{-1} = \mathfrak {I}-\frac{1}{1+\varvec{a}\cdot \varvec{b}} \varvec{b}^T \varvec{a}= \mathfrak {I}+ \varvec{\lambda }^T \varvec{a}\), which enables to write:

$$\begin{aligned} \varvec{c}= \varvec{p}\left( \mathfrak {I}+ \varvec{\lambda }^T \varvec{a}\right) = \varvec{p}+(\varvec{\lambda }\cdot \varvec{p})\varvec{a}. \end{aligned}$$

\(\square \)

Corollary 26

Given \(\varvec{u}, \varvec{v}\in \mathbb {F}_{q}^n\) the map \(\varvec{p}\mapsto \varvec{p}+ (\varvec{u}\cdot \varvec{p})\varvec{v}\) is an automorphism of \(\mathbb {F}_{q}^n\) if and only if \(\varvec{u}\cdot \varvec{v}\ne -1\).

Appendix 3: Proof of Proposition 15

This follows immediately from the fact that we can express \(\varvec{z}_i\) in terms of the \(\varvec{g}_j\)’s, say

$$\begin{aligned} \varvec{z}_i = \sum _{1 \leqslant j \leqslant k} a_{ij} \varvec{g}_j. \end{aligned}$$

We observe now that there exist three relations between the \(\varvec{z}_i \star \varvec{g}_j\)’s:

$$\begin{aligned} \sum _{1 \leqslant j \leqslant k} a_{2j} \varvec{z}_1 \star \varvec{g}_j - \sum _{1 \leqslant j \leqslant k} a_{1j} \varvec{z}_2 \star \varvec{g}_j = \varvec{z}_1 \star \varvec{z}_2 - \varvec{z}_2 \star \varvec{z}_1&= 0 \end{aligned}$$
(33)
$$\begin{aligned} \sum _{1 \leqslant j \leqslant k} a_{3j} \varvec{z}_1 \star \varvec{g}_j - \sum _{1 \leqslant j \leqslant k} a_{1j} \varvec{z}_3 \star \varvec{g}_j = \varvec{z}_1 \star \varvec{z}_3 - \varvec{z}_3 \star \varvec{z}_1&= 0 \end{aligned}$$
(34)
$$\begin{aligned} \sum _{1 \leqslant j \leqslant k} a_{3j} \varvec{z}_2 \star \varvec{g}_j - \sum _{1 \leqslant j \leqslant k} a_{2j} \varvec{z}_3 \star \varvec{g}_j = \varvec{z}_2 \star \varvec{z}_3 - \varvec{z}_3 \star \varvec{z}_2&= 0 \end{aligned}$$
(35)

It remains to prove that the three obtained identities relating the \(\varvec{z}_i \star \varvec{g}_j\)’s are independent under some conditions on the \(\varvec{z}_i\)’s. Actually, these relations are independent if and only if the \(\varvec{z}_i\)’s generate a space of dimension larger than or equal to \(2\). Indeed, sort the \(\varvec{z}_1 \star \varvec{g}_j\)’s as \(\varvec{z}_1 \star \varvec{g}_1, \ldots , \varvec{z}_1 \star \varvec{g}_k, \varvec{z}_2 \star \varvec{g}_1, \ldots , \varvec{z}_2 \star \varvec{g}_k, \varvec{z}_3 \star \varvec{g}_1, \ldots , \varvec{z}_3 \star \varvec{g}_k\). Then the system defined by Eqs. 33 to 35 is defined by the \( 3 \times 3k\) matrix

$$\begin{aligned} A:=\left( \begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} a_{21} &{} \cdots &{} a_{2k} &{} -a_{11} &{} \cdots &{} -a_{1k} &{} 0 &{} \cdots &{} 0 \\ a_{31} &{} \cdots &{} a_{3k} &{} 0 &{} \cdots &{} 0 &{} -a_{11} &{} \cdots &{} -a_{1k} \\ 0 &{} \cdots &{} 0 &{} -a_{31} &{} \cdots &{} -a_{3k} &{} a_{21} &{} \cdots &{} a_{2k} \end{array}\right) . \end{aligned}$$

Then, \(A\) has rank strictly less than \(3\) if there exists a vector \(\varvec{u} = (u_1, u_2, u_3)\) such that \(\varvec{u}A = 0\) which is equivalent to the system

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l@{\quad }l} u_1 \varvec{z}_2 + u_2 \varvec{z}_3 &{} = &{} 0\\ -u_1 \varvec{z}_1 - u_3 \varvec{z}_3 &{} = &{} 0\\ - u_2 \varvec{z}_1 + u_3 \varvec{z}_2 &{} = &{} 0 \end{array}\right. \end{aligned}$$

and such a system has a nonzero solution \(\varvec{u}=(u_1, u_2, u_3)\) if and only if the \(\varvec{z}_i\)’s are pairwise collinear i.e. generate a subspace of dimension lower than or equal to \(1\).

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Couvreur, A., Gaborit, P., Gauthier-Umaña, V. et al. Distinguisher-based attacks on public-key cryptosystems using Reed–Solomon codes. Des. Codes Cryptogr. 73, 641–666 (2014). https://doi.org/10.1007/s10623-014-9967-z

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