Abstract
The security of code-based constructions is usually assessed by Information Set Decoding (ISD) algorithms. In the quantum setting, amplitude amplification yields an asymptotic square root gain over the classical analogue. However, already the most basic ISD algorithm by Prange suffers enormous width requirements caused by the quadratic description length of the underlying problem. Even if polynomial, this need for qubits is one of the biggest challenges considering the application of real quantum circuits in the near- to mid-term.
In this work we overcome this issue by presenting the first hybrid ISD algorithms that allow to tailor the required qubits to any available amount while still providing quantum speedups of the form \(T^\delta \), \(0.5<\delta <1\), where T is the running time of the purely classical procedure. Interestingly, when constraining the width of the circuit instead of its depth we are able to overcome previous optimality results on constraint quantum search.
Further we give an implementation of the fully-fledged quantum ISD procedure and the classical co-processor using the quantum simulation library Qibo and SageMath.
M. Manzano—This work was conducted while the author was affiliated with Technology Innovation Institute.
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Notes
- 1.
Our implementation (available at https://github.com/qiboteam/qISD) also includes an implementation of the Lee-Brickel [21] ISD improvement.
- 2.
Note that in Algorithm 1 we model \(H_I\) as the first \(n-k\) columns of HP, where P is a random permutation matrix.
References
Alagic, G., et al.: Status report on the second round of the NIST post-quantum cryptography standardization process. US Department of Commerce, NIST (2020)
Aragon, N., et al.: BIKE: bit flipping key encapsulation (2020)
Bärtschi, A., Eidenbenz, S.: Deterministic preparation of Dicke states. In: Gąsieniec, L.A., Jansson, J., Levcopoulos, C. (eds.) FCT 2019. LNCS, vol. 11651, pp. 126–139. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-25027-0_9
Becker, A., Joux, A., May, A., Meurer, A.: Decoding random binary linear codes in \(2^{n/20}\): how \(1+1=0\) improves information set decoding. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 520–536. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29011-4_31
Bernstein, D.J.: Grover vs. McEliece. In: Sendrier, N. (ed.) PQCrypto 2010. LNCS, vol. 6061, pp. 73–80. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12929-2_6
Biasse, J.F., Bonnetain, X., Pring, B., Schrottenloher, A., Youmans, W.: A trade-off between classical and quantum circuit size for an attack against CSIDH. J. Math. Cryptol. 15(1), 4–17 (2020)
Biasse, J.F., Pring, B.: A framework for reducing the overhead of the quantum oracle for use with Grover’s algorithm with applications to cryptanalysis of SIKE. J. Math. Cryptol. 15(1), 143–156 (2020)
Both, L., May, A.: Decoding linear codes with high error rate and its impact for LPN security. In: Lange, T., Steinwandt, R. (eds.) PQCrypto 2018. LNCS, vol. 10786, pp. 25–46. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-79063-3_2
Canteaut, A., Chabaud, F.: A new algorithm for finding minimum-weight words in a linear code: application to McEliece’s cryptosystem and to narrow-sense BCH codes of length 511. IEEE Trans. Inf. Theory 44(1), 367–378 (1998)
Chou, T., et al.: Classic McEliece: conservative code-based cryptography, 10 October 2020 (2020)
Dumer, I.: On minimum distance decoding of linear codes. In: Proceedings of the 5th Joint Soviet-Swedish International Workshop on Information Theory, pp. 50–52 (1991)
Efthymiou, S., et al.: Qibo: a framework for quantum simulation with hardware acceleration. arXiv preprint arXiv:2009.01845 (2020)
Efthymiou, S., et al.: Quantum-TII/Qibo: Qibo (2020). https://doi.org/10.5281/zenodo.3997195
Esser, A., Bellini, E.: Syndrome decoding estimator. In: Hanaoka, G., Shikata, J., Watanabe, Y. (eds.) PKC 2022. LNCS, vol. 13177, pp. 112–141. Springer, Cham (2022). https://doi.org/10.1007/978-3-030-97121-2_5
Esser, A., May, A., Verbel, J., Wen, W.: Partial key exposure attacks on BIKE, Rainbow and NTRU. Cryptology ePrint Archive (2022)
Esser, A., Ramos-Calderer, S., Bellini, E., Latorre, J.I., Manzano, M.: An optimized quantum implementation of ISD on scalable quantum resources. arXiv preprint arXiv:2112.06157 (2021)
Gilbert, E.N.: A comparison of signalling alphabets. Bell Syst. Tech. J. 31(3), 504–522 (1952)
Jaques, S., Naehrig, M., Roetteler, M., Virdia, F.: Implementing Grover oracles for quantum key search on AES and LowMC. In: Canteaut, A., Ishai, Y. (eds.) EUROCRYPT 2020. LNCS, vol. 12106, pp. 280–310. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45724-2_10
Kachigar, G., Tillich, J.-P.: Quantum information set decoding algorithms. In: Lange, T., Takagi, T. (eds.) PQCrypto 2017. LNCS, vol. 10346, pp. 69–89. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-59879-6_5
Kirshanova, E.: Improved quantum information set decoding. In: Lange, T., Steinwandt, R. (eds.) PQCrypto 2018. LNCS, vol. 10786, pp. 507–527. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-79063-3_24
Lee, P.J., Brickell, E.F.: An observation on the security of McEliece’s public-key cryptosystem. In: Barstow, D., et al. (eds.) EUROCRYPT 1988. LNCS, vol. 330, pp. 275–280. Springer, Heidelberg (1988). https://doi.org/10.1007/3-540-45961-8_25
May, A., Meurer, A., Thomae, E.: Decoding random linear codes in \(\tilde{\cal{O}}(2^{0.054n})\). In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 107–124. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-25385-0_6
May, A., Ozerov, I.: On computing nearest neighbors with applications to decoding of binary linear codes. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 203–228. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46800-5_9
Melchor, C.A., et al.: Hamming quasi-cyclic (HQC) (2020)
Nielsen, M.A., Chuang, I.L.: Quantum Information and Quantum Computation, vol. 2, no. 8, p. 23. Cambridge University Press, Cambridge (2000)
Perriello, S., Barenghi, A., Pelosi, G.: A quantum circuit to speed-up the cryptanalysis of code-based cryptosystems. In: Garcia-Alfaro, J., Li, S., Poovendran, R., Debar, H., Yung, M. (eds.) SecureComm 2021. LNICST, vol. 399, pp. 458–474. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-90022-9_25
Prange, E.: The use of information sets in decoding cyclic codes. IRE Trans. Inf. Theory 8(5), 5–9 (1962)
Stern, J.: A method for finding codewords of small weight. In: Cohen, G., Wolfmann, J. (eds.) Coding Theory 1988. LNCS, vol. 388, pp. 106–113. Springer, Heidelberg (1989). https://doi.org/10.1007/BFb0019850
Varshamov, R.R.: Estimate of the number of signals in error correcting codes. Docklady Akad. Nauk, SSSR 117, 739–741 (1957)
Zalka, C.: Grover’s quantum searching algorithm is optimal. Phys. Rev. A 60(4), 2746 (1999)
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Esser, A., Ramos-Calderer, S., Bellini, E., Latorre, J.I., Manzano, M. (2022). Hybrid Decoding – Classical-Quantum Trade-Offs for Information Set Decoding. In: Cheon, J.H., Johansson, T. (eds) Post-Quantum Cryptography. PQCrypto 2022. Lecture Notes in Computer Science, vol 13512. Springer, Cham. https://doi.org/10.1007/978-3-031-17234-2_1
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