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POLKA: Towards Leakage-Resistant Post-quantum CCA-Secure Public Key Encryption

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Public-Key Cryptography – PKC 2023 (PKC 2023)

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Abstract

As for any cryptographic algorithm, the deployment of post-quantum CCA-secure public key encryption schemes may come with the need to be protected against side-channel attacks. For existing post-quantum schemes that have not been developed with leakage in mind, recent results showed that the cost of these protections can make their implementations more expensive by orders of magnitude. In this paper, we describe a new design, coined POLKA, that is specifically tailored to reduce this cost. It leverages various ingredients in order to enable efficient side-channel protected implementations such as: (i) the rigidity property (which intuitively means that the de-randomized encryption and decryption are injective functions) to avoid the very leaky re-encryption step of the Fujisaki-Okamoto transform, (ii) the randomization of the decryption thanks to the incorporation of a dummy ciphertext, removing the adversary’s control of its intermediate computations and making these computations ephemeral, (iii) key-homomorphic computations that can be masked against side-channel attacks with overheads that scale linearly in the number of shares, (iv) hard physical learning problems to argue about the security of some critical unmasked operations. Furthermore, we use an explicit rejection mechanism (returning an error symbol for invalid ciphertexts) to avoid the additional leakage caused by implicit rejection. As a result, the operations of POLKA can be protected against leakage in a cheaper way than state-of-the-art designs, opening the way towards schemes that are both quantum-safe and leakage-resistant.

B. Libert—This work was done when this author was a CNRS researcher at Laboratoire LIP (UMR CNRS - ENS Lyon - UCB Lyon 1 - INRIA 5668), Lyon, France.

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Notes

  1. 1.

    Informally, SPAs are side-channel attacks where the adversary can only observe the leakage of a few inputs to the target operation for a given secret. DPAs are attacks where the adversary can observe the leakage of many such inputs.

  2. 2.

    D’Anvers et al. [32] defined a homogeneous variant of \(\textsf {SIP}\text {-}\textsf {LWE}\) which is unconditionally hard, even in fully splitting rings. Still, relying on this variant incurs a partial re-encryption to enforce the equality \(c_2=c_2'\).

  3. 3.

    When the hybrid KEM-DEM framework is instantiated with an implicit rejection KEM, invalid ciphertexts are usually rejected during the symmetric decryption step as decrypting \(c_{sym}\) with a random key \(K'\) yields \(\perp \).

  4. 4.

    The underlying explicit rejection KEM can be proven CCA-secure secure in the ROM but we do not prove it CCA-secure in the QROM as we only consider the CCA security of the hybrid PKE scheme.

  5. 5.

    \(\textsf {Decrypt}_2\) still uses explicit rejection at step 1 because the secret key is not needed at this step and the goal of implicit rejection is to handle validity checks that depend on the secret key and the ciphertext.

  6. 6.

    We may assume that H outputs \(\perp \) on input of a triple \((r,e_1,e_2) \not \in D_E\). A hash function can always check domain membership before any computation.

  7. 7.

    As will be clear in conclusions, software implementations are left as an interesting open problem. In this case, the typical option to obtain security against SPA would be to emulate parallelism thanks to the shuffling countermeasure [66].

  8. 8.

    https://github.com/cmomin/polka_implem.

  9. 9.

    As mentioned in Subsect. 5.1, the security of the internal computations of \(t=\sum _{i=1}^d (p\cdot \overline{c_1}) \cdot s^i\) is obtained thanks to masking. So here, we only need to argue that the leakage of the recombined t does not lead to strong attacks.

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Acknowledgments

The authors thank Tobias Schneider for useful feedback on the design of \(\texttt{POLKA} \). Thomas Peters and François-Xavier Standaert are respectively research associate and senior research associate of the Belgian Fund for Scientific Research (F.R.S.-FNRS). This work has been funded in parts by the European Union through the ERC project 724725 (acronym SWORD) and the PROMETHEUS project (Horizon 2020 Research and Innovation Program, grant 780701), and by the Walloon Region Win2Wal project PIRATE.

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  1. Crypto Group, ICTEAM Institute, UCLouvain, Louvain-la-Neuve, Belgium

    Clément Hoffmann, Charles Momin, Thomas Peters & François-Xavier Standaert

  2. Zama, Annecy, France

    Benoît Libert

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Hoffmann, C., Libert, B., Momin, C., Peters, T., Standaert, FX. (2023). POLKA: Towards Leakage-Resistant Post-quantum CCA-Secure Public Key Encryption. In: Boldyreva, A., Kolesnikov, V. (eds) Public-Key Cryptography – PKC 2023. PKC 2023. Lecture Notes in Computer Science, vol 13940. Springer, Cham. https://doi.org/10.1007/978-3-031-31368-4_5

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