Abstract
The Montgomery Ladder is widely used for implementing the scalar multiplication in elliptic curve cryptographic designs. This algorithm is efficient and provides a natural robustness against (simple) side-channel attacks. Previous works however showed that implementations of the Montgomery Ladder using Lopez-Dahab projective coordinates easily leak the value of the most significant bits of the secret scalar, which led to a full key recovery in an attack known as LadderLeak [3]. In light of such leakage, we analyse further popular methods for implementing the Montgomery Ladder. We first consider open source software implementations of the X25519 protocol which implement the Montgomery Ladder based on the ladderstep algorithm from Düll et al. [15]. We confirm via power measurements that these implementations also easily leak the most significant scalar bits, even when implementing Z-coordinate randomisations. We thus propose simple modifications of the algorithm and its handling of the most significant bits and show the effectiveness of our modifications via experimental results. Particularly, our re-designs of the algorithm do not incurring significant efficiency penalties. As a second case study, we consider open source hardware implementations of the Montgomery Ladder based on the complete addition formulas for prime order elliptic curves, where we observe the exact same leakage. As we explain, the most significant bits in implementations of the complete addition formulas can be protected in an analogous way as we do for Curve25519 in our first case study.
E. Alpirez Bock—Work partially done while at Radboud University.
L. Chmielewski—This work was partially supported by the Technology Innovation Institute (https://www.tii.ae/) and by Ai-SecTools (VJ02010010) project.
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Notes
- 1.
In this paper we will use the term balanced value to refer to large values or bitstrings containing similar amounts of 0s and 1s. While we expect operations on such values to consume a notably larger amount of power than operations on small values like zero or one, this may not always be clearly visible due, e.g. to software optimisations.
- 2.
The source code from [4] is located in the following repository: https://github.com/sca-secure-library-sca25519/sca25519.
- 3.
Note that in the X25519 protocol, the most significant (254th) bit of the secret scalars is always set to 1; this is done by anding the most significant scalar byte with 0x7F|0x40 in [4]. However, since we consider the ECDSA protocol then the most significant scalar bits can be 0 and we need to consider fully random scalars.
- 4.
We refer to EdDSA with the parameters of Curve25519 as Ed25519 [7].
- 5.
Note that 256th bit of the scalar is always set to 0 since \(p = 2^{255}-19\).
- 6.
\(P_x= \texttt {0x67C5590EF5591AEEE312308D155579DC042E497FEC764BB3CAF3DE88597B8C24}\).
- 7.
We acknowledge that the traces in Fig. 2 look different than the ones collected from the first implementation. This is caused not only by differences in implementations, but also due to the fact that these new traces were collected later on with a new physical setup (although probes and oscilloscopes were equivalent models).
- 8.
We will provide a link to the code repository in the final version of the paper.
- 9.
The target has however an instruction cache. This caching mechanism is randomized, but since the sequence of instructions is always the same in our algorithms, this potential timing difference is independent from the scalar.
- 10.
The sequence of instructions performed by our algorithms is always the same but the instruction caching of our target seems to be random.
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Acknowledgments
The work of Estuardo Alpirez Bock was in part supported by MATINE, Ministry of Defence of Finland.
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Alpirez Bock, E., Chmielewski, L., Miteloudi, K. (2022). Protecting the Most Significant Bits in Scalar Multiplication Algorithms. In: Batina, L., Picek, S., Mondal, M. (eds) Security, Privacy, and Applied Cryptography Engineering. SPACE 2022. Lecture Notes in Computer Science, vol 13783. Springer, Cham. https://doi.org/10.1007/978-3-031-22829-2_7
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