Increasing the Robustness of the Montgomery kP-Algorithm Against SCA by Modifying Its Initialization

Bock, Estuardo Alpírez; Dyka, Zoya; Langendoerfer, Peter

doi:10.1007/978-3-319-47238-6_12

Cited by 12 publications

(2 citation statements)

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“…They showed how this small leakage can be exploited together with advanced approaches for solving the hidden number problem [11], leading thus to a complete recovery of the secret scalar. In an earlier work, the authors of [9] also identified the same leakage on a hardware implementa- 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.…”

Section: Algorithm 1 Montgomery Laddermentioning

confidence: 84%

“…Its robustness comes from the fact that for each loop iteration, we always perform a point addition followed by a point doubling, independent of the bit value we are processing for the scalar (see Algorithm 1). Nevertheless, previous works have shown that implementations of the Montgomery Ladder based on Lopez-Dahab projective coordinates [22] easily leak at least one bit of the scalar via simple side channel observations [3,9]. Lopez-Dahab projective coordinates represent the points on the curve only by means of their x-coordinate in the form x = X Z and allow for fast computation of the Montgomery Ladder since no divisions need to be performed during the main loop.…”

Section: Introductionmentioning

confidence: 99%

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Protecting the Most Significant Bits in Scalar Multiplication Algorithms

Bock

Chmielewski

Miteloudi

2022

Security, Privacy, and Applied Cryptography Engineering

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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) sidechannel 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.

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Section: Algorithm 1 Montgomery Laddermentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

Protecting the Most Significant Bits in Scalar Multiplication Algorithms

Bock

Chmielewski

Miteloudi

2022

Security, Privacy, and Applied Cryptography Engineering

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Resistance of the Montgomery Ladder Against Simple SCA: Theory and Practice

et al. 2021

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The Montgomery kP algorithm i.e. the Montgomery ladder is reported in literature as resistant against simple SCA due to the fact that the processing of each key bit value of the scalar k is done using the same sequence of operations. We implemented the Montgomery kP algorithm using Lopez-Dahab projective coordinates for the NIST elliptic curve B-233. We instantiated the same VHDL code for a wide range of clock frequencies for the same target FPGA and using the same compiler options. We measured electromagnetic traces of the kP executions using the same input data, i.e. scalar k and elliptic curve point P, and measurement setup. Additionally, we synthesized the same VHDL code for two IHP CMOS technologies, for a broad spectrum of frequencies. We simulated the power consumption of each synthesized design during an execution of the kP operation, always using the same scalar k and elliptic curve point P as inputs. Our experiments clearly show that the success of simple electromagnetic analysis attacks against FPGA implementations as well as the one of simple power analysis attacks against synthesized ASIC designs depends on the target frequency for which the design was implemented and at which it is executed significantly. In our experiments the scalar k was successfully revealed via simple visual inspection of the electromagnetic traces of the FPGA for frequencies from 40 to 100 MHz when standard compile options were used as well as from 50 MHz up to 240 MHz when performance optimizing compile options were used. We obtained similar results attacking the power traces simulated for the ASIC. Despite the significant differences of the here investigated technologies the designs’ resistance against the attacks performed is similar: only a few points in the traces represent strong leakage sources allowing to reveal the key at very low and very high frequencies. For the “middle” frequencies the number of points which allow to successfully reveal the key increases when increasing the frequency.

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