The Hidden Parallelepiped Is Back Again: Power Analysis Attacks on Falcon

Abstract

FALCON is a very efficient and compact lattice-based signature finalist of the NIST’s Post-Quantum standardization campaign. This work assesses Falcon’s sidechannel resistance by analyzing two vulnerabilities, namely the pre-image computation and the trapdoor sampling. The first attack is an improvement of Karabulut and Aysu (DAC 2021). It overcomes several difficulties inherent to the structure of the stored key like the Fourier representation and directly recovers the key with a limited number of traces and a reduced complexity. The main part of this paper is dedicated to our second attack: we show that a simple power analysis during the signature execution could provide the exact value of the output of a subroutine called the base sampler. This intermediate value does not directly lead to the secret and we had toadapt the so-called hidden parallelepiped attack initially introduced by Nguyen and Regev in Eurocrypt 2006 and reused by Ducas and Nguyen in Asiacrypt 2012. We extensively quantify the resources for our attacks and experimentally demonstrate them with FALCON’s reference implementation on the ELMO simulator (McCann, Oswald and Whitnall USENIX 2017) and on a ChipWhisperer Lite with STM32F3 target (ARM Cortex M4).These new attacks highlight the need for side-channel protection for one of the three finalists of NIST’s standardization campaign by pointing out the vulnerable parts and quantifying the resources of the attacks.

Determining the effect of a floating point on the Falcon digital signature algorithm security

“…When performing all calculations, this difference turns into some small integer factor (in practice, the largest value obtained is (7)) of polynomials at the lowest level of recursion. If the difference occurs in the polynomial z 0 , then, according to formula (11), the situation δ 0 ≠0, δ 1 =0 arises, which is a necessary condition for the operation of formula (12) and the implementation of the attack.…”

“…It has been shown that it is possible to protect against this attack with the help of the correct implementation of calculations. But in [11], a new attack on the implementation was proposed, which made it possible to recover the secret base using power analysis. The conditions for this attack were improved in work [12].…”

“…, . s s Substituting (10) into the expression for h, we get: z z δ = -, expression (11) implies expression (8), which had to be proved.…”

Masking Floating-Point Number Multiplication and Addition of Falcon

“…Besides, a masked implementation of the Gaussian sampler is provided in [EFG + 22]. Recently, Guerreau et al [GMRR22] proposed a simple power analysis attack on Falcon and a related light countermeasure, and their work was further improved in [ZLYW23], which provides more potential sources of leakage and reduces the number of required traces by new algorithms.…”

“…In [KA21], Karabulut and Aysu first performed an EM attack on such computation in Falcon, whose method in their setting can recover the secret key with a few thousand measured traces. Their attack was later improved by Guerreau et al [GMRR22], which reduces the guess space complexity from 2 27 to only 2 11 . However, the protection of the pre-image vector computation in Falcon has so far seldom been discussed.…”