Jan Jancar
Masaryk University, Brno, Czechia

Vojtech Suchanek
Masaryk University, Brno, Czechia

Petr Svenda
Masaryk University, Brno, Czechia

Vladimir Sedlacek
Rutgers University, Piscataway, New Jersey

Łukasz Chmielewski
Masaryk University, Brno, Czechia

Keywords: elliptic curve cryptography, black-box implementations, reverse engineering, ECDH, ECDSA

Abstract

Side-channel attacks on elliptic curve cryptography (ECC) often assume a white-box attacker who has detailed knowledge of the implementation choices taken by the target implementation. Due to the complex and layered nature of ECC, there are many choices that a developer makes to obtain a functional and interoperable implementation. These include the curve model, coordinate system, addition formulas, and the scalar multiplier, or lower-level details such as the finite-field multiplication algorithm. This creates a gap between the attack requirements and a real-world attacker that often only has black-box access to the target – i.e., has no access to the source code nor knowledge of specific implementation choices made. Yet, when the gap is closed, even real-world implementations of ECC succumb to side-channel attacks, as evidenced by attacks such as TPM-Fail, Minerva, the Side Journey to Titan, or TPMScan [MSE+20; JSS+20; RLM+21; SDB+24].
We study this gap by first analyzing open-source ECC libraries for insight into realworld implementation choices. We then examine the space of all ECC implementations combinatorially. Finally, we present a set of novel methods for automated reverse engineering of black-box ECC implementations and release a documented and usable open-source toolkit for side-channel analysis of ECC called pyecsca.
Our methods turn attacks around: instead of attempting to recover the private key, they attempt to recover the implementation configuration given control over the private and public inputs. We evaluate them on two simulation levels and study the effect of noise on their performance. Our methods are able to 1) reverse-engineer the scalar multiplication algorithm completely and 2) infer significant information about the coordinate system and addition formulas used in a target implementation. Furthermore, they can bypass coordinate and curve randomization countermeasures.