Same Values Analysis Attack over Binary Elliptic Curves

In this work, we are interested in studying a particular class of Side Channel Attacks on elliptic curves defined over binary fields. Side Channel Attacks exploit physical leakages such as power consumption or electromagnetic emanations during cryptographic computations in order to recover secret information. Among these attacks, the one we focus on is known as <i>Same Values Analysis (SVA)</i>. This method does not rely directly on distinguishing the sequence of operations, but rather on detecting situations where different inputs lead to identical intermediate values inside the formulas used for point addition and point doubling. Since these formulas are usually well known and publicly available, an adversary can exploit such collisions in order to reveal sensitive information, in particular the secret scalar used during scalar multiplication. The objective of our study is therefore to identify the points on elliptic curves that produce identical intermediate variables during addition or doubling steps, and to determine the algebraic conditions under which such coincidences occur. By analyzing these conditions, one can highlight vulnerabilities in scalar multiplication algorithms and evaluate their potential exploitation. We will focus our investigation on three important families of elliptic curves over binary fields: <i>Weierstrass curves, Edwards curves, and Hessian curves</i>. For completeness, we will also verify the effectiveness of the SVA attack on standardized curves recommended by NIST and SECG. The results of our analysis clearly show that all these curves are vulnerable to Same Values Analysis, which implies that their use in cryptographic protocols requires a careful reassessment of security guarantees.