On the Worst-Case Side-Channel Security of ECC Point Randomization in Embedded Devices

Abstract

Point randomization is an important countermeasure to protect Elliptic Curve Cryptography (ECC) implementations against side-channel attacks. In this paper, we revisit its worst-case security in front of advanced side-channel adversaries taking advantage of analytical techniques in order to exploit all the leakage samples of an implementation. Our main contributions in this respect are the following: first, we show that due to the nature of the attacks against the point randomization (which can be viewed as Simple Power Analyses), the gain of using analytical techniques over simpler divide-and-conquer attacks is limited. Second, we take advantage of this observation to evaluate the theoretical noise levels necessary for the point randomization to provide strong security guarantees and compare different elliptic curve coordinates systems. Then, we turn this simulated analysis into actual experiments and show that reasonable security levels can be achieved by implementations even on low-cost (e.g. 8-bit) embedded devices. Finally, we are able to bound the security on 32-bit devices against worst-case adversaries.

A Factorization of \(f_{\mathrm {add}}\)

The LPRM rules for information propagation for factors with multiple outputs are deduced from the factorization of a factor with two outputs into two factors with one output each, as shown by the diagram below for the addition operation:

Where in1 and in2 refer to the two inputs to the addition. out1 to the result of the addition and out2 to the output carry bit. Then the add factor is defined as:

$$\begin{aligned}&f_{\mathrm {add}} ( \mathsf {in1}, \mathsf {in2}, \mathsf {out1}, \mathsf {out2})\\&\quad =\left\{ \begin{array}{ll} 1 if \mathsf {out1} = ( \mathsf {in1}+\mathsf {in2}) \% 256 and \mathsf {out2} = ( \mathsf {in1}+\mathsf {in2}) /256 \\ 0 otherwise\\ \end{array} \right. \end{aligned}$$

The add factor can be factorized into $f_{\mathrm {add}}^1$ and $f_{\mathrm {add}}^2$ which are defined as:

$$\begin{aligned} f^1_{\mathrm {add}} ( \mathsf {in1}, \mathsf {in2}, \mathsf {out1})=\left\{ \begin{array}{ll} 1 if \mathsf {out1} = ( \mathsf {in1}+\mathsf {in2}) \% 256 \\ 0 otherwise\\ \end{array} \right. \end{aligned}$$

$$\begin{aligned} f^2_{\mathrm {add}} ( \mathsf {in1}, \mathsf {in2}, \mathsf {out2})=\left\{ \begin{array}{ll} 1 if \mathsf {out2} = ( \mathsf {in1}+\mathsf {in2}) /256 \\ 0 otherwise\\ \end{array} \right. \end{aligned}$$

The LRPM propagation rules applied to the factorized factor yield for the variable in1:

$$\begin{aligned} \mathsf {MI}_{ f^1_{\mathrm {add}} \rightarrow \mathsf {in1}} = \mathsf {MI_{in2}} \times \mathsf {MI_{out1}} \mathrm {and} \mathsf {MI}_{ f^2_{\mathrm {add}} \rightarrow \mathsf {in1}} = \mathsf {MI_{in2}} \times \mathsf {MI_{out2}} \end{aligned}$$

Since information at variable node is summed we have:

$$\begin{aligned} \mathsf {MI}_{( f^1_{\mathrm {add}}, f^2_{\mathrm {add}} ) \rightarrow \mathsf {in1}} = \mathsf {MI}_{ f_{\mathrm {add}} \rightarrow \mathsf {in1}} = \mathsf {MI_{in2}} \times ( \mathsf {MI_{out1}} + \mathsf {MI_{out2}} ) \end{aligned}$$