Unified point addition for computing elliptic curve point addition and doubling is considered to be resistant to simple power analysis. Recently, new side-channel attacks, such as recovery of secret exponent by triangular trace analysis and horizontal collision correlation analysis, have been successfully applied to elliptic curve methods to investigate their resistance to side-channel attacks. These attacks turn out to be very powerful since they only require leakage of a single power consumption trace. In this paper, using these side-channel attack analyses, we introduce two vulnerabilities of unified point addition on the binary Huff curve. Also, we propose a new unified point addition method for the binary Huff curve. Furthermore, to secure against these vulnerabilities, we apply an equivalence class to the side-channel atomic algorithm using the proposed unified point addition method.

Advanced collision-based single trace attacks which can be applied on simple power analysis resistant scalar multiplications become virtual threat on elliptic curve cryptosystems recently as their practical experimental results are increasingly reported in the literature. Since such attacks are based on detecting collisions of data dependent leakage caused by underlying long integer multiplications, so-called global shuffling countermeasure which breaks such collision correlation by independently randomizing the execution order of unit operations such as single precision multiplication and carry propagation, is considered as promising countermeasure if theoretical randomness of shuffling order is guaranteed. In this paper, we firstly analyze the practical security of the global shuffling long integer multiplications by exhibiting a combined single trace attack on software implementations on an ARM Cortex-M4 microcontroller. Our combined attack consists of a simple power analysis for revealing random permutation vectors which enables later collision-based single trace attack. First we demonstrate how to reveal random permutation vectors for carry propagation process of whole global shuffling long integer multiplications within a single power trace by simple power analysis accompanied with straightforward substitution of power consumption samples. Then we perform collision-based single trace attacks after rearranging the order of subtraces for unit carry propagations based on revealed permutation vectors. Since the vulnerability to simple power analysis is originated from the if-statement for selection of proper entries of the permutation vectors, we propose a novel countermeasure which eliminates such selection with simple addition and modulus operation and also demonstrate practical result achieving regularity in power trace patterns. INDEX TERMS Cryptography, digital signatures, elliptic curves, public key, side-channel attacks.

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