We consider in this paper scalar multiplication algorithms over a hyperelliptic curve which are immune against simple power analysis and timing attack. To reach this goal we adapt the regular modular exponentiation based on multiplicative splitting presented in JCEN 2017 to scalar multiplication over a hyperelliptic curve. For hyperelliptic curves of genus g = 2 and 3, we provide an algorithm to split the base divisor as a sum of two divisors with smaller degree. Then we obtain an algorithm with a regular sequence of doubling always followed by an addition with a low degree divisor. We also provide efficient formulas to add such low degree divisors with a divisor of degree g. A complexity analysis and implementation results show that the proposed approach is better than the classical Double-and-add-always approach for scalar multiplication
