I'm working on side channel attack on prime number generation by measuring power consumed by the cpu while doing that, let's say that the cpu process 10000 mpz_t integer ( size 512 ) does the number of set bits (Hamming weight) in those integers have anything to do with power consumption by the cpu?, in other words a 512 bits integer with 500 set bits is going to consume more power than a 512 bits integer with only 10 set bits?
Yes, that's the general idea. There are of course tricks to minimize the difference between a 0 and 1 bit, but without measurements against side channel attacks, a zero bit will be faster than a bit set to one during (modular) multiplication - basically because, for multiplication - nothing needs to be calculated.