I have a 2015 Mazda3 and today, for the first time, my security icon started flashing. It is the padlock intertwined with the car symbol, but I am not able to submit a picture in acceptable format for this forum.
Normally, that indicates the alarm system is armed. I don't think it should be flashing once you enter the car with the fob and start it. The manual says the car might not restart if you shut it off when the light is flashing as the immoblizer may be activated.