There's been a lot of talk over the years of implementing a data modem over GSM voice - as far as I can tell, this would solve basically all the issues with the current crop of encryption products. Instead of constantly polling or faffing about with third party servers, one phone could simply initiate a two-way connection to another phone the old fashioned way: via phone call. There's your "push messaging". You wouldn't even have to leave your data on. Your text messages would also probably arrive faster than an SMS.

I'm very disappointed that despite numerous papers on the subject in the past decade (even a DEFCON presentation), no implementation has yet appeared, not even a demo.