TWINKLE is a device devised by Adi Shamir to optimize the sieving step of GNFS. It consists of a cylinder, at the bottom of which are LEDs corresponding to factor base primes which blink with frequency 1/p. If more than a certain number of the LEDs blink at the same clock tick, you've found a GNFS relation (smooth integer pair). This is measured by a brightness sensor at the top of the cylinder: if the light level is above some threshold, the necessary number of LEDs must be lit.

My question is: why use LEDs instead of just measuring current? You could replace the LEDs with transistors that alternate current instead of blinking, and if the current sums to a certain level you have a hit. Why wouldn't this worK?

Leaving aside the WhatsApp aspect fo the question: the effect described can be achieved with hybrid encryption, where the bulk of the data is encrypted under a random secret symmetric key, and the result uploaded once; revealing that data to a recipient is performed by encrypting that symmetric key towards the recipient (e.g. using his/her public key), and thus requires very little data transfer from the sender for each additional recipient.