Methods and systems for linear key agreement with forward secrecy using an insecure shared communication medium

A method for shared secret agreement with forward secrecy comprising:

• generating, with a controller in a second node, a first plurality of bits of data using a cryptographically secure pseudo-random number generator (PRNG) applied to a first shared seed value stored in the memory of the second node and at least one index value, the first shared seed value being shared between the second node and a first node that are connected to a shared communication medium;
• transmitting, with the controller and a transceiver in the second node, each bit in the first plurality of bits of data through the shared communication medium simultaneously to transmission of a second plurality of bits of data from a third node connected to the shared communication medium and received by the transceiver in the second node;
• identifying, with the controller in the second node, first shared secret data with the third node using a portion of the first plurality of bits of data that are logical complements of the second plurality of bits of data that are transmitted simultaneously by the second node and the third node;
• generating, with the controller in the second node, a second shared seed value using a cryptographically secure one-way function applied to the first shared secret data, the second shared seed value being stored in the memory;
• and
• generating, with the controller in the second node, an updated first shared seed value using the cryptographically secure one-way function applied to the first shared seed value, the updated first shared seed value being stored in the memory to replace the first shared seed value.