 קטנה פרופקציה תפעה, פרופקציה סביגת, תפעה קטנה וצרופקציה סביגת. תפועה תפעה רצת. Where the input is split between Alice and Bob, who have to compute some target function with as little communication as possible. Our example for a target function is the equality predicate, where Alice and Bob have to figure out whether the inputs are identical or not. Anyway, most of the results apply to almost any predicate, there are some reasonable criteria. The central properties of communication complexity models are the network layout, whether the participants communicate directly or with a third party. And whether there exists a common random source. We are not considering here deterministic protocols. In this short introduction, we'll focus on the simultaneous messages model. In this model, Alice and Bob get their inputs and produce one message each. These messages are sent to a third party, a referee, who has to compute the target function with some constant high probability. For this model, with no common randomness, there is a known tight lower bound of square root of n. That was proven several times. See the paper of Baba and Kimmel for a description of a protocol that achieves the lower bound and references for more such protocols. Our central question is, can the communication complexity be reduced in a computational world? One motivation is to discuss settings closer to the real world. We discuss two computational variations. The first is the preset randomness model, where there is a common source of randomness, but the inputs are chosen by an adversary who sees the common randomness also. The second model is the free talk model that we will discuss later. In the preset randomness model, we show that breaking the known lower bound implies the existence of DCRH, Distributional Collision Resistant Hush Functions. DCRH is where only random collisions are guaranteed to be hard to find. That is, we show explicitly how such a protocol can be used to construct a DCRH function. That's for the simultaneous messages model. In the interactive model, we show a similar result for a different bound. We also show that there are no protocols of constant communication regardless of assumptions. We use techniques from Baba and Kimmel's proof to show those results. Note that assuming collision-resistant Hush Functions regular, one can break the lower bound of a square root of n easily. The second model is the stateful free talk model. In this model, Alice and Bob can communicate freely before the inputs are chosen. Then the inputs are chosen by the adversary who sees their communication, and now the communication is measured. In this model, we consider adversary with more power. He chooses the input for Bob at the last moment, and also we can wait for an opportunity to attack. In this model, we show that protocols that are both resilient to such a powerful adversary and also very efficient, where the error is exponentially smaller in the communication, imply the existence of secret-key-agreement protocols. A secret-key-agreement protocol is where two parties with nothing in common agree on a secret key, such that any adversary listening to the communication is unable to know anything about the agreed key. Note that using a secret-key-agreement protocol, such protocols exist, and of course even better. Thank you.