# Is a composition of clocked entities a clocked entity in the UC model?

In the book titled "Secure Multiparty Computation and Secret Sharing" the authors define an interactive agent and an interactive system. Further, the author defines a clocked entity for considering both synchronous and asynchronous protocols in the UC model. A clocked entity is an interactive agent which has certain rules that define how the activation token is passed. The exact formulation of interactive agents, interactive systems and clocked entities can be found in the above mentioned book.

I have two doubts regarding the concept of clocked entities, which are described below:

1. Is the composition of clocked entities a clocked entity? The authors claim that this is trivially true and does not require a proof, however, I'm not convinced with this claim. I have constructed the following counter example for the same:

• Consider an interactive system $\mathcal{IS}$ composed of three clocked entities A, B and C. Further, consider a clocked entity D outside the system $\mathcal{IS}$. Let say the activation token enters the interactive system $\mathcal{IS}$ at port A.infl. Further A calls B, which in turn calls D. At this stage, A, B and $IS$ are in the calling state by definition. Now let say D calls C, which is currently inactive, hence C is activated now. But if $\mathcal{IS}$ was a clocked entity in the calling state, it should have bounced back the activation token when it receives activation token on any port other than D.leak. Hence $\mathcal{IS}$ does not follow the satisfy all the requirements of a clocked entity.

Please let me know as to what is wrong with my construction above and how the composition lemma follows from the definition of clocked entities and interactive systems.

1. How do clocked entities help implement synchronous protocols? What does the word "clock" signify in the word "clocked entities"?

## 1 Answer

When you compose $2$ interactive systems $\mathcal{IS}_1$ and $\mathcal{IS}_2$ into $\mathcal{IS}_1 \circ \mathcal{IS}_2$, the book indicates that you're simply connecting the matching ports between them.

In your example $\mathcal{IS} \circ D$ is a clocked entity because when $B$ calls $D$ then $D$ makes a recursive call to $\mathcal{IS}$ by sending the token to $C$. Taking from the book: "When it is clocked it must eventually return the activation on the matching outport leak" not immediately.

Regarding your second question:

You can think of a synchronous execution as a single thread program execution which goes from one line to the next one. Each line represents an agent and the clock can be here the line number. The clock is just a flag to capture which agent is running at a given time.

• I did not understand as to how induction can be used for proving that composition of clocked entities is a clocked entitiy. Your argument seems to be for the fact that a composition of interactive systems is another interactive system. Also, the second sentence of your answer is incorrect as the composed interactive system can have open inports/outports of any of the composed systems. If your argument works for clocked entities, can you tell me as to what is erroneous with the construction I have presented above. May 21 '16 at 7:46
• You are right about the inports/outports thing. After re-reading your question several times, I am not sure why do you think B should bounce back immediately after A calls B. May 21 '16 at 11:17
• Sorry for the inconvenience. There was a typographical error in my question. I have corrected "B.leak" to "D.leak", hence implying C must bounce back the call made by D. May 21 '16 at 11:35
• I see now. What you described is a recursive call. D can send the activation token on whatever port he has as long as in the end it will pass the token back on D.leak. Is my answer clearer now? May 21 '16 at 12:16