In the RSA public key cryptosystem, the private and public keys are (e, n) and (d, n) respectively, where n = p$$ \times $$q and p and q are large primes. Besides, n is public and p and q are private. Let M be an integer such that 0 < M < n and f(n) = (p - 1)(q - 1). Now consider the following equations.

I. M'= M^e mod n
M = (M')^d mod n

II. ed ≡ 1 mod n

III. ed ≡ 1 mod $$\phi $$(n)

IV. M'= M^e mod $$\phi $$(n)
M = (M')^d mod $$\phi $$(n)

Which of the above equations correctly represent RSA cryptosystem?

How many $$32k$$ x $$1$$ $$RAM$$ chips are needed to provide a memory capacity of $$256$$ $$K$$-bytes?

Consider a $$4$$-way set associative cache (initially empty) with total $$16$$ cache blocks. The main memory consists of $$256$$ blocks and the request for memory blocks is in the following order:
0, 255, 1, 4, 3, 8, 133, 159, 216, 129, 63, 8, 48, 32, 73, 92, 155.

Which one of the following memory block will NOT be in cache if $$LRU$$ replacement policy is used?

Consider a $$4$$ stage pipeline processor. The number of cycles needed by the four instructions $${\rm I}1,$$ $${\rm I}2,$$ $${\rm I}3,$$ $${\rm I}4,$$ in stages $$S1, S2, S3, S4$$ is shown below.

What is the number of cycles needed to execute the following loop?
For $$\left( {i = 1} \right.$$ to $$\left. 2 \right)$$ $$\left\{ {{\rm I}1;{\rm I}2;{\rm I}3;{\rm I}4;} \right\}$$