A computer uses 46-bit virtual address, 32-bit physical address, and a three-level paged page table organization. The page table base register stores the base address of the first-level table ($T_1$), which occupies exactly one page. Each entry of $T_1$ stores the base address of a page of the second-level table ($T_2$). Each entry of $T_2$ stores the base address of a page of the third-level table ($T_3$). Each entry of $T_3$ stores a page table entry (PTE). The PTE is 32 bits in size. The processor used in the computer has a 1 MB 16-way set associative virtually indexed physically tagged cache. The cache block size is 64 bytes. What is the minimum number of page colours needed to guarantee that no two synonyms map to different sets in the processor cache of this computer?

To prevent cache aliasing (synonyms mapping to different sets) in a virtually indexed cache, page coloring is used.

1. Cache Parameters:

   • Cache size $= 1\text{ MB} = 2^{20}\text{ bytes}$

   • Block/Line size $= 64\text{ bytes} = 2^6\text{ bytes}$

   • Associativity $= 16 = 2^4$

   • Total number of lines $= \frac{2^{20}}{2^6} = 2^{14}$

   • Number of sets $= \frac{2^{14}}{2^4} = 2^{10} = 1024\text{ sets}$

2. Virtual Indexing Bits:

   • Cache set index requires $\log_2(1024) = 10\text{ bits}$.

   • Block offset requires $\log_2(64) = 6\text{ bits}$.

   • Total virtual address bits used for cache set index and block offset $= 10 + 6 = 16\text{ bits}$.

3. Page Offset Bits:

   • From Q.52, page size is $8\text{ KB} = 2^{13}\text{ bytes}$, which gives $13$ page offset bits.

4. Overlapping Bits & Page Coloring:

   • The cache index uses $16 - 13 = 3$ bits from the virtual page number (VPN).

   • These $3$ overlapping bits can differ for two virtual addresses mapping to the same physical page (synonyms).

   • The minimum number of colors required to ensure synonyms share the same cache set is $2^3 = 8$.

Hence, the correct option is (C).