Docs: hints docs

docs/docs/vm-fundamentals/hints.md

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-# Hints
 
-There are several hints, explain each of them here. How they interact with the VM and affects it.
+# Hint
+  ## HINTS  IN CAIRO VIRTUAL MACHINE
+Hints in Cairo are pieces of Python code only executed by the sequencer.
+
+Hints are used to help the Virtual Machine determine the next path of execution it should take.
+
+Below are some Cairo code with hints:
+
+```cairo
+      [ap] = 25, ap++;
+    %{
+        import math
+         memory[ap] = int(math.sqrt(memory[ap - 1]))
+     %}
+      [ap - 1] = [ap] * [ap], ap++;
+```
+
+
+```cairo
+      func main() {
+          %{ memory[ap] = program_input['secret'] %}
+         [ap] = [ap], ap++;
+         ret;
+      }
+``` 
+
+
+
+### WHY HINTS RUN ON SEQUENCERS/ PROVERS AND NOT VERIFIERS
+
+Sequencers are the leads that produces blocks by executing transaction and updating the blockchain state.
+During the proof generation process, the prover uses hints to generate additional constraints or assumptions about the program's behavior, which makes part of the proof sent to the verifier for verification. In the Cairo Virtual Machine, hints are sorted out during different stages of the compilation and execution process by the sequencer. However, they are not directly involved in the verification stage. Hints are primarily for performance optimization.
+
+The key is the implementation of relevant and accurate hints that assist the prover or sequencers in their respective tasks.
+
+
+
+Example:
+Let's consider a Cairo function that calculates the factorial of a given input *`n`*:
+
+```Cairo
+    @public
+   func factorial(n : felt) -> (result : felt):
+    if n <= 1:
+        result := 1
+    else:
+     result := n * factorial(n - 1)
+```
+
+In this code, the hint *@public* indicates that the *`n`* factorial should be publicly available in the Cairo program.
+The provers/sequencers will utilize the hint to generate a proof or sequence of instructions for the function to ensure maximum optimization. The verifier will check that the generated proof or sequence of instructions has correctly calculated the factorial of the input *`n`* or any other specification is satisfied.
+
+## IMPLEMENTING HINTS IN CAIRO VIRTUAL MACHINE
+
+ In Cairo Virtual Machine, hints are accepted and recognized by its code. 
+Hints are implemented using the *%{...%}* syntax, which allows you to insert Python code that is only executed by the prover. When implementing hints, the *'variable naming convention'* should be considered. The method to handle specific hints should be outlined like this; *'create hintName Hinter'*. Hints should be grouped by functionality in Cairo programs. The structure returned should implement the interface. Implementing the hint means you have to write code in Go that has the same behavior as the code of the hint in Python. Unit tests are added to check if the behavior of your code is correct.
+
+Examples of how to implement a hint in cairo:
+
+``` cairo
+      from starkware.cairo.common.hints import Hint
+
+        class FactorialHint(Hint):
+        def __init__(self, n):
+            self.n = n
+
+        def process(self, cairo_ctx):
+           result = 1
+        for i in range(2, self.n + 1):
+            result *= i
+        return [result]
+
+    //Usage   
+      [ap] = 5, ap++;
+    %{
+       hint_processor = FactorialHint(memory[ap - 1])
+        memory[ap] = cairo_runner.run_from_entrypoint(
+           entrypoint,
+           [2, (2,0)],
+           False,
+           vm,
+           hint_processor
+        )[0];
+    %}
+      [ap - 1] = [ap] * [ap], ap++;
+
+```
+
+```cairo
+   from starkware.cairo.common.math import assert_nn_le
+
+  struct KeyValue {
+      key: felt,
+      value: felt,
+    }
+
+       struct HintResult {
+       idx: felt,
+    }
+
+    func get_value_by_key{range_check_ptr}(
+        list: KeyValue*, size, key
+     ) -> (value: felt) {
+       alloc_locals;
+       local idx;
+       local hint_result: HintResult;
+
+    %{
+        # Define the hint function
+        def hint_get_index_by_key(list, size, key):
+            for i in range(size):
+                if list[i].key == key:
+                    return HintResult(i)
+            raise ValueError("Key not found")
+
+        # Call the hint function and store the result in memory
+        hint_result = hint_get_index_by_key(list, size, key)
+        memory[ap] = hint_result.idx
+        ap += 1
+    %}
+
+       # Load the result of the hint function from memory
+       hint_result = memory[ap - 1];
+         ap -= 1;
+
+       assert_nn_le(hint_result.idx, size);
+       return list[hint_result.idx].value;
+    }
+
+```
+
+
+