30-bit bus interaction ID (#2474)

Solves #2470. Depends on #2465. Approval will only works after merging
#2465.

The following is a response to #2470 and motivation of this PR: 

Chance of collission:
For 24-bits:
100 interactions: 0.000295000053831429
1000 interactions: 0.02933425835911685
10000 interactions: 0.9492338975723106

For 30-bits (theoretically the largest we could get to without changing
pil code due to field prime of bb and m31):
100 interactions: 4.610036260510597e-06
1000 interactions: 0.0004650875835883195
10000 interactions: 0.04549425469529611

Neither look ideal, but I think after
#2469, bus linker mode should
always have 2 bus interactions only. For user defined bus interactions,
they can intentionally input IDs that don't clash, so I think just using
30-bit interaction id should be good enough for 29 bits of security
(under 2 interactions linker case).

Author: qwang98

linker/src/bus.rs

@@ -19,13 +19,15 @@ use std::{
 use crate::{call, try_into_namespace_degree, DegreeMode, LinkerBackend, MAIN_OPERATION_NAME};
 
 /// Compute a unique identifier for an interaction
-fn interaction_id(link_to: &LinkTo) -> u16 {
+fn interaction_id(link_to: &LinkTo) -> u32 {
     let mut hasher = Sha256::default();
     hasher.update(format!("{}/{}", link_to.machine, link_to.operation));
     let result = hasher.finalize();
-    let mut bytes = [0u8; 2];
-    bytes.copy_from_slice(&result[..2]);
-    u16::from_le_bytes(bytes)
+    let mut bytes = [0u8; 4];
+    bytes.copy_from_slice(&result[..4]);
+    // Ensure that interaction_id is at most 30-bits, in order to
+    // fill in a single field element for BabyBear and M31.
+    u32::from_le_bytes(bytes) >> 2
 }
 
 pub struct BusLinker {
@@ -132,7 +134,7 @@ impl LinkerBackend for BusLinker {
                             .unwrap()
                             .into(),
                     )),
-                    arguments: vec![(interaction_id as u32).into(), tuple, selector],
+                    arguments: vec![interaction_id.into(), tuple, selector],
                 },
             ));
     }
@@ -274,7 +276,7 @@ impl BusLinker {
 
             let arguments = match selector_index {
                 // a selector index of None means this operation is called via lookup
-                None => vec![(interaction_id as u32).into(), latch, tuple, 0.into()],
+                None => vec![interaction_id.into(), latch, tuple, 0.into()],
                 // a selector index of Some means this operation is called via permutation
                 Some(selector_index) => {
                     let call_selector_array = namespaced_reference(
@@ -287,12 +289,7 @@ impl BusLinker {
                     let call_selector =
                         index_access(call_selector_array, Some((*selector_index).into()));
                     let rhs_selector = latch * call_selector;
-                    vec![
-                        (interaction_id as u32).into(),
-                        rhs_selector,
-                        tuple,
-                        1.into(),
-                    ]
+                    vec![interaction_id.into(), rhs_selector, tuple, 1.into()]
                 }
             };
 
@@ -355,7 +352,7 @@ mod test {
     pc' = (1 - first_step') * pc_update;
     pol commit call_selectors[0];
     std::array::map(call_selectors, std::utils::force_bool);
-    std::protocols::bus::bus_multi_send([std::protocols::bus::BusInteraction::Send(12064, [0, pc, instr__jump_to_operation, instr__reset, instr__loop, instr_return], 1)]);
+    std::protocols::bus::bus_multi_send([std::protocols::bus::BusInteraction::Send(454118344, [0, pc, instr__jump_to_operation, instr__reset, instr__loop, instr_return], 1)]);
 namespace main__rom(4);
     pol constant p_line = [0, 1, 2] + [2]*;
     pol constant p_instr__jump_to_operation = [0, 1, 0] + [0]*;
@@ -364,7 +361,7 @@ namespace main__rom(4);
     pol constant p_instr_return = [0]*;
     pol constant operation_id = [0]*;
     pol constant latch = [1]*;
-    std::protocols::bus::bus_multi_receive_batch_lookup_permutation([(12064, main__rom::latch, [main__rom::operation_id, main__rom::p_line, main__rom::p_instr__jump_to_operation, main__rom::p_instr__reset, main__rom::p_instr__loop, main__rom::p_instr_return], 0)]);
+    std::protocols::bus::bus_multi_receive_batch_lookup_permutation([(454118344, main__rom::latch, [main__rom::operation_id, main__rom::p_line, main__rom::p_instr__jump_to_operation, main__rom::p_instr__reset, main__rom::p_instr__loop, main__rom::p_instr_return], 0)]);
 "#;
 
         let file_name = "../test_data/asm/empty_vm.asm";