feat: equality avm circuit (#4595)

Avm equality circuit

---------

Co-authored-by: Ilyas Ridhuan <ilyas@aztecprotocol.com>

barretenberg/cpp/pil/avm/avm_alu.pil

@@ -16,6 +16,7 @@ namespace avm_alu(256);
     pol commit alu_op_mul;
     pol commit alu_op_div;
     pol commit alu_op_not;
+    pol commit alu_op_eq;
 
     // Flattened boolean instruction tags
     pol commit alu_ff_tag;
@@ -223,4 +224,31 @@ namespace avm_alu(256);
     #[ALU_OP_NOT]
     alu_op_not * (alu_ia + alu_ic - UINT_MAX) = 0;
 
+    // ========= EQUALITY Operation Constraints ===============================
+    // TODO: Note this method differs from the approach taken for "equality to zero" checks
+    // in handling the error tags found in avm_main and avm_mem files. The predicted relation difference
+    // is minor and when we optimise we will harmonise the methods based on actual performance.
+
+    // Equality of two elements is found by performing an "equality to zero" check.
+    // This relies on the fact that the inverse of a field element exists for all elements except zero
+    // 1) Given two values x & y, find the difference z = x - y
+    // 2) If x & y are equal, z == 0 otherwise z != 0
+    // 3) Field equality to zero can be done as follows
+    //   a) z(e(x - w) + w) - 1 + e = 0;
+    //   b) where w = z^-1 and e is a boolean value indicating if z == 0
+    //   c) if e == 0; zw = 1 && z has an inverse. If e == 1; z == 0 and we set w = 0;
+
+    // Registers Ia and Ib hold the values that equality is to be tested on
+    pol DIFF = alu_ia - alu_ib;
+
+    // Need an additional helper that holds the inverse of the difference;
+    pol commit alu_op_eq_diff_inv;
+
+    // If EQ selector is set, ic needs to be boolean
+    #[ALU_RES_IS_BOOL]
+    alu_op_eq * (alu_ic * (1 - alu_ic)) = 0;
+
+    #[ALU_OP_EQ]
+    alu_op_eq * (DIFF * (alu_ic * (1 - alu_op_eq_diff_inv) + alu_op_eq_diff_inv) - 1 + alu_ic) = 0;
+
 

barretenberg/cpp/pil/avm/avm_main.pil

@@ -36,6 +36,8 @@ namespace avm_main(256);
     pol commit sel_op_div;
     // NOT
     pol commit sel_op_not;
+    // EQ
+    pol commit sel_op_eq;
 
     // Instruction memory tag (0: uninitialized, 1: u8, 2: u16, 3: u32, 4: u64, 5: u128, 6:field)
     pol commit in_tag;
@@ -64,7 +66,7 @@ namespace avm_main(256);
     pol commit rwc;
 
     // Memory index involved into a memory operation per pertaining intermediate register
-    // We should range constrain it to 32 bits ultimately. For first mini-AVM,
+    // We should range constrain it to 32 bits ultimately. For first version of the AVM,
     // we will assume that these columns are of the right type.
     pol commit mem_idx_a;
     pol commit mem_idx_b;
@@ -82,6 +84,7 @@ namespace avm_main(256);
     sel_op_mul * (1 - sel_op_mul) = 0;
     sel_op_div * (1 - sel_op_div) = 0;
     sel_op_not * (1 - sel_op_not) = 0;
+    sel_op_eq * (1 - sel_op_eq) = 0;
 
     sel_internal_call * (1 - sel_internal_call) = 0;
     sel_internal_return * (1 - sel_internal_return) = 0;
@@ -175,7 +178,7 @@ namespace avm_main(256);
 
     //===== CONTROL_FLOW_CONSISTENCY ============================================
     pol INTERNAL_CALL_STACK_SELECTORS = (first + sel_internal_call + sel_internal_return + sel_halt);
-    pol OPCODE_SELECTORS = (sel_op_add + sel_op_sub + sel_op_div + sel_op_mul + sel_op_not);
+    pol OPCODE_SELECTORS = (sel_op_add + sel_op_sub + sel_op_div + sel_op_mul + sel_op_not + sel_op_eq);
 
     // Program counter must increment if not jumping or returning
     #[PC_INCREMENT]