feat(avm): to_radix gadget (AztecProtocol#6368)

Please read [contributing guidelines](CONTRIBUTING.md) and remove this
line.

barretenberg/cpp/pil/avm/avm_main.pil

@@ -3,6 +3,7 @@ include "avm_mem.pil";
 include "avm_alu.pil";
 include "avm_binary.pil";
 include "avm_kernel.pil";
+include "gadgets/avm_conversion.pil";
 
 namespace avm_main(256);
     // Kernel lookup selector opcodes
@@ -25,6 +26,9 @@ namespace avm_main(256);
     pol commit sel_op_coinbase;
     pol commit sel_op_timestamp;
 
+    //===== Gadget Selectors ======================================================
+    pol commit sel_op_radix_le;
+
     //===== CONSTANT POLYNOMIALS ==================================================
     pol constant clk(i) { i };
     pol constant first = [1] + [0]*; // Used mostly to toggle off the first row consisting
@@ -167,6 +171,8 @@ namespace avm_main(256);
     sel_op_fee_per_da_gas * (1 - sel_op_fee_per_da_gas) = 0;
     sel_op_transaction_fee * (1 - sel_op_transaction_fee) = 0;
 
+    sel_op_radix_le * (1 - sel_op_radix_le) = 0;
+
     sel_op_add * (1 - sel_op_add) = 0;
     sel_op_sub * (1 - sel_op_sub) = 0;
     sel_op_mul * (1 - sel_op_mul) = 0;
@@ -451,6 +457,11 @@ namespace avm_main(256);
     is
     avm_binary.start {avm_binary.clk, avm_binary.acc_ia, avm_binary.acc_ib, avm_binary.acc_ic, avm_binary.op_id, avm_binary.in_tag};
 
+    #[PERM_MAIN_CONV]
+    sel_op_radix_le {clk, ia, ic, id}
+    is
+    avm_conversion.to_radix_le_sel {avm_conversion.clk, avm_conversion.input, avm_conversion.radix, avm_conversion.num_limbs};
+
     #[PERM_MAIN_MEM_A]
     mem_op_a {clk, mem_idx_a, ia, rwa
             , r_in_tag, w_in_tag, sel_mov_a, sel_cmov}

barretenberg/cpp/pil/avm/gadgets/avm_conversion.pil

@@ -0,0 +1,21 @@
+include "../avm_main.pil";
+
+namespace avm_conversion(256);
+
+    pol commit clk;
+
+    // Selector for Radix Operation
+    pol commit to_radix_le_sel;
+    to_radix_le_sel * (1 - to_radix_le_sel) = 0;
+
+    // ===== DRAFT: Planned Constraints for To Radix LE
+    // Similar to the binary trace; multi-row decomposition of the input using the number of limbs specified as the row count.
+    // (1) limb_ctr' - limb_ctr + 1 = 0; // Next row decrements the limb_ctr
+    // (2) Check equality to 0 of limb_ctr to terminate the operations.
+    // (3) An accumulation column to track the partial re-composition of the limbs
+    // (4) Range check each row.limb < radix
+    // (5) TODO: Is there a risk of over/under flow.
+
+    pol commit input;
+    pol commit radix;
+    pol commit num_limbs;

barretenberg/cpp/src/barretenberg/relations/generated/avm/avm_conversion.hpp

@@ -0,0 +1,47 @@
+
+#pragma once
+#include "../../relation_parameters.hpp"
+#include "../../relation_types.hpp"
+#include "./declare_views.hpp"
+
+namespace bb::Avm_vm {
+
+template <typename FF> struct Avm_conversionRow {
+    FF avm_conversion_to_radix_le_sel{};
+};
+
+inline std::string get_relation_label_avm_conversion(int index)
+{
+    switch (index) {}
+    return std::to_string(index);
+}
+
+template <typename FF_> class avm_conversionImpl {
+  public:
+    using FF = FF_;
+
+    static constexpr std::array<size_t, 1> SUBRELATION_PARTIAL_LENGTHS{
+        3,
+    };
+
+    template <typename ContainerOverSubrelations, typename AllEntities>
+    void static accumulate(ContainerOverSubrelations& evals,
+                           const AllEntities& new_term,
+                           [[maybe_unused]] const RelationParameters<FF>&,
+                           [[maybe_unused]] const FF& scaling_factor)
+    {
+
+        // Contribution 0
+        {
+            Avm_DECLARE_VIEWS(0);
+
+            auto tmp = (avm_conversion_to_radix_le_sel * (-avm_conversion_to_radix_le_sel + FF(1)));
+            tmp *= scaling_factor;
+            std::get<0>(evals) += tmp;
+        }
+    }
+};
+
+template <typename FF> using avm_conversion = Relation<avm_conversionImpl<FF>>;
+
+} // namespace bb::Avm_vm