Ops id trait (privacy-scaling-explorations#1758)

### Description
to address discussion in PR
privacy-scaling-explorations#1748

### Rationale

implement OpsIdentity for word<T> so that we no longer need to type hint
`::<Expression<F>>` or `::<F>` every time.

Author: ChihChengLiang

eth-types/src/lib.rs

@@ -24,9 +24,12 @@ pub use keccak::{keccak256, Keccak};
 
 pub use bytecode::Bytecode;
 pub use error::Error;
-use halo2_proofs::halo2curves::{
-    bn256::{Fq, Fr},
-    ff::{Field as Halo2Field, FromUniformBytes, PrimeField},
+use halo2_proofs::{
+    halo2curves::{
+        bn256::{Fq, Fr},
+        ff::{Field as Halo2Field, FromUniformBytes, PrimeField},
+    },
+    plonk::Expression,
 };
 
 use crate::evm_types::{memory::Memory, stack::Stack, storage::Storage, OpcodeId};
@@ -42,9 +45,58 @@ pub use ethers_core::{
 use serde::{de, Deserialize, Serialize};
 use std::{collections::HashMap, fmt, str::FromStr};
 
+/// trait to retrieve general operation itentity element
+pub trait OpsIdentity {
+    /// output type
+    type Output;
+    /// additive identity
+    fn zero() -> Self::Output;
+    /// multiplicative identity
+    fn one() -> Self::Output;
+}
+
+impl<F: Field> OpsIdentity for Expression<F> {
+    type Output = Expression<F>;
+    fn zero() -> Self::Output {
+        Expression::Constant(F::ZERO)
+    }
+
+    fn one() -> Self::Output {
+        Expression::Constant(F::ONE)
+    }
+}
+
+// Impl OpsIdentity for Fr
+impl OpsIdentity for Fr {
+    type Output = Fr;
+
+    fn zero() -> Self::Output {
+        Fr::zero()
+    }
+
+    fn one() -> Self::Output {
+        Fr::one()
+    }
+}
+
+// Impl OpsIdentity for Fq
+impl OpsIdentity for Fq {
+    type Output = Fq;
+
+    fn zero() -> Self::Output {
+        Fq::zero()
+    }
+
+    fn one() -> Self::Output {
+        Fq::one()
+    }
+}
+
 /// Trait used to reduce verbosity with the declaration of the [`PrimeField`]
 /// trait and its repr.
-pub trait Field: Halo2Field + PrimeField<Repr = [u8; 32]> + FromUniformBytes<64> + Ord {
+pub trait Field:
+    Halo2Field + PrimeField<Repr = [u8; 32]> + FromUniformBytes<64> + Ord + OpsIdentity<Output = Self>
+{
     /// Gets the lower 128 bits of this field element when expressed
     /// canonically.
     fn get_lower_128(&self) -> u128 {

zkevm-circuits/src/evm_circuit/execution/begin_tx.rs

@@ -27,7 +27,7 @@ use crate::{
     },
 };
 use bus_mapping::state_db::CodeDB;
-use eth_types::{evm_types::PRECOMPILE_COUNT, keccak256, Field, ToWord, U256};
+use eth_types::{evm_types::PRECOMPILE_COUNT, keccak256, Field, OpsIdentity, ToWord, U256};
 use halo2_proofs::{
     circuit::Value,
     plonk::{Error, Expression},

zkevm-circuits/src/evm_circuit/execution/callop.rs

@@ -31,7 +31,9 @@ use bus_mapping::{
     evm::OpcodeId,
     precompile::{is_precompiled, PrecompileCalls},
 };
-use eth_types::{evm_types::GAS_STIPEND_CALL_WITH_VALUE, Field, ToAddress, ToScalar, U256};
+use eth_types::{
+    evm_types::GAS_STIPEND_CALL_WITH_VALUE, Field, OpsIdentity, ToAddress, ToScalar, U256,
+};
 use halo2_proofs::{circuit::Value, plonk::Error};
 use std::cmp::min;
 

zkevm-circuits/src/evm_circuit/execution/create.rs

@@ -34,7 +34,7 @@ use bus_mapping::{
 };
 use eth_types::{
     evm_types::{GasCost, INIT_CODE_WORD_GAS},
-    Field, ToBigEndian, ToScalar, ToWord, U256,
+    Field, OpsIdentity, ToBigEndian, ToScalar, ToWord, U256,
 };
 use ethers_core::utils::keccak256;
 use gadgets::util::{and, select};

zkevm-circuits/src/evm_circuit/execution/end_block.rs

@@ -14,7 +14,7 @@ use crate::{
     table::{CallContextFieldTag, TxContextFieldTag},
     util::{word::Word, Expr},
 };
-use eth_types::Field;
+use eth_types::{Field, OpsIdentity};
 use gadgets::util::select;
 use halo2_proofs::{circuit::Value, plonk::Error};
 

zkevm-circuits/src/evm_circuit/execution/error_precompile_failed.rs

@@ -17,7 +17,7 @@ use crate::{
     witness::{Block, Call, ExecStep, Transaction},
 };
 use bus_mapping::evm::OpcodeId;
-use eth_types::{Field, U256};
+use eth_types::{Field, OpsIdentity, U256};
 use halo2_proofs::{circuit::Value, plonk::Error};
 
 #[derive(Clone, Debug)]

zkevm-circuits/src/evm_circuit/execution/error_write_protection.rs

@@ -16,7 +16,7 @@ use crate::{
         Expr,
     },
 };
-use eth_types::{evm_types::OpcodeId, Field, ToAddress, U256};
+use eth_types::{evm_types::OpcodeId, Field, OpsIdentity, ToAddress, U256};
 use halo2_proofs::{circuit::Value, plonk::Error};
 
 #[derive(Clone, Debug)]

zkevm-circuits/src/evm_circuit/execution/stop.rs

@@ -21,7 +21,7 @@ use crate::{
     },
 };
 use bus_mapping::evm::OpcodeId;
-use eth_types::Field;
+use eth_types::{Field, OpsIdentity};
 use halo2_proofs::{circuit::Value, plonk::Error};
 
 #[derive(Clone, Debug)]

zkevm-circuits/src/evm_circuit/util/common_gadget.rs

@@ -28,7 +28,9 @@ use crate::{
     witness::{Block, Call, ExecStep},
 };
 use bus_mapping::state_db::CodeDB;
-use eth_types::{evm_types::GasCost, Field, ToAddress, ToLittleEndian, ToScalar, ToWord, U256};
+use eth_types::{
+    evm_types::GasCost, Field, OpsIdentity, ToAddress, ToLittleEndian, ToScalar, ToWord, U256,
+};
 use gadgets::util::{select, sum};
 use halo2_proofs::{
     circuit::Value,

zkevm-circuits/src/evm_circuit/util/constraint_builder.rs

@@ -18,7 +18,7 @@ use crate::{
 use bus_mapping::{
     circuit_input_builder::FeatureConfig, operation::Target, state_db::EMPTY_CODE_HASH_LE,
 };
-use eth_types::Field;
+use eth_types::{Field, OpsIdentity};
 use gadgets::util::{not, sum};
 use halo2_proofs::{
     circuit::Value,