docs: minor improvement to readme, tutorial

README.md

@@ -1,15 +1,15 @@
 # Overview
 
-The project is a halo2 verifier written in Move language. It provides on-chain verification logic, helper utilities, and SDK tooling for verifying halo2 proofs on chain.
+The project is a halo2 zk-proofs verifier written in Move language. It provides on-chain verification api, helper utilities, and SDK tooling for verifying halo2 proofs on any Move blockchains.
 
 It supports two verifier variants:
 - A pure Move verifier for maximum portability.
 - A fast verifier backed by native functions for higher performance.
 
 Key components:
 - `packages/common`: shared Move utilities (e.g., field/serialization helpers).
-- `packages/verifier`: the core Move verifier implementation.
+- `packages/verifier`: the core verifier implementation.
 - `packages/api`: public-facing Move APIs and integration helpers.
-- `crates/*`: Rust tooling that supports the verifier (serialization, deserialization, SDK helpers, and examples).
+- `crates/*`: Rust tooling that supports the verifier (serialization/deserialization, SDK helpers, and examples).
 
 See [TUTORIAL.md](./TUTORIAL.md) for step-by-step usage and workflows.

TUTORIAL.md

@@ -55,15 +55,15 @@ Run the script to publish the verifier contracts to the local network:
 PROFILE=<your-profile-for-contracts> ./publish_contracts.sh
 ```
 
-## Create verifier for specific circuit
+## Deploy verifier for specific circuit
 
 Two verifier variants are available:
 - Native: uses native functions for faster verification.
 - Pure Move: implements verification entirely in Move, offering better portability.
 
-This section deploys the native verifier first. We use the Fibonacci circuit (project_root/examples/fibonacci) as an example to show how to deploy the verifier.
+This section deploys the native verifier first, and then deploys the pure move verifier as an option. We use the Fibonacci circuit (project_root/examples/fibonacci) as an example to show how to deploy the verifier.
 
-### Deploy halo2 verifier (native functions)
+### Deploy the native halo2 verifier
 
 Publish KZG params:
 
@@ -95,7 +95,7 @@ aptos move run --json-file test_fibonacci-1747793629098-publish-vk-native.txn --
 aptos move run --json-file test_fibonacci-1747793629098-publish-circuit-native.txn --profile <your-profile-for-verifier>
 ```
 
-### Deploy halo2 verifier (pure Move)
+###  (Optional) Deploy the pure move verifier
 
 Publish KZG params:
 
@@ -134,18 +134,19 @@ Build a verify-proof transaction for the native verifier:
 zkmove aptos build-verify-proof-native-aptos-txn --pubs-path example/proofs/test_fibonacci-1754384516414.instance --proof-path example/proofs/test_fibonacci-1754384516414.proof --k <your_parameter_k> --native-verifier-contract-address <address-of-your-profile-for-contracts> --params-address <address-of-your-profile-for-params> --native-verifier-address <address-of-your-profile-for-verifier>
 ```
 
-Or, build a verify-proof transaction for the pure Move verifier:
+Submit the verify-proof transaction (any user can submit the transaction):
 
 ```shell
-zkmove aptos build-verify-proof-aptos-txn --pubs-path example/proofs/test_fibonacci-1754384516414.instance --proof-path example/proofs/test_fibonacci-1754384516414.proof --verifier-contract-address <address-of-your-profile-for-contracts> --params-address <address-of-your-profile-for-params> --verifier-address <address-of-your-profile-for-verifier>
+aptos move run --json-file test_fibonacci-1747793629098-verify-proof-native.txn --profile <any-profile>
 ```
 
-Submit the verify-proof transaction (anyone can submit the transaction):
+Or, build a verify-proof transaction for the pure Move verifier:
 
 ```shell
-aptos move run --json-file test_fibonacci-1747793629098-verify-proof-native.txn --profile <any-profile>
+zkmove aptos build-verify-proof-aptos-txn --pubs-path example/proofs/test_fibonacci-1754384516414.instance --proof-path example/proofs/test_fibonacci-1754384516414.proof --verifier-contract-address <address-of-your-profile-for-contracts> --params-address <address-of-your-profile-for-params> --verifier-address <address-of-your-profile-for-verifier>
 ```
-Or
+Submit the transaction:
+
 ```shell
 aptos move run --json-file test_fibonacci-1747793629098-verify-proof.txn --profile <any-profile>
 ```