chore(doc): add doc for dedicated CPK parameters and Cast for faster ZK

tfhe/docs/guides/zk-pok.md

@@ -58,14 +58,92 @@ pub fn main() -> Result<(), Box<dyn std::error::Error>> {
 }
 ```
 
-In terms of performance one can expect the following numbers:
-
-* Encrypting and proving a `CompactFheUint64` takes **6.9 s** on a `Dell XPS 15 9500` (simulating a client machine).
-* Verification takes **123 ms** on an `hpc7a.96xlarge` AWS instances.
-
 Performance can be improved by setting `lto="fat"` in `Cargo.toml`
 ```toml
 [profile.release]
 lto = "fat"
 ```
 and by building the code for the native CPU architecture and in release mode, e.g. by calling `RUSTFLAGS="-C target-cpu=native" cargo run --release`.
+
+{% hint style="info" %}
+You can choose a more costly proof with `ZkComputeLoad::Proof`, which has a faster verification time.  Alternatively, you can select `ZkComputeLoad::Verify` for a faster proof and slower verification.
+{% endhint %}
+
+## Using dedicated Compact Public Key parameters
+
+### A first example
+You can use dedicated parameters for the compact public key encryption to reduce the size of encrypted data and speed up the zero-knowledge proof computation.
+
+This works essentially in the same way as before. Additionally, you need to indicate the dedicated parameters to use:
+
+```rust
+use rand::prelude::*;
+use tfhe::prelude::FheDecrypt;
+use tfhe::set_server_key;
+use tfhe::zk::{CompactPkeCrs, ZkComputeLoad};
+
+pub fn main() -> Result<(), Box<dyn std::error::Error>> {
+    let mut rng = thread_rng();
+
+    let params = tfhe::shortint::parameters::PARAM_MESSAGE_2_CARRY_2_KS_PBS_TUNIFORM_2M64;
+    // Indicate which parameters to use for the Compact Public Key encryption
+    let cpk_params = tfhe::shortint::parameters::compact_public_key_only::PARAM_PKE_MESSAGE_2_CARRY_2_KS_PBS_TUNIFORM_2M64;
+    // And parameters allowing to keyswitch/cast to the computation parameters.
+    let casting_params = tfhe::shortint::parameters::key_switching::PARAM_KEYSWITCH_MESSAGE_2_CARRY_2_KS_PBS_TUNIFORM_2M64;
+    // Enable the dedicated parameters on the config
+    let config = tfhe::ConfigBuilder::with_custom_parameters(params, None)
+        .use_dedicated_compact_public_key_parameters((cpk_params, casting_params));
+
+    // Then use TFHE-rs as usual
+    let client_key = tfhe::ClientKey::generate(config.clone());
+    // This is done in an offline phase and the CRS is shared to all clients and the server
+    let crs = CompactPkeCrs::from_config(config.into(), 64).unwrap();
+    let public_zk_params = crs.public_params();
+    let server_key = tfhe::ServerKey::new(&client_key);
+    let public_key = tfhe::CompactPublicKey::try_new(&client_key).unwrap();
+
+    let clear_a = rng.gen::<u64>();
+    let clear_b = rng.gen::<u64>();
+
+    let proven_compact_list = tfhe::ProvenCompactCiphertextList::builder(&public_key)
+        .push(clear_a)
+        .push(clear_b)
+        .build_with_proof_packed(public_zk_params, ZkComputeLoad::Verify)?;
+
+    // Server side
+    let result = {
+        set_server_key(server_key);
+
+        // Verify the ciphertexts
+        let mut expander = proven_compact_list.verify_and_expand(&public_zk_params, &public_key)?;
+        let a: tfhe::FheUint64 = expander.get(0).unwrap()?;
+        let b: tfhe::FheUint64 = expander.get(1).unwrap()?;
+
+        a + b
+    };
+
+    // Back on the client side
+    let a_plus_b: u64 = result.decrypt(&client_key);
+    assert_eq!(a_plus_b, clear_a.wrapping_add(clear_b));
+
+    Ok(())
+}
+```
+
+### Benchmarks
+Benchmarks for the proofs have been run on a `m6i.4xlarge` with 16 cores to simulate an usual client configuration.  The verification are done on a `hpc7a.96xlarge` AWS instances to mimic a powerful server. 
+
+Timings in the case where the workload is mainly on the prover, i.e., with the  `ZkComputeLoad::Proof` option.
+
+| Inputs       | Proving | Verifying |
+|--------------|---------|-----------|
+| 1xFheUint64  | 2.79s   | 197ms     |
+| 10xFheUint64 | 3.68s   | 251ms     |
+ 
+
+Timings in the case where the workload is mainly on the verifier, i.e., with the  `ZkComputeLoad::Verify` option.
+
+| Inputs       | Proving | Verifying |
+|--------------|---------|-----------|
+| 1xFheUint64  | 730ms   | 522ms     |
+| 10xFheUint64 | 1.08s   | 682ms     |