//!
//! ## Overview
//!
//! This module implements a field gadget for a prime field `Fp` over another prime field `Fq` where `p != q`.
//!
//! When writing constraint systems for many cryptographic proofs, we are restricted to a native field (e.g., the scalar field of the pairing-friendly curve).
//! This can be inconvenient; for example, the recursive composition of proofs via cycles of curves requires the verifier to compute over a non-native field.
//!
//! The library makes it possible to write computations over a non-native field in the same way one would write computations over the native field. This naturally introduces additional overhead, which we minimize using a variety of optimizations. (Nevertheless, the overhead is still substantial, and native fields should be used where possible.)
//!
//! ## Usage
//!
//! Because [`NonNativeFieldVar`] implements the [`FieldVar`] trait in arkworks, we can treat it like a native field variable ([`FpVar`]).
//!
//! We can do the standard field operations, such as `+`, `-`, and `*`. See the following example:
//!
//! ```rust
//! # fn main() -> Result<(), ark_relations::r1cs::SynthesisError> {
//! # use ark_std::UniformRand;
//! # use ark_relations::{ns, r1cs::ConstraintSystem};
//! # use ark_r1cs_std::prelude::*;
//! use ark_r1cs_std::fields::nonnative::NonNativeFieldVar;
//! use ark_bls12_377::{Fr, Fq};
//!
//! # let mut rng = ark_std::test_rng();
//! # let a_value = Fr::rand(&mut rng);
//! # let b_value = Fr::rand(&mut rng);
//! # let cs = ConstraintSystem::<Fq>::new_ref();
//!
//! let a = NonNativeFieldVar::<Fr, Fq>::new_witness(ns!(cs, "a"), || Ok(a_value))?;
//! let b = NonNativeFieldVar::<Fr, Fq>::new_witness(ns!(cs, "b"), || Ok(b_value))?;
//!
//! // add
//! let a_plus_b = &a + &b;
//!
//! // sub
//! let a_minus_b = &a - &b;
//!
//! // multiply
//! let a_times_b = &a * &b;
//!
//! // enforce equality
//! a.enforce_equal(&b)?;
//! # Ok(())
//! # }
//! ```
//!
//! ## Advanced optimization
//!
//! After each multiplication, our library internally performs a *reduce* operation,
//! which reduces an intermediate type [`NonNativeFieldMulResultVar`] to the normalized type [`NonNativeFieldVar`].
//! This enables a user to seamlessly perform a sequence of operations without worrying about the underlying details.
//!
//! However, this operation is expensive and is sometimes avoidable. We can reduce the number of constraints by using this intermediate type, which only supports additions. To multiply, it must be reduced back to [`NonNativeFieldVar`]. See below for a skeleton example.
//!
//! ---
//!
//! To compute `a * b + c * d`, the straightforward (but more expensive) implementation is as follows:
//!
//! ```ignore
//! let a_times_b = &a * &b;
//! let c_times_d = &c * &d;
//! let res = &a_times_b + &c_times_d;
//! ```
//!
//! This performs two *reduce* operations in total, one for each multiplication.
//!
//! ---
//!
//! We can save one reduction by using the [`NonNativeFieldMulResultVar`], as follows:
//!
//! ```ignore
//! let a_times_b = a.mul_without_reduce(&b)?;
//! let c_times_d = c.mul_without_reduce(&d)?;
//! let res = (&a_times_b + &c_times_d)?.reduce()?;
//! ```
//!
//! It performs only one *reduce* operation and is roughly 2x faster than the first implementation.
//!
//! ## Inspiration and basic design
//!
//! This implementation employs the standard idea of using multiple **limbs** to represent an element of the target field. For example, an element in the TargetField may be represented by three BaseField elements (i.e., the limbs).
//!
//! ```text
//! TargetField -> limb 1, limb 2, and limb 3 (each is a BaseField element)
//! ```
//!
//! After some computation, the limbs become saturated and need to be **reduced**, in order to engage in more computation.
//!
//! We heavily use the optimization techniques in [\[KPS18\]](https://akosba.github.io/papers/xjsnark.pdf) and [\[OWWB20\]](https://eprint.iacr.org/2019/1494).
//! Both works have their own open-source libraries:
//! [xJsnark](https://github.com/akosba/xjsnark) and
//! [bellman-bignat](https://github.com/alex-ozdemir/bellman-bignat).
//! Compared with these, this module works with the `arkworks` ecosystem.
//! It also provides the option (based on an `optimization_goal` for the constraint system) to optimize
//! for constraint density instead of number of constraints, which improves efficiency in
//! proof systems like [Marlin](https://github.com/arkworks-rs/marlin).
//!
//! ## References
//! \[KPS18\]: A. E. Kosba, C. Papamanthou, and E. Shi. "xJsnark: a framework for efficient verifiable computation," in *Proceedings of the 39th Symposium on Security and Privacy*, ser. S&P ’18, 2018, pp. 944–961.
//!
//! \[OWWB20\]: A. Ozdemir, R. S. Wahby, B. Whitehat, and D. Boneh. "Scaling verifiable computation using efficient set accumulators," in *Proceedings of the 29th USENIX Security Symposium*, ser. Security ’20, 2020.
//!
//! [`NonNativeFieldVar`]: crate::fields::nonnative::NonNativeFieldVar
//! [`NonNativeFieldMulResultVar`]: crate::fields::nonnative::NonNativeFieldMulResultVar
//! [`FpVar`]: crate::fields::fp::FpVar

#![allow(
    clippy::redundant_closure_call,
    clippy::enum_glob_use,
    clippy::missing_errors_doc,
    clippy::cast_possible_truncation,
    clippy::unseparated_literal_suffix
)]

use ark_std::fmt::Debug;

/// Utilities for sampling parameters for non-native field gadgets
///
/// - `BaseField`:              the constraint field
/// - `TargetField`:            the field being simulated
/// - `num_limbs`:              how many limbs are used
/// - `bits_per_limb`:          the size of the limbs
pub mod params;
/// How are non-native elements reduced?
pub(crate) mod reduce;

/// a macro for computing ceil(log2(x)) for a field element x
macro_rules! overhead {
    ($x:expr) => {{
        use ark_ff::BigInteger;
        let num = $x;
        let num_bits = num.into_bigint().to_bits_be();
        let mut skipped_bits = 0;
        for b in num_bits.iter() {
            if *b == false {
                skipped_bits += 1;
            } else {
                break;
            }
        }

        let mut is_power_of_2 = true;
        for b in num_bits.iter().skip(skipped_bits + 1) {
            if *b == true {
                is_power_of_2 = false;
            }
        }

        if is_power_of_2 {
            num_bits.len() - skipped_bits
        } else {
            num_bits.len() - skipped_bits + 1
        }
    }};
}

pub(crate) use overhead;

/// Parameters for a specific `NonNativeFieldVar` instantiation
#[derive(Clone, Debug)]
pub struct NonNativeFieldConfig {
    /// The number of limbs (`BaseField` elements) used to represent a `TargetField` element. Highest limb first.
    pub num_limbs: usize,

    /// The number of bits of the limb
    pub bits_per_limb: usize,
}

mod allocated_field_var;
pub use allocated_field_var::*;

mod allocated_mul_result;
pub use allocated_mul_result::*;

mod field_var;
pub use field_var::*;

mod mul_result;
pub use mul_result::*;