import type { Reduce } from '../../types/function/operator/module.f.ts';
import { type PrimeField } from '../prime_field/module.f.ts';
/**
 * A 2D point represented as a pair of `bigint` values `[x, y]`.
 */
type Point2D = readonly [bigint, bigint];
/**
 * A 2D point on an elliptic curve, represented as a pair of `bigint` values.
 * `null` represents the point at infinity (`O`).
 */
export type Point = Point2D | null;
/**
 * Initialization parameters for an elliptic curve.
 */
export type Init = {
    readonly p: bigint;
    readonly a: readonly [bigint, bigint];
    readonly g: readonly [bigint, bigint];
    readonly n: bigint;
};
/**
 * Represents an elliptic curve and its associated operations.
 */
type Curve = {
    readonly pf: PrimeField;
    readonly nf: PrimeField;
    readonly y2: (x: bigint) => bigint;
    readonly y: (x: bigint) => bigint | null;
    readonly neg: (a: Point) => Point;
    readonly add: Reduce<Point>;
    readonly mul: (a: Point) => (n: bigint) => Point;
};
/**
 * Constructs an elliptic curve with the given initialization parameters.
 *
 * @example
 *
 * ```js
 * const curveParams = {
 *     p: 23n,
 *     a: [0n, 1n],
 *     g: [1n, 1n],
 *     n: 19n
 * };
 * const curveInstance = curve(curveParams);
 *
 * // Access curve operations
 * const point = curveInstance.add([1n, 1n])([2n, 5n]); // Add two points
 * const negPoint = curveInstance.neg([1n, 1n]); // Negate a point
 * const mulPoint = curveInstance.mul([1n, 1n])(3n); // Multiply a point by 3
 * ```
 */
export declare const curve: ({ p, a: [a0, a1], n }: Init) => Curve;
export declare const eq: (a: Point) => (b: Point) => boolean;
/**
 * https://neuromancer.sk/std/secg/secp192r1
 */
export declare const secp192r1: Init;
/**
 * https://en.bitcoin.it/wiki/Secp256k1
 * https://neuromancer.sk/std/secg/secp256k1
 */
export declare const secp256k1: Init;
/**
 * https://neuromancer.sk/std/secg/secp256r1
 */
export declare const secp256r1: Init;
/**
 * https://neuromancer.sk/std/secg/secp384r1
 */
export declare const secp384r1: Init;
/**
 * https://neuromancer.sk/std/secg/secp521r1
 */
export declare const secp521r1: Init;
export {};