import type { Unary, Reduce } from '../../types/bigint/module.f.ts';
/**
 * A type representing a prime field and its associated operations.
 */
export type PrimeField = {
    readonly p: bigint;
    readonly middle: bigint;
    readonly max: bigint;
    readonly neg: Unary;
    readonly sub: Reduce;
    readonly add: Reduce;
    readonly abs: Unary;
    readonly mul: Reduce;
    readonly reciprocal: Unary;
    readonly div: Reduce;
    readonly pow: Reduce;
    readonly pow2: Unary;
    readonly pow3: Unary;
};
/**
 * Creates a prime field with the specified prime modulus and associated operations.
 *
 * @param p - A prime number to define the field.
 * @returns The prime field object.
 */
export declare const prime_field: (p: bigint) => PrimeField;
/**
 * Computes the square root of a number in a prime field.
 *
 * @throws If the prime modulus `p` does not satisfy `p % 4 == 3`.
 *
 * @example
 *
 * ```js
 * const field = prime_field(7n);
 * const root = sqrt(field)(4n);
 * if (root !== 2n) { throw root }
 * ```
 */
export declare const sqrt: ({ p, mul, pow }: PrimeField) => (a: bigint) => bigint | null;