import React from "react";
import Entry from "../../components/entry";
import Link from "../../components/Link";
import CodeBlock from "../../components/CodeBlock";
import { Styled } from "theme-ui";
import { BlockMath, InlineMath } from "react-katex";

export default ({ data, location }) => {
  let blsCode = `class BLS(object):
    @classmethod
    def generate_prime_order(cls, zbits):
        while True:
            z = randint(2^(zbits - 1), 2^zbits)
            pz = int(cls.p(z))
            if not is_prime(pz):
                continue
            rz = int(cls.r(z))
            if not is_prime(rz):
                continue
            break
        K = GF(pz)
        b = 1
        while True:
            curve = EllipticCurve(K, [0, b])
            card = curve.cardinality()
            if card % rz == 0:
                break
            b += 1
        return curve


class BLS12(BLS):
    @staticmethod
    def p(z):
        return (z - 1)^2 * (z^4 - z^2 + 1)/3 + z
    @staticmethod
    def r(z):
        return z^4 - z^2 + 1
    @staticmethod
    def t(z):
        return z + 1


class BLS24(BLS):
    @staticmethod
    def p(z):
        return (z - 1)^2 * (z^8 - z^4 + 1)/3 + z
    @staticmethod
    def r(z):
        return z^8 - z^4 + 1
    @staticmethod
    def t(z):
        return z + 1`;
  return (
    <Entry data={data} location={location} title={"BLS"}>
      <Styled.h2>Barreto-Lynn-Scott curves</Styled.h2>
      <Styled.p>
        A class of pairing-friendly curves with embedding degree{" "}
        <InlineMath>{`k \\in \\{12, 24\\}`}</InlineMath>.
      </Styled.p>
      <Styled.h3>BLS12</Styled.h3>

      <Styled.p>
        Given an integer <InlineMath>{`z \\in \\mathbb{N}`}</InlineMath> the BLS
        curve with embedding degree <InlineMath>12</InlineMath> can be
        constructed over a prime field{" "}
        <InlineMath>{`\\mathbb{F}_p`}</InlineMath> with the number of points{" "}
        <InlineMath>r</InlineMath> and a trace of Frobenius{" "}
        <InlineMath>t</InlineMath>.
      </Styled.p>
      <BlockMath>
        {`\\begin{aligned}
          p(z) &= (z - 1)^2 (z^4 - z^2 + 1)/3 + z\\\\
          r(z) &= z^4 - z^2 + 1\\\\
          t(z) &= z + 1
          \\end{aligned}`}
      </BlockMath>
      <Styled.h3>BLS24</Styled.h3>
      <Styled.p>
        Given an integer <InlineMath>{`z \\in \\mathbb{N}`}</InlineMath> the BLS
        curve with embedding degree <InlineMath>24</InlineMath> can be
        constructed over a prime field{" "}
        <InlineMath>{`\\mathbb{F}_p`}</InlineMath> with the number of points{" "}
        <InlineMath>r</InlineMath> and a trace of Frobenius{" "}
        <InlineMath>t</InlineMath>.
      </Styled.p>
      <BlockMath>
        {`\\begin{aligned}
          p(z) &= (z - 1)^2 (z^8 - z^4 + 1)/3 + z\\\\
          r(z) &= z^8 - z^4 + 1\\\\
          t(z) &= z + 1
          \\end{aligned}`}
      </BlockMath>
      <Styled.p>The class of curves has the Short-Weierstrass form:</Styled.p>
      <BlockMath>y^2 \equiv x^3 + b</BlockMath>
      <Styled.p>
        where given <InlineMath>z</InlineMath> such that{" "}
        <InlineMath>p(z)</InlineMath> is prime, a curve with a prime order
        subgroup of <InlineMath>r(z)</InlineMath> points can be found either via
        complex multiplication or by exhaustively trying small coefficients{" "}
        <InlineMath>b</InlineMath> until a curve is found. Some generate curves
        can be found in the <Link to={"/bls/"}>BLS</Link> category.
      </Styled.p>
      <Styled.p>
        The following SageMath code generates BLS curves with embedding degree{" "}
        <InlineMath>12</InlineMath> and <InlineMath>24</InlineMath>.
      </Styled.p>
      <CodeBlock code={blsCode} language="python" />

      <Styled.h4>References</Styled.h4>
      <ul>
        <li>
          Paulo S. L. M. Barreto, Ben Lynn, Michael Scott:{" "}
          <Link to="https://eprint.iacr.org/2002/088.pdf">
            Constructing Elliptic Curves with Prescribed Embedding Degrees
          </Link>
        </li>
        <li>
          Diego F. Aranha, Laura Fuentes-Castaneda, Edward Knapp, Alfred
          Menezes, Francisco Rodríguez-Henríquez:{" "}
          <Link to="https://eprint.iacr.org/2012/232.pdf">
            Implementing Pairings at the 192-bit Security Level
          </Link>
        </li>
      </ul>
    </Entry>
  );
};