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The ECDLP is the inverse operation to multiplication by a scalar; if K=kP, given K and P, find k.

Elliptic curve cryptography ECC is an asymmetric cryptosystem based on the elliptic curve discrete log problem. The ECDLP arises in Abelian groups defined on elliptic curves. -R. P. Q. R. y 2 =x 3 – ax + b.

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The ECDLP is the inverse operation to multiplication by a scalar; if K=kP, given K and P, find k.

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  1. Elliptic curve cryptographyECC is an asymmetric cryptosystem based on the elliptic curve discrete log problem.The ECDLP arises in Abelian groups defined on elliptic curves.

  2. -R P Q R y2 =x3 – ax + b The ECDLP is the inverse operation to multiplication by a scalar; if K=kP, given K and P, find k. P + Q = R. Defining addition (and doubling) defines multiplication by a scalar.

  3. The ECDLP is intractable; for a given field size, it is vastly harder to find k from kP and P than it is to find kP from k and P.k is thus used as the private key; kP is used as the public.The ECDLP is widely believed to be resistant to Number Field Sieve attacks. The best known attack is Pollard’s Rho—whose difficulty grows more rapidly with the field size than do NFS methods.

  4. Equivalent key sizes

  5. Four primitives/protocols — ECDSA — ECDH — ECIES — ECMQV

  6. ECDSAElliptic Curve Digital Signature AlgorithmECDSA provides sign and verify operations; it is analogous to DSA.

  7. ECDHElliptic Curve Diffie-HellmanECDH is analogous to conventional Diffie-Hellman; p(qG) = q(pG); qG, pG are public values; p and q are private.

  8. ECIESElliptic Curve Integrated Encryption Scheme.ECIES is analogous to public key encryption.The initiator performs an ECDH-type transform with the respondent’s static key pair; the generated key is then used for encryption/signing.

  9. ECMQVElliptic Curve Menezes-Qu-Vanstone.MQV is intended as replacement for Signed DH.ECMQV is endorsed in the NSA’s Suite B.Both parties must use their private key in an AVF to generate the shared key; this functions as an implicit signature, proving possession of the private key.

  10. Uses in MIKEYECDH drops in for DH.ECDSA drops in for DSA.*ECIES and 1-pass MQV drop in for public key encryption methods.*2-pass MQV could drop in for signed DH; this is not in the current draft.** These methods require EC certs.

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