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What is Elliptic-curve cryptography?

A family of public-key cryptosystems that use arithmetic on points of an elliptic curve over a finite field. Their security rests on the difficulty of recovering the secret integer k from the point kG, the curve point G multiplied by k (the elliptic-curve discrete logarithm problem). For the same security they need much shorter keys than RSA: a 256-bit key is the guideline for about the same security as 3072-bit RSA (128-bit security). They are used for key exchange (ECDH and ECDHE) and signatures (ECDSA and EdDSA), with curves such as P-256 and Curve25519 in wide use. Because Shor's algorithm breaks them, they are among the schemes to be replaced by post-quantum cryptography.

Japanese name: 楕円曲線暗号

Elliptic-curve cryptography appears in 4 practice questions across 2 topics, including Applied Cryptography, Quantum Computing Hardware.

Field: Applied Cryptography

Try it: questions about Elliptic-curve cryptography

In ECDSA digital signatures, a secret random number k is used for each signature. Which of the following correctly describes how k must be handled?

  1. Ak is included in the signature and made public, so using the same value any number of times does not affect security
  2. BIf the same k is used for signatures on different messages, the private key can be computed from the two signatures
  3. CEven if k is a predictable value, signatures cannot be forged unless the private key itself leaks
Show answer and explanation

Answer: B. If the same k is used for signatures on different messages, the private key can be computed from the two signatures

An ECDSA signature (r, s) is computed from k, the hash value of the message and the private key. If two messages are signed with the same k, k can be eliminated from the two equations for s, and both k and the private key can be found. It is known that even a partial bias or predictability in the bits of k can let the private key be recovered from many signatures. Therefore either a good-quality random number is used, or k is generated deterministically from the private key and the message, as in RFC 6979.

Elliptic-curve cryptography in the explanations

Questions that mention Elliptic-curve cryptography (4)

  1. In ECDSA digital signatures, a secret random number k is used for each signature. Which of the following correctly describes how k must be handled?
  2. Once large fault-tolerant quantum computers are built, which ciphers are expected to have their security seriously undermined by Shor's algorithm?
  3. Which correctly describes post-quantum cryptography (PQC)?
  4. In the table of NIST SP 800-57, which of the following is the combination of RSA key length (the number of bits of n) and elliptic-curve key length given as the guideline for security comparable to a 128-bit symmetric-key cipher (AES-128)?

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Related terms

RSA cryptosystem Shor's algorithm AES (Advanced Encryption Standard) Post-quantum cryptography ML-KEM Public-key cryptography Symmetric-key cryptography Authenticated encryption (AEAD) Birthday attack Block cipher mode of operation

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