Elliptic Curve Cryptography: Basics and Applications

This lecture covers the fundamentals of elliptic curve cryptography, including the use of elliptic curves, elliptic curve cryptography, and pairing-based cryptography. It delves into the ECM factorization algorithm, standard curves, manipulating elliptic curves in practice, domain parameters, NIST standard curves, and other standards. The lecture also explores practical examples such as secp192r1 and sect163r2, as well as real-world applications like secp256r1 in Swiss biometric passports and Curve25519 in various technologies. Additionally, it discusses key concepts like ECDH, ECDSA, and ECIES, along with common protocols and activities related to elliptic curve cryptography.

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Elliptic-curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC allows smaller keys compared to non-EC cryptography (based on plain Galois fields) to provide equivalent security. Elliptic curves are applicable for key agreement, digital signatures, pseudo-random generators and other tasks. Indirectly, they can be used for encryption by combining the key agreement with a symmetric encryption scheme.

Elliptic Curve Digital Signature Algorithm

In cryptography, the Elliptic Curve Digital Signature Algorithm (ECDSA) offers a variant of the Digital Signature Algorithm (DSA) which uses elliptic-curve cryptography. As with elliptic-curve cryptography in general, the bit size of the private key believed to be needed for ECDSA is about twice the size of the security level, in bits. For example, at a security level of 80 bits—meaning an attacker requires a maximum of about operations to find the private key—the size of an ECDSA private key would be 160 bits.

Curve25519

In cryptography, Curve25519 is an elliptic curve used in elliptic-curve cryptography (ECC) offering 128 bits of security (256-bit key size) and designed for use with the elliptic curve Diffie–Hellman (ECDH) key agreement scheme. It is one of the fastest curves in ECC, and is not covered by any known patents. The reference implementation is public domain software. The original Curve25519 paper defined it as a Diffie–Hellman (DH) function. Daniel J.

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Transport Layer Security (TLS) is a cryptographic protocol designed to provide communications security over a computer network. The protocol is widely used in applications such as email, instant messaging, and voice over IP, but its use in securing HTTPS remains the most publicly visible. The TLS protocol aims primarily to provide security, including privacy (confidentiality), integrity, and authenticity through the use of cryptography, such as the use of certificates, between two or more communicating computer applications.

Curve448

In cryptography, Curve448 or Curve448-Goldilocks is an elliptic curve potentially offering 224 bits of security and designed for use with the elliptic-curve Diffie–Hellman (ECDH) key agreement scheme. Developed by Mike Hamburg of Rambus Cryptography Research, Curve448 allows fast performance compared with other proposed curves with comparable security. The reference implementation is available under an MIT license. The curve was favored by the Internet Research Task Force Crypto Forum Research Group (IRTF CFRG) for inclusion in Transport Layer Security (TLS) standards along with Curve25519.

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