Home > Cryptography > ECC: Elliptic-Curve Cryptography

In contrast to Symmetric Key schemes such as AES (Advanced Encryption Standard), the asymmetric key scheme contains a key pair, the public key, and the private key. This is also known as public key cryptography.

Asymmetric Key

Complex mathematical problems underpin the asymmetric key construction, which allows the private key to derive a corresponding public key easily while keeping reverse computation infeasible. Several popular cryptographic schemes have been constructed using the mathematical relation between the public and private keys, such as a digital signature or a key agreement.

ECC Standards

The elliptic curve cryptography (ECC) is an asymmetric key tool based on the elliptic curve discrete logarithm problem (ECDLP). SEC 1 presents the ECC-based signature scheme, encryption and key transport scheme, and key agreements scheme. To assist others with adopting the ECC, SEC 2 proposed 15 domain parameters categorized into five security levels: 80-bit, 112-bit, 128-bit, 192-bit, and 256-bit.

Classical Public key Cryptography

RSA and Diffie-Hellman-Merkle are the first two public key constructions. RSA is based on the integer factorization problem, and Diffie-Hellman-Merkle is based on the finite field discrete logarithm problem (DLP). Due to the existence of sub-exponential time algorithms solving these problems, the key size of these two public key systems is longer than the ECC key size. The ECC cryptography therefore is considered a natural modern successor of RSA and Diffie-Hellman-Merkle, due to the smaller keys and signatures producing the same level of security, while also providing faster key generation, key agreement, and signatures.


SM2 is a bunch of ECC standards proposed by China. It defines an ECC-based digital signature scheme, an encryption scheme, and a Key exchange scheme. In addition, it also presents its own domain parameters of security level 128-bit, which is different from those in SEC 2.

PUFcc-provided ECC Functions

PUFcc Crypto Coprocessors implements the big number arithmetic computation as the building block for ECC-based cryptographic operations. Additionally, the private key blinding technique can be introduced to the ECC operations to resist a side-channel attack on private key related calculations. Based on the underlying mathematical computation, PUFcc also supports the generation of an ECC private key and public key, the validation of a public key, the elliptic curve digital signature algorithm (ECDSA) elliptic curve Diffie-Hellman (ECDH), and SM2-related functions. These implementations are all certified by NIST CAVP and the office of the state commercial cryptography administration (OSCCA).