The following section of this document is related to the usage of ECC – 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 curves are also used in several integer factorization algorithms based on elliptic curves that have applications in cryptography, such as Lenstra elliptic-curve factorization.
Listing the available curves
The first think that is necessary to do is to select the appropriate curve to work with. In order to do that, the following command can be used:
openssl ecparam -list_curves
As a result, we obtain a list of possible curves (this is just a small sample):
secp112r1 : SECG/WTLS curve over a 112 bit prime field
secp112r2 : SECG curve over a 112 bit prime field
secp128r1 : SECG curve over a 128 bit prime field
secp128r2 : SECG curve over a 128 bit prime field
secp160k1 : SECG curve over a 160 bit prime field
secp160r1 : SECG curve over a 160 bit prime field
secp160r2 : SECG/WTLS curve over a 160 bit prime field
secp192k1 : SECG curve over a 192 bit prime field
secp224k1 : SECG curve over a 224 bit prime field
secp224r1 : NIST/SECG curve over a 224 bit prime field
secp256k1 : SECG curve over a 256 bit prime field
secp384r1 : NIST/SECG curve over a 384 bit prime field
secp521r1 : NIST/SECG curve over a 521 bit prime field
prime192v1: NIST/X9.62/SECG curve over a 192 bit prime field
prime192v2: X9.62 curve over a 192 bit prime field
prime192v3: X9.62 curve over a 192 bit prime field
prime239v1: X9.62 curve over a 239 bit prime field
prime239v2: X9.62 curve over a 239 bit prime field
prime239v3: X9.62 curve over a 239 bit prime field
prime256v1: X9.62/SECG curve over a 256 bit prime field
Generate a private key using ECC
Let us generate a private key using ECC and an appropriate curve (in this case, we are going to use prime256v1 [1][2]).
-----BEGIN PUBLIC KEY-----
MFkwEwYHKoZIzj0CAQYIKoZIzj0DAQcDQgAEGY20uk/tma1JkfNiXbWPshXVxfw+
Y2GoycH2t5tOcf70dztMaXmozKHyQu8EvcrVaBN86Y+9q+d2gsh/WUJ4GA==
-----END PUBLIC KEY-----
As mentioned before, ECC can be used to do some key agreement between two parties, to derive a common key that might be used for encryption and decryption, using an appropriate symmetric cryptography algorithm.
Let’s imagine that you have Alice and Bob that want to use ECC to create a common shared key. The following should be done.
Alice, creates a private key and extracts the public key:
