Cryptography Overview
66 RSA BSAFE Crypto-C Developer’s Guide
• Elliptic Curve Signature Schemes (ECDSA)
• Elliptic Curve Authenticated Encryption Scheme (ECAES)
• Elliptic Curve Diffie-Hellman key agreement (ECDH)
Crypto-C also allows you to generate precomputed acceleration tables to speed up
certain elliptic curve operations. For more information, see the example “Generating a
Public-Key Acceleration Table” on page 277.
Elliptic Curve Parameters
A number of parameters are necessary for elliptic curve cryptosystems. These
parameters must be generated before you generate a key pair, create an acceleration
table, initiate encryption, or perform key agreement with these systems. You can use
the same parameters to generate more than one key. These parameters include:
• The finite field, F
q
, over which the elliptic curve is defined.
• Two elements of F
q
, a and b, which define the elliptic curve; a and b are also called
the coefficients of the curve.
• A point P of prime order on the elliptic curve E .
• The order, n, of P .
• The cofactor h=#E(F
q
)/n. Here, E(F
q
) means the set of points on the elliptic curve
and #E(F
q
) means the number of points in that set. See “The Order of an Elliptic
Curve” on page 70 for more information.
Note: In all discussions of elliptic curves, the upper case letters P and Q are used to
denote points on an elliptic curve. The lower case letter p is used to denote a
prime.
The next section discusses these terms in detail. We will try to give enough of the
math to give you a feel for what the underlying concepts are without going too deeply
into the details. A full discussion of elliptic curve cryptography is far beyond the
scope of this manual. For background on elliptic curves, see the book by J. Silverman
and J. Tate, Rational Points on Elliptic Curves [20]. For more information on elliptic
curves in cryptography, see the ANSI X9.62 and X9.63 standards [13], the IEEE
Standard Specifications for Public-Key Cryptography [14], and A. Menezes’s book, Elliptic
Curve Public Key Cryptosystems [19].
The Finite Field
The elliptic curves used in cryptography are always defined over a finite field, denoted
F
q
. There are two choices for this field:
