The ECP group structure.
We consider two types of curve equations:
-
Short Weierstrass:
y^2 = x^3 + A x + B mod P
(SEC1 + RFC-4492)
-
Montgomery:
y^2 = x^3 + A x^2 + x mod P
(Curve25519, Curve448)
In both cases, the generator (G
) for a prime-order subgroup is fixed.
For Short Weierstrass, this subgroup is the whole curve, and its cardinality is denoted by N
. Our code requires that N
is an odd prime as mbedtls_ecp_mul() requires an odd number, and mbedtls_ecdsa_sign() requires that it is prime for blinding purposes.
For Montgomery curves, we do not store A
, but (A + 2) / 4
, which is the quantity used in the formulas. Additionally, nbits
is not the size of N
but the required size for private keys.
If modp
is NULL, reduction modulo P
is done using a generic algorithm. Otherwise, modp
must point to a function that takes an mbedtls_mpi
in the range of 0..2^(2*pbits)-1
, and transforms it in-place to an integer which is congruent mod P
to the given MPI, and is close enough to pbits
in size, so that it may be efficiently brought in the 0..P-1 range by a few additions or subtractions. Therefore, it is only an approximative modular reduction. It must return 0 on success and non-zero on failure.
- Note
- Alternative implementations of the ECP module must obey the following constraints.
- Group IDs must be distinct: if two group structures have the same ID, then they must be identical.
- The fields
id
, P
, A
, B
, G
, N
, pbits
and nbits
must have the same type and semantics as in the built-in implementation. They must be available for reading, but direct modification of these fields does not need to be supported. They do not need to be at the same offset in the structure.
Definition at line 232 of file ecp.h.