A private key is secret data controlled by its owner in an asymmetric key pair. It derives a public key, creates shared secrets, and signs documents; security depends on keeping it secret. section 1.3 section 4.1.3
Use the private key in three closely related flows:
- Create a public key by multiplying the private scalar by a public base point: A = k_A G. section 4.1.1
- Derive a shared secret by multiplying the other party’s public key by the local private key; both sides obtain k_A k_B G. section 1.3.1 section 4.1.1
- Sign a document together with its hash so anyone can verify authorship and detect even a one-bit modification using the public key. section 1.3.2
For elliptic-curve systems, choose a secure curve and public base point G. Convert a pass phrase into an integer, commonly by hashing modulo the curve order, or use another bit string. The resulting scalar is the private key; elliptic-curve multiplication produces the public key. section 4.1.1 section 4.1.3
Alice and Bob
Alice keeps k_A secret and publishes A = k_A G. Bob does the same with k_B and B. Alice computes k_A B, while Bob computes k_B A; both equal the shared point S and can derive symmetric-key material from its x component. Only public keys and the encrypted message cross the open network. section 1.3.1 section 4.1.1
Listing 4.3 Key generation
void gen_key(mpz_t sk, POINT *PK, unsigned char *phrase, BASE_SYSTEM bse)
{
long np;
np = 0; #1
while(phrase[np]) np++; #1
hash(sk, phrase, np, bse.order); #2
elptic_mul(PK, bse.Base, sk, bse.E);
}
Shows the compact implementation flow: measure the pass phrase, hash it into the private scalar, and multiply the base point to obtain the public key. section 4.1.3
Listing 18.2 Key generation for pairing signatures
void keygen(mpz_t sk, POLY_POINT *PK, POLY_POINT G2, POLY_CURVE Ex,
unsigned char *ptr, int len)
{
#1
mpz_import(sk, len, -1, 1, 0, 0, ptr); #1
poly_elptic_mul(PK, G2, sk, Ex); #2
} #2
Extends the same pattern to pairing-based signatures by importing input bytes as a scalar and multiplying a field-extension base point. section 18.3.1
