# Is the generator G of the keypair always the same number for all different tx messages in Bitcoin?

If so, what is that number and why can't you divide the public key by G to obtain the private key?

• G's not publicized. – Nick ODell Mar 4 '13 at 17:22
• i thought the generator point G was publicly known? – joe Mar 4 '13 at 23:08

No sufficiently efficient algorithm for division is known. Multiplication over a closed group is a lossy process.

For an imperfect analogy, consider multiplication over the group of 100-digit numbers where you just keep the last 100 digits of the result. Since you don't know what digits were discarded, you can't naively reverse a multiplication.

The corresponding algorithm in DSA is modular exponentiation -- `(G^X) mod n`, which is also, for practical purposes with large numbers, irreversible even if G and n are known.

• Multiplication in a closed group is not lossy. It is hard to go back, but with a*G = P, if you know P and G, there will be exactly one a (modulo the order of the group) that satisfies it. It's just very hard to compute. – Pieter Wuille May 15 '14 at 8:27