# Can I generate two similar ECDSA public keys?

I am using a system that relies on base64 encoded ECDSA public keys. I have managed to brute-force a public key that when encoded starts with a word I like. Is it possible for me, given the private key, generate similar public keys?

• Think in this direction: if one keeps doing what they have done when they "managed to brute-force a public key that when encoded starts with a word (they) like", will they succeed again? And if so, what are the odds that they generate the same public key?
– fgrieu
Apr 4, 2022 at 7:03
• Are you asking in the context of cryptocurrency? Apr 5, 2022 at 18:21
• No, I guess that for most cryptocurrency people want their hashed public key to look cool which seems like a different problem as similar public keys would result in different hashes. Apr 7, 2022 at 5:39

It's possible to generate one extremely similar public key. If your private key is $$s$$ and your public key is the point $$(x,y)$$ then another valid private/public key pairs is $$\ell-s$$ and $$(x,p-y)$$. Here $$\ell$$ is the curve order and $$p$$ is the characterstic of the field. If you write your public key in compressed form, the representations will differ only in a single bit (the compression bit)!
Some elliptic curves also permit other simple endomorphisms due to their complex multiplication structure e.g. curves of the form $$Y^2=X^3-aX$$ allow one to generate the additional private/public pair $$js$$ and $$(-x,\pm iy)$$ where $$i$$ is a square root of -1 mod $$p$$ and $$j$$ is a square root of -1 mod $$\ell$$.