Firstly, Monero relies on the safety of Ed25519 (as opposed to secp256k1 for Bitcoin) and EdDSA. Note that if EdDSA/Ed25519 breaks, a lot more than Monero will be impacted, including SSH, Tor, I2P, OpenBSD, GnuPG and many more
From the Surae Noether CryptoNote white paper review
Implementation and use key images to prevent double spending
The CN protocol implements a piece of cryptography unseen in cryptocurrencies before, in particular, the idea of using key images to protect against double spending. This is boldly treading on dangerous ground; no matter how deeply I, or any mathematician scrutinizes an algorithm in any white paper, it’s possible some 16 year old in South Africa will figure out a way to crack the encryption.
New PoW (CryptoNight)
Implementing an entirely new Proof-of-Work algorithm could be just as vulnerable to exploitation as implementing any new piece of software. To be frank, without any sort of explicit, clear explanation of how it’s been done, it can’t necessarily be trusted. With Bitcoin,the task was clear: find the nonce so that the SHA hash is small. With this algorithm? I have no idea
Elliptic curve contants
My single biggest question after reading the entire paper
is the “how did they choose their elliptic curve constants?” The protocol appears sound; who chose the constants? Will there be a plan for choosing new constants in the future if needed? How can I choose other constants if I decide to fork it? Did the NSA come up with CryptoNote and choose these constants so any CryptoNote network has 10% the entropy of any other coin? Who knows. It’s probably not a big deal, and every coin has this as a critical point. Indeed, it’s a centralization point. If we all go happily forking the CryptoNote code left and right, we are trusting those developers to have made good decisions on the constants.
However, having said all that, CryptoNote is absolutely spectacular. If you have a problem with the constants, and if you can figure out how to generate new ones, I say go for it. The protocol looks secure and tight.