## cryptographic primitives

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### bls signatures

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* used in the beacon chain to verify large numbers of signtures.
* invented by dan boneh, ben lynn, and hovav shacham.
* in optimistic rollups such as arbitrum and optimism, each tx must be accompanied by its own signature. these signatures are stored on l1 calldata, a read-only format that's commited as a part of a transaction rather than to (expensive) contract storage.
* storing txs and signatures as calldata is the cheapst method available for rollups to keep data on l1.
* the key property of bls signatures is that multiple signatures can be combined into one - so only one aggregate signature needs to be verified and stored on-chain (meaning less gas fees).

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### shamir's secret sharing

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* secret sharing algorithm to distribute private information among a group, and the secret cannot be revealed unless a quorum of the groups acts together to pool their knowledge.
* the secret is matematically divided into parts. if an attacker steals some shares, it's impossible for the attacker to reconstrcut the secret unless they have stolen a quorum number of shares.
* uses cases: password managers, encrypted emails, and crypto wallets.

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### cool resources

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* **[number theory course by stanford](https://crypto.stanford.edu/pbc/notes/numbertheory/)**
* **[pairing-friendly elliptic curves of prime order](https://eprint.iacr.org/2005/133.pdf)**
* **[intro to differential power analysis](https://link.springer.com/content/pdf/10.1007/s13389-011-0006-y.pdf)**
* **[the 12 commandments of synchronization](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.227.3871&rep=rep1&type=pdf)**
* **[ring confidential transactions](https://eprint.iacr.org/2015/1098.pdf)**
* **[galois fields, part one](https://www.youtube.com/watch?v=yBVqk4YM2VY)**
* **[bls multi-signatures with pub-key aggregation](https://crypto.stanford.edu/~dabo/pubs/papers/BLSmultisig.html)**