# Account signatures and keys in Polkadot

#1

We believe Polkadot accounts should primarily use Schnorr signatures with both public keys and the R point in the signature encoded using the Ristretto point compression for the Ed25519 curve. We should collaborate with the dalek ecosystem for which Ristretto was developed, but provide a simpler signature crate, for which schnorr-dalek provides a first step.

I’ll write a another comment giving more details behind this choice, but the high level summary goes:

Account keys must support the diverse functionality desired of account keys on other systems like Ethereum and Bitcoin. As such, our account keys shall use Schnorr signatures because these support fast batch verification and hierarchical deterministic key derivation ala BIP32. All features from the Bitcoin Schnoor wishlist provides a case for Schnorr signatures matter too, like

• interactive threshold and multi-signaturtes, as well as
• adaptor, and perhaps even blind, signatures for swaps and payment channels.

We make conservative curve choices here because account keys must live for decades. In particular, we avoid pairing-based cryptography and BLS signatures for accounts, at the cost of true aggregation of the signatures in a block when verifying blocks, and less interactive threshold and multi-signaturtes. [1].

In the past, there was a tricky choice between the more secure curves:

• miss-implementation resistance is stronger with Edwards curves, including the Ed25519 curve, but
• miss-use resistance in stronger when curves have cofactor 1, like secp256k1.

In fact, miss-use resistance was historically a major selling point for Ed25519, which itself is a Schnorr variant, but this miss-use resistance extends only so far as the rudimentary signature scheme properties it provided. Yet, any advanced signature scheme functions, beyond batch verification, break precisely due to Ed25519’s miss-use resistance. In fact, there are tricks for doing at least hierarchical deterministic key derivation on Ed25519, as implemented in hd-ed25519, but almost all previous efforts produced insecure results.

We observe that secp256k1 provides a good curve choice from among the curves of cofactor 1, which simplify make implementing fancier protocols. We do worry that such curves appear at least slightly weaker than Edwards curves. We worry much more than such curves tend to be harder to implement well, due to having incomplete addition formulas, and thus require more review (see safecurves.cr.yp.to). We could select only solid implementations for Polkadot itself, but we cannot control the implementations selected elsewhere in our ecosystem, especially by wallet software.

In short, we want an Edwards curve but without the cofactor, which do not exist, except…

In Edwards curve of with cofactor 4, Mike Hamburg’s Decaf point compression only permits serialising and deserialising points on the subgroup of order l, which provides a perfect solution. Ristretto pushes this point compression to cofactor 8, making it applicable to the Ed25519 curve. Implementations exist in both Rust and C. If required in another language, the compression and decompression functions are reasonable to implement using an existing field implementation, and fairly easy to audit.

In the author’s words, “Rather than bit-twiddling, point mangling, or otherwise kludged-in ad-hoc fixes, Ristretto is a thin layer that provides protocol implementors with the correct abstraction: a prime-order group.”

[1] Aggregation can dramatically reduce signed message size when applying numerous signatures, but if performance is the only goal then batch verification techniques similar results, and exist for mny signature schemes, including Schnorr. There are clear advantages to reducing interactiveness in threshold and multi-signaturtes, but parachains can always provide these on Polkadot. Importantly, there are numerous weaknesses in all known curves that support pairings, but the single most damning weakness is the pairing e : G_1 \times G_2 \to G_T itself. In essence, we use elliptic curves in the first palce because they insulate us somewhat from mathematicians ever advancing understanding of number theory. Yet, any known pairing maps into a group G_T that re-exposes us, so attacks based on index-calculus, etc. improve more quickly. As a real world example, there were weaknesses found in BN curve of the sort used by ZCash during development, so after launch they needed to develop and migrate to a new curve. We expect this to happen again for roughly the same reasons that RSA key sizes increase slowly over time.

#2

## Arguments for Schnorr signatures:

We prefer Schnorr signatures because they satisfy the Bitcoin Schnoor wishlist and work fine with extremely secure curves, like secp256k1 or the Ed25519 curve. You could do fancier tricks, including like aggregation, with a pairing based curve like BLS12-381 and the BLS signature scheme. These curves are slower for single verifications, and worse accounts should last decades while pairing friendly curves should be expected become less secure as number theory advances.

There is one sacrifice we make by choosing Schnorr signatures over ECDSA signatures for account keys: Both require 64 bytes, but only ECDSA signatures communicate their public key. There are obsolete Schnorr variants that support recovering the public key from a signature, but they break important functionality like hierarchical deterministic key derivation. In consequence, Schnorr signatures often take an extra 32 bytes for the public key.

In exchange, we gain a slightly faster signature scheme with far simpler batch verification than ECDSA batch verification and more natural threshold and multi-signatures, as well as tricks used by payment channels. I also foresee the presence of this public key data may improve locality in block verification, possibly openning up larger optimisations.

Yet most importantly, we can protect Schnorr signatures using both the derandomization tricks of EdDSA along with a random number generator, which gives us stronger side-channel protections than conventional ECDSA schemes provide. If we ever do want to support ECDSA as well, then we would first explore improvements in side-channel protections like rfc6979, along with concerns like batch verification, etc.

In this vein, “Biased Nonce Sense: Lattice Attacks against Weak ECDSA Signatures in Cryptocurrencies” by Joachim Breitner and Nadia Heninger gives a recent example of a practical attack due to ECDSA’s complexity and randomness issues.

#3

## Arguments for Ristretto compressed Ed25519

There are two normal curve choices for accounts on a blockchain system, either secp256k1 or the Ed25519 curve, so we confine our discussion to them. If you wanted slightly more speed, you might choose FourQ, but it sounds excessive for blockchains, implementations are rare, and it appears covered by older but not quite expired patents. Also, you might choose Zcash’s JubJub if you wanted fast signature verification in zkSNARKs, but that’s not on our roadmap for Polkadot, and Jubjub also lacks many implementations.

### How much secp256k1 support?

We need some minimal support for secp256k1 keys because token sale accounts are tied to secp256k1 keys on Ethereum, so some “account” type must necessarily use secp256k1 keys. At the same time, we should not encourage using the same private keys on Ethereum and Polkadot. We might pressure users into switching key types in numerous ways, like secp256k1 accounts need not support balance increases, or might not support anything but replacing themselves with an ed25519 key. There are conceivable reasons for fuller secp256k1 support though, like wanting ethereum smart contracts to verify some signatures on Polkadot. We might support secp256k1 accounts with limited functionality, but consider expanding that functionality if such use cases arise.

### Is secp256k1 risky?

There are two theoretical reasons for preferring an twisted Edwards curve over secp256k1: First, secp256k1 has a small CM field discriminant, which might yield better attacks in the distant future. Second, secp256k1 has fairly rigid paramater choices but not the absolute best. I do not believe either to be serious cause for concern. Among more practical curve weaknesses, secp256k1 does have twist security which eliminates many attack classes.

I foresee only one substantial reason for avoiding secp256k1: All short Weierstrass curves like secp256k1 have incomplete addition formulas, meaning certain curve points cannot be added to other curve points. As a result, addition code must check for failures, but these checks make writing constant time code harder. We could examine any secp256k1 library we use in Polkadot to ensure it both does these checks and has constant-time code. We cannot however ensure that all implementations used by third party wallet software do so.

I believe incomplete addition formulas looks relatively harmless when used for simple Schnorr signatures, although forgery attacks might exist. I’d worry more however if we began using secp256k1 for less well explored protocols, like multi-signaturtes and key derivation. We ware about such use cases however, especially those listed in the Bitcoin Schnoor wishlist.

### Is Ed25519 risky? Aka use Ristretto

Any elliptic curve used in cryptography has order h*l where l is a big prime, normally close to a power of two, and h is some very small number called the cofactor. Almost all protocol implementations are complicated by these cofactors, so implementing complex protocols is safer on curves with cofactor h=1 like secp256k1.

The Ed25519 curve has cofactor 8 but a simple convention called “clamping” that makes two particularly common protocols secure. We must restrict or drop “clamping” for more complex protocols, like multi-signaturtes and key derivation, or anything else in the Bitcoin Schnoor wishlist.

If we simple dropped “clamping” then we’d make implementing protocols harder, but luckily the Ristretto encoding for the Ed25519 curve ensures we avoid any curve points with 2-torsion. I thus recommend:

• our secret key continue being Ed25519 “expanded” secret keys, while
• our on-chain encoding, aka “point compression” becomes Ristretto for both public keys and the R component of Schnoor signatures.

In principle, we could use the usual Ed25519 “mini” secret keys for simple use cases, but not when doing key derivation. We could thus easily verify standrad Ed25519 signatures with Ristretto encoded public keys. We should ideally use Ristretto throughout instead of the standard Ed25519 point compression.

In fact, we can import standard Ed25519 compressed points like I do here but this requires the scalar exponentiation done in the is_torsion_free method, which runs slower than normal signature verification. We might ideally do this only for key migration between PoCs.

Ristretto is far simpler than the Ed25519 curve itself, so Ristretto can be added to Ed25519 implementations, but the curve25519-dalek crate already provides a highly optimised rust implementation.

### Zero-knowledge proofs in the dalek ecosystem

In fact, the dalek ecosystem has an remarkably well designed infrastructure for zero-knowledge proofs without pairings. See:
https://medium.com/interstellar/bulletproofs-pre-release-fcb1feb36d4b
https://medium.com/interstellar/programmable-constraint-systems-for-bulletproofs-365b9feb92f7

All these crates use Ristretto points so using Ristretto for account public keys ourselves gives us the most advanced tools for building protocols not based on pairings, meaning that use our account keys. In principle, these tools might be abstracted for twisted Edwards curves like FourQ and Zcash’s Jubjub, but yu might loose some batching operations in abstracting them for short Weierstrass curves like secp256k1.