Score:2

# Why do we need to convert hashes to points on an elliptic curve?

In order to sign message m, m must be mapped to a point in G.1

However, Point can be multiplied. Why can't I simply do $$mG$$?

Example:

• $$pk = [sk]G_1$$
• $$m = hash(message)$$
• Signing: $$s = m [sk] G_2$$
• Verify: $$e(G_1, s) == e(pk, m G_2)$$

Oh, if it is possible to convert G1 to G2, it seems possible to multiply the public key by m. then,

• Signing: $$k = random, r = kG_2, s = [sk] (m + k)$$
• Verify: $$e(G_1, sG_2) == e(pk, r + mG_2)$$
• ($$[sk](m + k) == [sk](k + m)$$)
Score:2

Because under such a scheme possession of a single valid signature allows an adversary to forge arbitrary messages.

Suppose I have a signature $$S=m[sk]G_2$$ for the message $$m$$ and wish to forge a signature for a message $$m'$$. Using the Euclidean algorithm I compute $$x=\frac{m'}m$$ modulo the order of $$G_2$$ and then create the new signature $$S'=xS=m'[sk]G_2$$. Verifier could confirm that $$e(G_1,S')=e(PK,mG_2)$$, but this signature was produced without knowledge of the private key.

Score:2

In order to sign message m, m must be mapped to a point in G

In BLS, we do (not that's all that difficult) - other EC based signature algorithms, such as ECDSA and EdDSA, have no such need.

To answer the specific BLS-type proposals you made:

Why can't I simply do $$mG$$?

Well, your first suggestion would allow anyone to perform a forgery; it had:

• Signing: $$s = m [sk] G_2$$

Then, suppose someone had a valid signature $$s$$ for a known message $$m$$. Then, they could compute $$m^{-1} s = [sk] G_2$$; with that, they can sign any message they wanted.

Your second suggestion would also allow forgeries (given a valid signature of a known message); if the attacker had a valid $$(r, s)$$ pair for a message $$m$$, then to sign a message $$m'$$, he can construct $$r' = (r + (m - m')G_2, s' = s$$; on the rhs, the validator would compute $$e(pk, r' + m'G_2) = e(pk, r + mG_2)$$, which would match the lhs (because the attacker used the same $$s$$).

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