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Clarify tpke default.
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derekpierre committed Jul 20, 2023
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3 changes: 1 addition & 2 deletions book/src/tpke.md
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Expand Up @@ -25,7 +25,7 @@ Check that \\(e(U, H_{\mathbb{G}_2} (U))= e(G, W)\\) for ciphertext validity.
1. Check ciphertext validity.
2. Let \\(s = e(U, Z)\\)

### Threshold Decryption (simple method)
### Threshold Decryption (simple method - default)

1. Check ciphertext validity.
2. Each decryption share is \\(C_i = e(U, Z_i)\\).
Expand Down Expand Up @@ -65,4 +65,3 @@ Suppose there is an adversary that wins the IND-CCA2 game. Then on input \\((G,
The shared secret \\(s\\) can be rekeyed with respect to the secret key \\(Z_1\\) to a new secret key \\(\hat{Z} = [\alpha] Z_1 + Z_2\\), as the new shared secret \\(\hat{s} = s^{\alpha} e(U, Z_2) = e(U, [\alpha] Z_2)e(U, Z_2) = e(U, [\alpha]Z_1 + Z_2)\\).

The shared secret \\(s\\) can be rekeyed with respect to the public key \\(Y_1\\) to a new public key \\(\hat{Y} = [\alpha] Y_1 + Y_2\\) as the new shared secret \\(\hat{s} = s^{\alpha} e([r] Y_2, H) = e([r\alpha] Y_1, H)e([r]Y_2, H) = e([r]([\alpha]Y_1 + Y_2), H)\\).

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