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@ -25,7 +25,7 @@ |
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\title{Notes on BLS Signatures} |
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\title{Notes on BLS Signatures} |
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\author{arnaucube} |
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\author{arnaucube} |
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\date{} |
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\date{July 2022} |
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\begin{document} |
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\begin{document} |
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@ -70,13 +70,13 @@ Identical to verification of a normal signature as long as we use the same corre |
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$$e(g_1, \sigma_{aggr})==e(pk_{aggr}, H(m))$$ |
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$$e(g_1, \sigma_{aggr})==e(pk_{aggr}, H(m))$$ |
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Unfold: |
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Unfold: |
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$$\fbox{e(pk_{aggr}, H(m))}= e(pk_1 + pk_2 + \ldots + pk_n, H(m)) =$$ |
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$$e(pk_{aggr}, H(m))= e(pk_1 + pk_2 + \ldots + pk_n, H(m)) =$$ |
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$$=e([sk_1] \cdot g_1 + [sk_2] \cdot g_1 + \ldots + [sk_n] \cdot g_1, H(m))=$$ |
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$$=e([sk_1] \cdot g_1 + [sk_2] \cdot g_1 + \ldots + [sk_n] \cdot g_1, H(m))=$$ |
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$$=e([sk_1 + sk_2 + \ldots + sk_n] \cdot g_1, H(m))=$$ |
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$$=e([sk_1 + sk_2 + \ldots + sk_n] \cdot g_1, H(m))=$$ |
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$$=[sk_1 + sk_2 + \ldots + sk_n]~\cdot~e(g_1, H(m))=$$ |
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$$=[sk_1 + sk_2 + \ldots + sk_n]~\cdot~e(g_1, H(m))=$$ |
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$$=e(g_1, [sk_1 + sk_2 + \ldots + sk_n] \cdot H(m))=$$ |
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$$=e(g_1, [sk_1 + sk_2 + \ldots + sk_n] \cdot H(m))=$$ |
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$$=e(g_1, [sk_1] \cdot H(m) + [sk_2] \cdot H(m) + \ldots + [sk_n] \cdot H(m))=$$ |
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$$=e(g_1, [sk_1] \cdot H(m) + [sk_2] \cdot H(m) + \ldots + [sk_n] \cdot H(m))=$$ |
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$$=e(g_1, \sigma_1 + \sigma_2 + \ldots + \sigma_n)= \fbox{e(g_1, \sigma_{aggr})}$$ |
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$$=e(g_1, \sigma_1 + \sigma_2 + \ldots + \sigma_n)= e(g_1, \sigma_{aggr})$$ |
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Note: in the current notes $pk \in G_1$ and $\sigma, H(m) \in G_2$, but we could use $\sigma, H(m) \in G_1$ and $pk \in G_2$. |
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Note: in the current notes $pk \in G_1$ and $\sigma, H(m) \in G_2$, but we could use $\sigma, H(m) \in G_1$ and $pk \in G_2$. |
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