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Element Size When choosing elliptic curve parameters, there is a lot of freedom. For the size of elements, the two parameters worth noting are the prime, $p$, and the embedding degree, $k$. If $\mathbb{G}_1$ is an elliptic curve over $F_p$,1 then $\mathbb{G}_2$ is an elliptic curve over $F_{p^k}$, and $\mathbb{G}_T$ is a subgroup of $F_{p^k}$. So elements of ...
The proposed digital signature scheme is not secure! More precisely, it is not existentially unforgeable under an adaptive chosen-message attack. Let's consider the following efficient adversary $\mathcal{A}$: it queries the $\mathsf{Sign}_{sk}(\cdot)$ oracle for the digital signatures on $m_1,m_2$, where $m_2:=m_1+1$. The received signatures are $\sigma_1=... 4 For the purposes of elliptic curves and pairings with affine coordinates, functions are rational functions (ratios of two polynomials) in the two variables$X$and$Y$with coefficients in compatible fields. Curves are the set of points where a particular function is zero. Lines are curves where the underlying function is a polynomial of total degree 1. A ... 3 If I understand your quantifies (for any given irreducible$f(x)$, there does not exist such an algorithm), then it’s a stronger assumption and one that is unlikely to be true as$n$grows. First, note that if we write$x_i$for$g^{s_i}$then the degree (at most)$n$polynomial$\sum c_is^i\$ gives $$g^{\sum c_is^i}=\prod x_i^{c_i}$$ as easily calculable. ...