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Apr
10
comment Hill cipher key space
Actually, they're the same: $p^{k^2}(1-p^{-1})\cdots(1-p^{-k}) = p^{k^2}\prod_{i=1}^k(1-p^{-i}) = \prod_{i=1}^k p^k(1-p^{-i}) = \prod_{i=1}^k (p^k - p^{k-i})$ (this common value is indeed the number of invertible $k\times k$ matrices over $\mathbb{Z}/p\mathbb{Z}$).
Feb
22
comment Standardized parameters for elliptic curve cryptography
One can infer that indeed, but on the other hand, the Digital Signature Standard says that NIST-approved curves are randomly generated (and even “provably” so!) in the sense that the curve coefficients are chosen using a hash function (even though the base fields are quite special). So it seems difficult to tell what exactly randomly generated means in this context.
Feb
22
comment Standardized parameters for elliptic curve cryptography
Very useful, thanks.
Jan
18
comment Pairing-friendly curves in small characteristic fields
I should perhaps note, however (and sorry for commenting twice), that in principle, it might be possible to construct pairing-friendly curves over extension fields of a form like $\mathbb{F}_{p^2}$ with the CM method (see e.g. the discussion in 4.1 of Barreto and Naehrig's paper). But $p$ still has to be large and you cannot fix it in advance, so it doesn't solve the problem in small characteristic.
Jan
18
comment Pairing-friendly curves in small characteristic fields
If you check the output of Algorithms 4.2 and 5.1 in Freeman's paper, you'll find that all of his curves (and hence their Jacobians) are defined over prime fields. The $J(\mathbb{F}_{q^k})$ in the abstract is about finding where the full r-torsion of the Jacobian in contained.