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Jul
24
comment Group Theory: group under the composition multiplication modulo $p$
In addition, this can be easily generalised to compute inverses in any finite field.
Jul
24
comment Explaining elementary arithmetic in terms of group theory
For example $(\mathbf{R},\times)$ is monoid, but not a group (as is any ring under its multiplicative law).
Jul
24
comment Explaining elementary arithmetic in terms of group theory
You may be looking for monoids, which are essentially to groups what rings are to fields: the requirement for inverses is dropped.
Jul
24
comment Is 1 always an element in multiplicative group?
For reference, this horrible notation $\mathbb{G}_T$ is something many cryptographers have adopted in the context of pairing-based cryptography. In typical scenarios, it is a subgroup of the multiplicative group of a finite field.
Jul
24
comment One divided by infinity is not zero?
Just because something can happen doesn't mean it has non-zero probability. (It's also impossible to generate a uniform probability distribution over the reals, by the way.)
Jul
24
comment Is the subset of squares of a group a subgroup?
The free group on $\{g,h\}$ will do.
Jul
23
answered Must an algorithm that decides a problem in NP also produce a solution?
Jul
23
comment Can I choose the larger of 2 arbitrary numbers with P>0.5 after only seeing 1?
Well, yes and no. The basic idea is sound, as demonstrated by the discrete case where you do get an advantage. It's just that going to the reals introduces complications (which I am not qualified to discuss in depth). It is possible that a rigorous argument will yield a "nice" generalisation of the discrete case.
Jul
23
comment Can I choose the larger of 2 arbitrary numbers with P>0.5 after only seeing 1?
Basically, just because something can happen doesn't mean it has non-zero probability. I think the original problem is not very interesting, it is more useful to consider the "bonus question" where the numbers are chosen from $\{0,1,\dots,10\}$.
Jul
23
comment Can I choose the larger of 2 arbitrary numbers with P>0.5 after only seeing 1?
Nobody said those numbers are random, they are arbitrarily chosen.
Jul
23
answered Extension of field automorphism to automorphism of algebraic closure
Jul
23
comment Extension of field automorphism to automorphism of algebraic closure
Since in general the algebraic closure has infinite degree, you need Zorn's lemma to extend an isomorphism to it.
Jul
23
comment What is $0\div0\cdot0$?
+1 for the last two sentences. Just don't do it, it's not rocket science.
Jul
22
comment Which background is more suitable to study “Cryptography”
Cryptography is vast, it would be helpful to describe in more detail what kind of material you are studying (or, even better, what the assigned textbook is, if any). For example, is it a course in the foundations (one-way functions, pseudorandomness, etc.), a more standard "intro to cryptography course" where you study classical ciphers, Diffie-Hellman, RSA, or something else?
Jul
22
revised If you remove an element from an infinite set, does the set remain infinite?
added 387 characters in body
Jul
22
comment If you remove an element from an infinite set, does the set remain infinite?
Yes, true, I was misled by the text of the question which mentions only the set of natural numbers...
Jul
22
answered If you remove an element from an infinite set, does the set remain infinite?
Jul
22
comment General Topology on Complex field?
General topology by definition deals with abstract topological spaces, it is equally valid on the field of complex numbers (with a suitable topology). For that matter, even the special case of metric spaces can apply to the complex field (again, with a suitable metric).
Jul
22
comment Is there a powerset with the cardinality of the natural numbers?
Such a set would need to be infinite (since the power set of a finite set is finite), and I believe $\aleph_0$ is the smallest infinity, so probably not.
Jul
22
revised Pairwise independence of binary vectors
edited body