Can someone explain how the Gallant-Lambert-Vanstone method works (or which literature explains it)?  It is also unclear to me how the Frobenius endomorphism can be used in some cases for a speedup.  Also: how does it make sure that an attack remains infeasible (by using this method)?  (I am especially interested because of the sec256k1 curve which uses the method)
121809
Is there the way to generate new Koblitz curves, over $\mathbb F_{2^n}$ and $\mathbb F_p$?  The  says:     The recommended parameters associated with a Koblitz curve were chosen by repeatedly selecting parameters admitting an efficiently computable endomorphism until a prime order curve was found.   What does "efficiently computable endomorphism" mean? Maybe it means that trace of the Frobenius map should be equal 1? I am confused about all the Frobenius maps, endomorphisms, traces, etc.  So, is there the way to generate Koblitz curves, or find efficiently computable endomorphisms?
60010
Can someone explain what exactly the Jacobian matrix is (specifically in its application to dynamical systems) and maybe give an example of how to compute it? It really confuses me...and I haven't been able to find any good resources online.
86422
Hi my course specifically talks about :  Cartesian and Polar Coordinates in 3 Dim, second Degree eqns in 3 vars, reduction to canonical forms, straight lines, shortest distance between 2 skew lines, Plane, sphere, cone, cylinder, paraboloid, ellipsoid,hyperboloid of one and two sheets and their properties.  This is merely indicative, and I would really like if the book talks about these (atleast) and some more. Would appreciate if it contained solved examples and exercise problems as well  Help Much Appreciated, Soham
1362
I know that the Lagrange multiplier method helps us evaluate critical points of $f$ on the closed boundary of the restriction.   In other words we solve:$$\nabla f=\lambda \nabla g$$  But why does this method actually work? Can someone please give me an explanation. Thank you!
88195
I am actually trying to study homomorphic encryption (on lattices) but I'm facing a problem. Every paper that I have read so far talk about writing the function to evaluate on ciphertexts as a circuit, either boolean or arithmetic according to our scheme if I understood this part well. My problem is that I couldn't find explicit documentation on how to turn a function to a circuit.   Especially, in circuits, we can't have loops or conditions and I would have liked to apply my FHE scheme to the weights of a neural network, but I'm kinda stuck on how to represent the activation function (I think I would use the sigmoid, that can be represented with its Taylor expansion) and simple gradient descent.  Does anyone have a clue or a good paper to help me understand how to write such functions to circuits?
60921
I'm curious to whether you guys have any tips on book concerning classical euclidean geometry. I'd like somewhat of an advanced treatment, around the same level as Coxeter's "Geometry revisited". I'd however want something more comprehensive, therefore I'm asking you for tips!  Hopeful for answers ^^
86572
How should I approach obvious proofs like these    I have been trying but couldn't find an elegant way to work these. Any help is highly appreciated ! Especially looking for a nice proof/hint for first problem
36077
Prove that for any piecewise smooth curve it is possible to find the parametrisation $\phi$ that is consistent with its length, ie.  length of a curve segment between $\phi(a)$ and $\phi(b)$ is equal to $|b-a|$.  It seems intuitive, but I don't have any idea how can I show this in the formal way.
6260
I'm trying to work through the exercises in Otto Forster's book on Riemann Surfaces. While most of them seemed not that hard, this one gives me a headache:  Let $X=\mathbb{C}\setminus\{\pm1\}$  and $Y = \mathbb{C}\setminus\{\frac{\pi}{2}+k\pi~|~k\in\mathbb{Z}\}$. Then $\sin: Y\rightarrow X$ is a covering map. This is clear.  Now consider the following curves:  $$ u: [0,1] \rightarrow X, u(t) = 1-e^{2\pi i t}$$  and  $$ v: [0,1] \rightarrow X, v(t) = -1+e^{2\pi i t} = -u(t)$$  Let $w_1: [0,1]\rightarrow Y$ be the lifting of $u\cdot v$ and $w_2$ be the lifting of $v\cdot u$ with $w_1(0)=w_2(0)=0$.  Show that $w_1(1) = 2\pi$ while $w_2(1) = -2\pi$.  As of now, I don't have many ideas how to solve this. Of course, the inverse sine is a multi-valued function so the two liftings should use different branches. But I don't see how exactly they come into play.  I'm also not sure whether the solution can be done by calculations alone or if there is some more general principle underlying. Any suggestions would be welcome.
70415
I know the basic differences between Symmetric vs. Asymmetric cryptography, but I'd love to know more details:    Exactly why is the asymmetric approach slower than the symmetric?  Why does it make use of longer keys?  Why does it require more processing power, etc, etc? In what ways does it trump the symmetric approach?   Thanks, in anticipation, for your answers - I'm writing a BSC project on this issue.
11804
The problem statement is:  Part(a)  Assume P(z) is a non-constant polynomial with all of its roots in some half plane H.  Show that the derivative P'(z) must also have all of its roots in H.  Part (b)  Show that the roots of P' always lie within the closed convex hull of the set of roots of P.  The hint given is:   After obvious reductions of the problem, look at the behavior of $\frac{P'}{P}$ on the complementary half-plane.  My thoughts:   So I think that, whatever half-plane H that we have to work with, we can always just look at $P(e^{i\theta}z)$, which has its zeros in the UHP, for a suitable rotation by angle theta.    Now we can just show that $P'(e^{i\theta}z)e^{i\theta}$ also has its roots in the UHP -- and then undoing the rotations shows that both P and P' had its roots in the same half-plane H.  Using the hint regarding looking at the complementary half-plane, we then want to show that P' does not vanish in the LHP (we already know P doesn't vanish in the LHP, by assumption.)    Again following the hint we look at P'/P in the LHP and show that it does not vanish.  I am stuck at this point.  How can I proceed?  And regarding part(b) what is a "convex hull"?  I am guessing that the answer to part (b) is immediate, after figuring out part(a).  Any ideas are welcome.  Thanks,
97016
It seems like there is no trivial isomorphism between them and both are abelian and non-cyclic, so, it makes it even harder to conclude anything. I need some hints.  If I had to make a guess, I'd probably say there is no isomorphism between them.
100864
I'm supposed to show the complement property of DES ($c=DES(p,k) =&gt; \bar c = DES(\bar p, \bar k)$).  My idea was to just start the algorithm with $\bar p$ and see what happens (ignoring the initial permutation for now). I looked at it step by step and got the the point where $R1 = \bar L1 \oplus f(\bar k1, \bar R0)$. So I figured I had to show that $f(\bar k1, \bar R0) = \bar f(k1,R0)$. Is that even right so far? Because I got stuck at the point where the S-Boxes are used. Why would $S(\bar x) = \bar S(x)$?  Somewhere else I read something about a formal prove using induction, but I don't know how I would do that either...
8142
I'm looking for an understandable proof of this theorem, and also a complex one involving beautiful math techniques such as analytic number theory, or something else. I hope you can help me on that. Thank you very much
38990
That is,     why is $\varphi(AB)=\varphi(A)\varphi(B)$, if $A$ and $B$ are two coprime positive integers?   It's not just a technical trouble—I can't see why this should be, intuitively: I bellyfeel that its multiplicativity should be an approximation at most. And why the minimum value for $\varphi (n)$ should be $ \frac{4n}{15} $ completely passes me by.
89159
What would the steps be for someone new trying to learn how to solve this? I feel like I understand the concepts and what they mean but not how to prove it. Thanks
49366
Lately, I am doing an investigation on Stirling's formula and its applications. So I thought I could use it to prove that the mode of the Poisson model is approximately equal to the mean. Of course, you do that by considering the curve that is formed by connecting the points of the probabilities of occurrence and the different values of the discrete random variable. Then you differentiate the p.d.f. where for $x!$ you use Stirling's formula $x!\approx \sqrt{2\pi x}~x^xe^{-x}$. The result is $\lnλ-1/(2x)-\ln x$ whose roots cannot be found analytically, but by iterative methods we find that as λ is larger and larger, the mode~mean.   Problem is, I found the following paper online, which seems to be the solution from a Harvard's undergraduate problem set.     It reads "You can also show this by taking the derivative of eq. (2), with Stirling’s expression in place of the $x!$. Furthermore, you can show that $x = a[=λ ~\text{in my case}~]-1/2$ leads to a maximum $P(x)$ value of $P_\max\approx1/\sqrt{2\pi a}$."   Does this puzzle you as much as it puzzles me? My main concern is over the "=" sign: how does this hold? The derivative=0 equation cannot have such an exact solution. Furthermore at such x, how does $P(X=a-1/2)$ give $1/\sqrt{2\pi a}$?  Am I (and my professor) missing something rather obvious or is the solution wrong?  Discuss!  PS: This sort of question might have been asked before, but still, I am really curious that somebody reads the paper in the link above, so that I can figure out what's going on.