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How many bits long should the prime modulus $M$ be in order to be secure? The modulus $M$ should be long enough to prevent discrete logarithms from being computable. As of 2015 this means 2048 bits length is fine, but for other (official) recommendations you should consult keylength.com Should the $M$ be secret? You can make $M$ secret but ...

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Yes. For fixed $a\in\mathbb Z/n\mathbb Z$ and $b\in(\mathbb Z/n\mathbb Z)^\ast$ — note that $b$ must be invertible modulo $n$, which need not necessarily be a prime (but if it is, invertibility is equivalent to $b\neq0$), the map $$f\colon\;\mathbb Z/n\mathbb Z\to\mathbb Z/n\mathbb Z,\; r \mapsto a+br$$ is a bijection, hence it preserves uniform ...

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In the document the P-256 parameters describe the curve P-256 on which you want to perform the operations. Traditionally a curve is represented in Weierstrass-form as the set of points for which $$y^2\equiv x^3 + ax+b \pmod p$$ holds. Where $p$ is the prime defining the field for the operation and $a$ and $b$ define the shape of the curve. A point on the ...

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I provide a specific example. Say $p=11.$ You want to find the points of the elliptic curve $y^2=x^3+1$ over the finite field $L=GF(p^2).$ Also set $K=GF(p).$ Then a defining polynomial of the quadratic extension $L/K,$ will be an irreducible quadratic polynomial over $K.$ Suffices to take $f(z)=z^2+7z+2.$ Now you have to take every element of $L$ as the ...

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1. The equation $-x^2+y^2=1-(121665/121666)x^2y^2$ defining the curve $E$ is quadratic in $x$, hence for any given $y\in\mathbb F_q$, there are at most two points on $E$ which have $y$ as their second coordinate. In this case, the two possible $x$-coordinates for a point on $E$ with $y$-coordinate $4/5\in\mathbb F_q$ are the solutions to the equation $$... 2 Note: For y^2 \equiv x^3 + ax + b, you want$$m\equiv(3x^2+a)*(2y)^{-1}$$which is a shortcut that only makes sense once you understand how elliptic curves work. "Why does it say y=1 and 4 for x=1?" Like you said, plug in & solve:$$y^2 \equiv 1^3 + 2(1) + 3 \equiv 6 \equiv 1\space(mod\space5) Ask yourself "Is there a square such that it is ...

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