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3

I'm not quite sure what you mean by $p$ and I'm not sure what you meant for your code to print out. However, it looks like you are trying to find integer valued points on the elliptic curve $y^2=x^3+7$ by exhausting over $x$ values and I can't spot a bug other than the print statement. BUT Siegel showed that in general elliptic curves over the rational ...


0

This answer restricts to groups under the multiplication operation modulo a large prime $p$, because the question does (others groups are increasingly common in cryptography, including Elliptic Curve Groups). So we'll be working within the group $\mathbb Z_p^*$, a notation for the subset of the ring $\mathbb Z/p\mathbb Z$ formed by elements that have a ...


1

We will need an efficient primality test to produce $p$ and $q$. If you're happy with probable primes then the Miller-Rabin test will suffice for most practical purposes. Write IsPrime() for the test. Firstly choose $t$ according to the security requirements of your scheme. There's a chance of $2^{-t}$ that a deceiver can subvert the scheme at random, so a ...


1

Note that in any group, the exponent is computed modulo the order of the group. Thus $\alpha^{-a} = \alpha^{-a \bmod q} = \alpha^{q-a}$.


6

Index calculus is based on two simple ideas: Every integer can be written as a product of primes. A system of linear equations in a small number of variables can be solved with enough independent equations. Take for example the cyclic group $\mathbb{Z}/p$ with $p$ a prime and primitive root c. The elements $c^i$ (for $i=0,1,2,...,p-1$) are congruent modulo ...


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