All Questions
Tagged with cryptography elliptic-curves
11 questions with no upvoted or accepted answers
5
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Elliptic curve sequences needed for universal forgery
Elliptic Curve Digital Signature Algorithm (ECDSA) admits universal forgery (UF) if the Attacker can solve the equation
$$z=\frac{f_{k-1}(x,y)f_{k+1}(x,y)}{f_{k}(x,y)^2},$$
where $k$ is unknown, $f_{k}...
- 12.3k
3
votes
0
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215
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Why we are interested in p>3 Schoof's algorithm
In the Schoof's algorithm we are particularly interested in $char(K)>3$, where $K$ is the field. I know Schoof's algorithm is mostly used over large prime fields. Also, when we are transforming ...
- 149
2
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0
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110
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How to apply Pohlig Hellman using a very limited set of auxiliary inputs in that case?
So I was reading about Talotti, Paier, and Miculan - ECC’s Achilles’ Heel: Unveiling Weak Keys in Standardized Curves. The underlying idea is to lift the discrete logarithm problem to $\mathrm{prime}−...
- 87
2
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116
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Pollard's rho algorithm for ECDLP using supersingular elliptic curves over a field with characteristic equal to a Mersenne prime
I have been playing with Pollard's rho algorithm for elliptic curves over finite fields. I have noticed after some experimenting, that the algorithm almost always 'fails' for supersingular elliptic ...
- 121
2
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0
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309
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Merel's theorem on uniform bound for torsion of all elliptic curves
I am reading Silverman's book on Arithmetic of elliptic curves. There he mentions a theorem of Merel (Thm 7.5.1) which reads like this.
Thm: For every integer $d \geq 1$, there is a constant $N(d)$ ...
- 313
1
vote
0
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69
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Is it in theory possible to perform general Miller’s algorithm inversion as used with the optimal ate pairing with large trace in subexponential time?
Let’s I have the following :
2 curves $G_1$ defined on $F_p$ and $G_2$ being the $G_1$ curve’s twist defined on $F_p^2$ both having the same prime order ; a large trace ; and $F_p^{12}$ as their ...
- 87
1
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0
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133
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Select random point on elliptic curve
If I have an elliptic curve $E$ over some finite field $F_p$ what is a step by step algorithm to pick a random point that lays on this curve? There is definitely a naive approach to brute force all ...
- 11
1
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0
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109
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Why is the kernel cyclic if and only if the walk does not backtrack?
I'm reading Mathematics of Isogeny Based Cryptography by Luca De Feo. At some point (pg. 32), he says
"A walk of length $e_A$ in the $l_A$-isogeny graph corresponds to a kernel of size $l_A^{e_A};...
- 73
1
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0
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212
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Generate algorithmically an elliptic curve with its exact class group structure?
Is it possible to generate an elliptic curve $E$ (randomly), together with knowing its class group $\mathrm{Cl}(\mathcal{O})$ structure? where $\mathcal{O}$ is its endomorphism rings $\mathsf{End}(E)$ ...
- 181
0
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141
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State of the art on attempts to solve the elliptic curve discrete logarithm problem through transfering the problem to a weaker curve
Let an elliptic curve $E$, and 2 points on such curve $P$ and $O$ the methods I’m talking about consist in creating a weaker elliptic curve $F$ and mapping $P$ and $O$ to $F$ while successfully ...
- 87
0
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0
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57
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lower bound for solve ECDLP
Let $P$ and $Q$ are two points of NIST elliptic curve $E$ (defined over $F_{2^m}$ with prime $m$) and $k$ is a private key such that $k.P=Q$. Suppose $\ell$ be the number of bits in $k$, and let $k_i$ ...
