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1

Here's a justification for the estimate in CodesinChaos' comment. Helmut Hasse proved in his 1936 series of papers "Zur Theorie der abstrakten elliptischen Funktionenkörper" that any elliptic curve $E$ over a finite field $\mathbb F_q$ satisfies the inequality $$ \lvert q+1-\#E(\mathbb F_q)\rvert\leq2\sqrt q \text, $$ that is, the number of points is ...


3

You can't encrypt a message with ECDH alone, because all it gives you is a shared secret that you can't really control. Rather, you use that secret in a symmetric scheme like AES (generally after passing it through a KBKDF to convert from an ECDH result to a proper-length and less-structured symmetric key, which you then use as the key for symmetric crypto). ...


2

It sounds like you are thinking of performing static-static Diffie-Hellman. If that is performed naively then it will indeed derive the same secret time and time again. At least one of the key pairs needs to be non-static or ephemeral, or an additional variable (nonce) should be introduced. For instance in NIST SP 800-56A there is section 6.2.1: "Initiator ...


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The elliptic curve discrete logarithm—like integer factorization and the classic finite field discrete logarithm—is an instance of the abelian hidden subgroup problem. Any abelian (commutative) instance of the hidden subgroup problem can be efficiently solved in quantum computers with (variants of) the Shor algorithm; therefore all of the above problems ...


2

Curve secp256r1 is not a type of curve; it is a curve, and is standardized under that name by SECG, under the name P-256 by NIST, and under the name prime256v1 by ANSI. It also happens to be the by far the most common elliptic curve used in cryptography. The field size, curve equation, and generator point are all part of the curve spec; the point of having a ...


0

If $G$ is the distinguished point on your curve, you can perform pre-computations to speed up multiplication of $G$ by a scalar. For instance, if you need a transient key pair $\{K_{PRI},K_{PUB}\}$, you can use these pre-computed points to compute $K_{PUB} = [K_{PRI}]G$ quickly. But if you need to compute a shared secret using another public key $K'_{PUB}$, ...


3

Yes, you are correct. There are various methods for scalar multiplication on elliptic curves. Some of them are optimised for fixed base-point scalar multiplication, i.e., where you a-priori know that you will mostly/exclusively perform scalar multiplications with respect to a fixed base point on the curve. Thus, one can make (extensive) pre-computations ...


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Based on a quick Google search, I assume you're reading the paper "Modified Koblitz Encoding Method for ECC" by Kodali and Sarma (Int.J.Rec.Tr.Eng.Tech., vol. 8, no. 1, Jan 2013). The fact that the paper doesn't actually cite any source for the encoding scheme, or even cite anything by Koblitz, makes me a bit skeptical of the paper's quality. For that ...



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