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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 ...


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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 ...


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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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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 ...



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