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2

FE2OSP and OS2FEP convert between finite field element (such as the X or Y coordinate of a point on an Elliptic Curve) and octet string (equivalently bytestring). They were originally defined by IEEE Std 1363-2000 and revised by IEEE P1363a-2004. Finite field are sets with $q$ elements noted $\operatorname{GF}(q)$ or $\mathbb F_q$, where $q$ is a large ...


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The impact on security from 1 and 2 can be considered as not significant. Say you have 256 Bit security, that implies $2^{256}$ different keys to chose from. If you know the key is odd, this is reduced to $2^{255}$ different keys. For 2 it is similar. Knowing how many ones and zeros there are can be considered as a significant risk. Let's assume you have a ...


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NSA removed EC-256 and SHA-256 from CNSA recently--should we be alarmed by this? No. There is one overwhelming reason why, as stated in the document: The cryptographic systems that NSA produces, certifies, and supports often have very long lifecycles. NSA has to produce requirements today for systems that will be used for many decades in the future, and ...


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Let $E$ be an elliptic curve defined over a finite field $\mathbb{F}_p$: $$E : y^2 = x^3 + Ax + B \text{ with } A, B \in \mathbb{F}_p$$ Let $P$ and $T$ be points in $E(\mathbb{F}_p)$. Find an integer $a$ so that $$ P =aG$$ This is the problem for elliptic curves. The Problem can be rewritten using a logarithm: $$ a = log_G (P)$$ You can compare that to the &...


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The discrete logarithm problem can be defined for any finite cyclic group, not just the multiplicative group modulo a prime number. The most famous instance is the problem that you describe, but it is not the only one. Groups can be written with multiplicative notation (as with the multiplicative group modulo $p$), so that the group operation is written as $*...


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There is no substitute for modular multiplication in the question's cryptosystems. Some languages like Python make that easy for educational purposes, only. In RSA and DSA, and to a lesser degree ECC crypto on secp256k1 or secp256r1, one needs to compute $b^e\bmod m$ for large $e$. The fastest algorithms (e.g. sliding window exponentiation) perform about $\...


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If the private key is available: see this answer. If it's not (which appear to be the hypothesis): forget about it using the computing power available. The best known attack (Pollard's rho) would require in the order of $2^{129}$ point additions. It the machine could perform these at a rate of $2^{36}$ per second (which is wildly optimistic), the division ...


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Before looking into PEM/ASN.1: EC private key is a prime field element, almost a bignumber. EC public key is a point on the curve, (most likely) encoded as two field elements, either prime-field (that is, modulo another prime) or binary extension field. So you look into curve description for background. Just pick two prime numbers for secp521r1 (from your ...


4

The private key data is encoded in ASN.1, so you need to decode that to get the various fields out. openssl asn1parse can do this, but by default it'll parse the "EC PARAMETERS" section of the file (since that comes before the "EC PRIVATE KEY" section), so you need to strip that off first. You can do that with sed, and then pipe the ...


3

First of all your description is not quite right. There are usually very few points on an elliptic curve which have integer coordinates. The points where the curve equation is satisfied modulo some number typically do not correspond to a point on the continuous curve with integer coordinates. On the wider point of curve wrapping, think of wrapping a piece of ...


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