Questions tagged [prime-field]

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14 questions
46 views

Elliptic curve over prime field with high order roots of unity

Suppose I have an elliptic curve defined over a prime field $\operatorname{GF}(p)$ where $p$ is a large prime (e.g. 256-bit). Suppose also that $p = kn +1$, where $n$ is a relatively large power of $2$...
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Why is the strength of an Elliptic Curve Cryptography (ECC) half the size of the prime field size?

I've looked around and couldn't find a direct answer. As a general rule, I've read from various sources (here here, and here) that the strength of an elliptical curve key is half of the size of the ...
101 views

Why does libSTARK use binary fields as opposed to prime fields for zk-SNARKs?

zk-STARKs make use of FRI for low degree testing of polynomials. The zk-STARKs paper states on page 11: we stress that ZK-STARK could also operate over prime fields but we have not realized this ...
58 views

Randomizing Prime Field Elements

I need my code to generate random elements from $GF(p)$ ($F_p$ or $Z_p$, if you will). The Crypto API I have available provides one with random bit strings. To tailor that to my needs, I can think of ...
172 views

I'm trying to understand the implementation of the following function: Please note questions in comments. ...
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Secret sharing over $Z_n*$

Shamir's secret sharing protocol generates a polynomial over a prime field $F_p$. Is it possible to generates it over $Z_n^*$ as well? The reason is that I want to combine a secret shared value in ...
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Are there any (asymmetric) cryptographic primitives not relying on arithmetic over prime fields and/or finite fields?

Trying to figure out if any (asymmetric) cryptographic primitives exists, which do not rely on arithmetic over a prime field and/or arithmetic over a finite field, some people might get lost in ...
197 views

Does this computation lead to solving DL?

Imagine we have $g^x$, $a$ and $g^b \in \mathbb{Z}_p$. Is it realistic to compute $g^{(a+x)b}$ without knowing $x$ and $b$? Or is it equivalent to solving the Discrete Logarithm problem?
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Order of the curve and generator

Does the order of the curve and the order of generator should be coprime for an elliptic curve defined over a prime field?
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Is the prime P is fixed for an elliptic curve defined over a particular prime field F_p?

I have seen the NIST, SEC and Brainpool standards. They have used same prime for a particular bit curve (128,192,256,521). Is the prime value fixed for a particular security (field size)?