# 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

### Questions about the Curve25519-donna implementation

I'm trying to understand the implementation of the following function: Please note questions in comments. ...
55 views

### 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 ...
2k views

### 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?
444 views

### 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)?