Questions tagged [finite-field]

A finite field is a mathematical construct based on a set of axioms which are held to be true. A number of interesting and useful properties arise from finite fields that makes them particularly suitable for use in cryptography, notably in block ciphers. Questions concerning finite fields should use this tag. Your question may concern finite fields if you are asking about AES, block ciphers or modular arithmetic.

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1answer
602 views

Sextic twist optimization of BN pairing - cubic root extraction required?

I found the following paper really interesting: http://www.researchgate.net/publication/220378229_A_family_of_implementation-friendly_BN_elliptic_curves/file/79e4150b3a773beecd.pdf It allows ...
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1answer
123 views

Does $i^n=j^n$ for $i, j \in GF(2^q)$ and $i \neq j$ for some $n<2^q-1$

Let $i, j \in GF(2^q)$ and $i \neq j$ and $i,j\neq0$. Is that possible that $i^n=j^n$ for some $n$ such that $0 < n < 2^q-1$? I am looking for a proof if the answer is no, or for a method to ...
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1answer
185 views

Standard basis representation of elements in binary field

In Remark B.1 from this paper it says: We assume canonical representation for binary fields $\mathbb{F}$, given by an irreducible polynomial and a primitive element $g \in \mathbb{F}$ for it (i.e., ...
3
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1answer
1k views

Why does FIPS 186-4 require specific sizes for keys?

In FIPS 186-4, page 32, about FFC crypto it is required that the length of $p$ will be exactly 1024 bit and the length of $q$ will be exactly 160 bit. Why is the requirement not stated in terms of ...
3
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1answer
83 views

Question about block erasure codes

I have a question about linear block erasure codes that are described in this paper. I briefly describe the idea behind the linear erasure codes and then I ask my question. Given a set $d=\langle x_i ...
2
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3answers
443 views

How can I get the binary form of AES's MDS matrix in MixColumns tranformation?

I need to write a procedure for calculating the MixColumns's operation result in the following form: $M*X^T,$ where $M$ is a 128x128 binary matrix, $X$ is a 128-bit vector (the state). My question ...
1
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1answer
244 views

Why only non-prime order fields have small subgroup attacks?

Why don't prime-order curves have small subgroup attacks? It seems that I can choose a Generator such that it has a small order, maybe 2 points, and so an attack could generate all of the points in ...
1
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1answer
547 views

Key size and finite fields in ECC (References)

So somehow I know that the key size in ECC is defined over the number of elements in a finite field or that it is almost equivalent to that (Correct me if I am wrong). However, other than on Wikipedia ...
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1answer
743 views

Finding $x$'s parity in the discrete log problem

Suppose $p$ is prime, and $g, a\in\mathbb F_p$ are given elements with $g$ a primitive root. The discrete log problem poses the task of finding an integer $x$ such that $g^x=a$. Show that even if $x$ ...
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1answer
91 views

Explanation of trace function $\operatorname{Tr}_m(x) = x^{2^{m}} \oplus x$

The following statement is from a paper (Partitions in the S-Box of Streebog and Kuznyechik) about S-Boxes: For all $ x \in \operatorname{GF}(2^{n})$, it holds that $x^{2^{n}} \oplus x = 0$. If $...
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0answers
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A related question to “relationship between generating elements given by cycles in Cayley graphs”

I am writing this question with reference to the post at Relationship between generating elements given by cycles in Cayley graph When considering the generating elements $g_qg_p$, does it have the ...
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42 views

Cryptographic properties of field multiplication (Continued)

This question follows-up from this question/comment. Suppose, you are given $X \odot Y$, where $X~(\neq 0) \in \operatorname{GF}(2^{128})$ is random, $Y~(\neq 0) \in \operatorname{GF}(2^{128})$ is ...

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