Talk:Finite field

message: I feel there should be a section about composite fields. A common example is mapping of GF(2^8) to GF(((2^2)^2)^2) for inversion (1/x) of a GF(2^8) number for AES encryption hardware where there may be 10 or more encoders/decoders on a single card, and the composite field greatly reduces gate count. Prior to carryless multiply instructions, composite fields allowed for faster computations for GF(2^32) and larger fields. However, with the carryless multiply pclmulqdq instruction on a X86 (since 2008), a GF(2^32) or GF(2^64) multiply can be done with 3 pclmulqdq and 1 or 2 xors. Rcgldr (talk) 22:58, 23 May 2020 (UTC)
replies: text: Can you point to any appropriate sources that discuss this?  (At the level of mathematics, the objects GF(2^8) and GF(((2^2)^2)^2) are not different, though obviously the second notation emphasizes a particular idea of building it by three quadratic extensions.  It seems plausible that there might be advantages of one representation versus another in CS, but we need reliable sources for that.) —JBL (talk)11:21, 24 May 2020 (UTC)
replies: text:  As mentioned above, there's been a lot of effort put into reducing gate count for AES S boxes, via GF(2^8) to GF(((2^2)^2)^2). Example references: GF(((2^2)^2)^2)(AES) S-box , A Very Compact S Box pdf. Other examples are about reducing large fields GF(2^n) for n >= 32: Efficient Software Implementations of Large Finite Fields pdf, which describes mapping GF(2^32) to GF((2^16^2), defines GF((2^16^2) and GF(2^16), but doesn't define GF(2^32) (it's probably jerasure standard x^32 + x^22 + x^2 + x + 1), and doesn't define the mapping matrix or it's inverse. I had to do an optimized brute force search to determine the missing data, to answer a question at a forum GF(2^32) to GF((2^16)^2) mapping. Rcgldr (talk) 12:50, 24 May 2020 (UTC)

text:  As for generating mapping matrices, let k = m · n, and GF(2^k) is to be mapped to GF((2^m)^n). Let e = the number of elements to map at a time, formatted as a matrix, elements[k][e]. Define the GF(2) mapping matrix as map[k][k]. Mapping e elements from GF(2^k) to GF((2^m)^n) is done via map[k][k] x elements[k][e].  The inverse of map is used to map back. The column indexes (right to left) of map[][] correspond to bit indexes of elements, map[][{...,4,3,2,1,0}] correspond to element bits {...,4,3,2,1,0}. Let α(x) = a primitive element of GF(2^k), and β(x) = a primitive element of GF((2^m)^n). The column values of map[][{...,4,3,2,1,0}] are pow(β(x),log_α(x)({...,16,8,4,2,1})). I'm looking for a reference for this, but generally the articles I find assume the reader is aware of this and just show an image of the k by k bit matrix. Rcgldr (talk) 12:50, 24 May 2020 (UTC)

text: For a given GF(2^k), GF((2^m)^n) has to be determined to follow two basic rules. Let a and b be elements of GF(2^k), then while operating in GF((2^m)^n) map(a + b) = map(a) + map(b) and map(a · b) = map(a) · map(b). In the case where α(x) = x + 0 and β(x) = x + 0, GF((2^m)^n) can be any of the m factors of GF(2^k) reinterpreted to have coefficients of GF(2^m), essentially GF((2^m)^k). For better optimization, α(x) can be any primitive element of GF(2^k), and an optimized brute force for the "best" combination of α(x) and GF((2^m)^k) is done. This is described in the paper I linked to before: A Very Compact S Box pdf. Rcgldr (talk) 12:50, 24 May 2020 (UTC)
replies: text: As far as I understand your post, you are concerned by computation is finite fields. For this, a representation of the field is needed, either as polynomials over GF(2) modulo an irreducible polynomial, or as a tower such representations. Your problems seems 1/ to determine the representation that leads to the most efficient computation 2/ (possibly) getting efficient algorithms for changing of representation. Both problems could be considered in the article (for the representation by a simple extension, the best choice of the defining polynomial is already sketched (with a reliable source) in section "non-prime field". For describing the known solutions of the other problems, we need sources. Those that you provide are not convenient for Wikipedia, as they are original research and consist mainly in proposed solutions, and they do not provide any synthesis of the knowledge in the subject. We need absolutely such a synthesis because, otherwise the presentation would be biased by giving emphasis to results for which we do not know whether they are generally accepted solutions. This is for this reason that original research and original syntheses are forbidden in Wikipedia (see WP:OR). As this subject seems to evolve rather quickly, this seems to early for presenting it in WP. D.Lazard (talk) 17:55, 24 May 2020 (UTC)
replies: text:  "Too early?" - there have been hardware implementations of composite (sub-field) mapping since the 1990's. I have a 1995 report from E J Weldon Jr for implementing hardware inversion (1/x) for GF(2^8) x^8+x^7+x^2+x+1 (hex 187, a primitive polynomial) using composite field (AKA sub-field mapping AKA "residue over a polynomial of degree 2 over GF(2^4)"). Usage of composite fields was fairly well known by the people involved in Reed Solomon codes or AES encryption by that time and probably dates back another 5 to 10 years prior to that report. In his 1995 report, he describes a mapping to GF((2^4)^2) x^2 + 4x + 2. Other implementations further reduced gate count by mapping to GF(((2^2)^2)2) and using a quadratic of the form x^2 + x + c. Some of the references in A Very Compact S Box pdf should qualify as reliable sources. Rcgldr (talk) 09:20, 25 May 2020 (UTC)
replies: text: Again, the references that you provide do not allow writing the section that you are asking for, without doing WP:original synthesis. By "too early", I meant that we must wait a published synthesis on the subject, which is apparently still lacking. D.Lazard (talk) 09:41, 25 May 2020 (UTC)

text: Here is a link to a portion of that report by professor E J Weldon Jr Finite Field Isomorphisms - Inversion pdf. I found more complete examples and explanations of using isomorphism for calculating a multiplicative inverse in GF(2^8). Starting at section E in aes sbox pdf and section 4.2 in  Compact Rijndael Hardware Architecture pdf. Perhaps this should be also be added to Finite field arithmetic#Multiplicative inverse Rcgldr (talk) 16:56, 25 May 2020 (UTC)

text: Composite or sub-mapped fields are isomorphic in addition and multiplication: map(a + b) = map(a) + map(b) and map(a · b) = map(a) · map(b). This is explained starting at section 7.8.2, page 95 of Standford - Introduction to finite fields pdf. For binary field mapping of the form GF(2^k) to GF((2^m)^n), generally GF(2^k) is fixed, the parameters for GF((2^m)^n) are pre-selected, and a search is done for a primitive element of GF(2^k), which is used to generate the mapping matrix, so that the resulting field GF((2^m)^n) is isomorphic to GF(2^k) in addition and multiplication. In the case of minimizing gate count for a given GF(2^8) such as AES x^8 + x^4 + x^3 + x + 1, nested mapping may be used, such as GF(2^8) to GF(((2^2)^2)^2), and a search for a primitive element of GF(2^k) as well as some of the constants for GF(((2^2)^2)^2) is done to result in a minimal gate count. This is the focus of the "very compact s box" articles. Rcgldr (talk) 03:55, 26 May 2020 (UTC)
thread_title:  Proposal to add a section for isomorphisms aka composite fields aka sub-field mapping 

message: I don't know all too much about fields, but upon browsing YouTube, I was reminded of a Numberphile video about finite fields (2021) (extra footage: here) which mentioned a concept of 'Parker' and 'Non-Parker' finite fields. The video cites the paper mentioned: Gaussian Integers, Rings, Finite Fields, and the Magic Square of Squares, and I was just wondering if this was significant enough to be mentioned in this article or not?
The video has almost 300,000 views, so I think it might be worth a mention maybe? However, I do recognise that a 'Parker Square' is a bit more of a meme than a recognised mathematical idea, but since a research paper has been released, maybe the article should mention the concept of 'Parker Finite Fields' and 'Non-Parker Finite Fields'?
Just a suggestion. :) – Pigeon <3 (talk) 13:54, 15 February 2022 (UTC)
replies: text: This is not really about finite fields, it is about Magic squares. So there is no reason for mentioning it here. Also this is published only in Youtube and ArXiV, which are not considered as WP:Reliable sources in Wikipedia. There is no WP:secondary sources for establishing the notability of the concept and related results. So, this is considered here as WP:original research, and Wikipedia policy is to not publishing original research. So, the answer is no for both questions. D.Lazard (talk) 14:49, 15 February 2022 (UTC)
replies: text:  I agree with D.Lazard. --JBL (talk) 15:01, 15 February 2022 (UTC)
replies: text: Thought as much, but thanks for letting me know! It was only a thought anyway :) – Pigeon <3 (talk) 19:59, 15 February 2022 (UTC)
thread_title:  Mention of Parker and Non-Parker finite fields? 

message: This is a great example of trying to convey a point with a minimum of jargon or presupposed knowledge, and then providing the technical term.  This makes it easier for those newer to a topic to grasp the concepts and learn the jargon without having to first look up many unfamiliar terms. Kudos to those who developed this entry! D1717 (talk) 11:15, 22 June 2023 (UTC)
thread_title:  great use of plain language 

message: Section  was illogically presented by starting from the algebraic closure for deducing properties already described in the preceding sections, instead of starting from these properties for deducing the properties of the algebraic closure. I have fixed this. 
By the way, I removed the paragraph about the absolute Galois group for the following reasons:
*It is completely unsourced, even when taken into account all linked articles.
*It is very confusing, as asserting implicitly that the Frobenius homorphism is an automorphism of the algebraic closure (this is probably true, but nowere explicited in Wikipedia)
*It is definitively out of scope of this article, since it involves many theories that the readers of this article (interested in fields) are not supposed to know (infinite Galois groups, profinite groups, Krull topology, topological groups, etc.)
The subject absolute Galois group of a finite field is interesting, but it is not adapted to the readers of this article. It must either be the object of a specific article, or be a detailed section in Absolute Galois group. The involved formalism makes it definitively unsuited for this article. D.Lazard (talk) 18:59, 2 December 2023 (UTC)
thread_title:  Algebraic closure