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July 17

Fractions of metric units

Is there any application that uses thirds, sixths and twelfths of metric units? Most metric units are in base 10, and do not divide evenly by these numbers. Are there any metric units (other than units of time) that are not in base 10, and divide evenly by these numbers? --40bus (talk) 11:26, 17 July 2025 (UTC)

Units of measure, including the metric units, are not numbers and are not "in a base". Numbers come into play when reporting the value of a physical quantity. Such values are presented in two parts: a numerical value together with a unit of measurement. The numerical value is customarily presented using decimal numbers (and the SI system prescribes their use), but this is independent of the system of units used.  ​‑‑Lambiam 20:21, 17 July 2025 (UTC)

Since altbn254 is using an isomorphism, how to map points from $\mathbb {F} _{p}^{12}$ back to $\mathbb {F} _{p}^{2}$ using SageMath provided they have a suitable order ?

Given the curve’s specifications, and the following code :

#set up the F_p^2 = F_p[i] / (i^2 + 1) field
p=21888242871839275222246405745257275088696311157297823662689037894645226208583
F2.<z2> = GF(p^2,modulus=x^2+1)
ec = EllipticCurve([F2(0),F2(3/(9+z2))]) # set up the curve over F_p^2 = F_p[i] / (i^2 + 1)​
X,Y = ( # set up the generator
21280594949518992153305586783242820682644996932183186320680800072133486887432 * z2 +
150879136433974552800030963899771162647715069685890547489132178314736470662,
1081836006956609894549771334721413187913047383331561601606260283167615953295 * z2 +
11434086686358152335540554643130007307617078324975981257823476472104616196090
)
pt = ec([X,Y,1])
#declare the isomorphsim to map the point to F_p¹²
G2.<w2> = GF(p^2,modulus=x^2-18*x+82)
F12.<z12> = GF(p^12,modulus=x^12-18*x^6+82)
X1=(X[0]-X[1]*9) + (X[1])*w2
Y1=(Y[0]-Y[1]*9) + (Y[1])*w2
print(Y1^2-X1^3-3/w2)
X2=X1[0] + X1[1]*z12^6
Y2=Y1[0] + Y1[1]*z12^6
print(Y2^2-X2^3-3/z12^6)
X3 = X2*z12^2
Y3 = Y2*z12^3
ec12 = EllipticCurve([F12(0),F12(3)]) # declare the F_p¹² curve linked to the isomorphism.
P0=ec12(X3,Y3) # X3 and Y3 represent the converted generator coordinates. See https://github.com/ethereum/py_pairing/blob/dd5ede17919c2afd042e4fbb7fda06d250df09be/py_ecc/bn128/bn128_curve.py#L100

but this is for mapping a point to the $y=x^{3}+3$ curve defined over $\mathbb {F} _{p}^{12}$. How to perform the reverse by going from $\mathbb {F} _{p}^{12}$ to the curve defined as ${\frac {Y^{2}=X^{3}+3}{i+9}}$ over the field $\mathbb {F} _{p}^{2}={\frac {F_{p}[i]}{i^{2}+1}}$ 2A01:E0A:ACF:90B0:0:0:A03F:E788 (talk) 15:54, 17 July 2025 (UTC)

Where are you getting this 254 and 12th power from? Neither of those are described above; they all mention altbn128, not 254, and the base field and its quadratic extension, not a 12th power extension Sesquilinear (talk) 18:28, 17 July 2025 (UTC)

$\mathbb {F} _{p}^{12}$ is the field’s for pairings because of the embedding degree while the curve over $\mathbb {F} _{p}^{2}={\frac {F_{p}[i]}{i^{2}+1}}$ is used as notation for point compression. There’re different bn128 curves wearing the same name, hence the one used with cryptocurrencies is sometimes labelled as altbn254. Everything is written in the specifications here. According to the implementation at https://github.com/ethereum/py_pairing/blob/dd5ede17919c2afd042e4fbb7fda06d250df09be/py_ecc/bn128/bn128_curve.py#L100 the mapping although only implemented for going from $\mathbb {F} _{p}^{12}$ to the curve defined as $\mathbb {F} _{p}^{2}={\frac {F_{p}[i]}{i^{2}+1}}$ is an isomorphism which suggest that doing in the other direction is possible and my question is to do it in SageMath. 2A01:E0A:ACF:90B0:0:0:A03F:E788 (talk) 20:00, 17 July 2025 (UTC)

Okay, I think I see the confusion. The Ethereum spec there is quite terse and jumps through the intermediate steps assuming you already know them.

My exercise for you is: describe as many groups relevant to the pairing and its implementation as you can, and tell me how many you think are isomorphic to each other, as well as which ones will be strict subgroups of each other; this last part is the most important part and I don't think you'll be able to understand anything in elliptic curve cryptography without being able to tell me that last part. Sesquilinear (talk) 20:52, 17 July 2025 (UTC)

This is about maths : as shown by https://github.com/ethereum/py_pairing/blob/dd5ede17919c2afd042e4fbb7fda06d250df09be/py_ecc/bn128/bn128_curve.py#L100, you need to map points to embedding field in order to perform the pairings.

As for the isomorphism : as you said the specification don’t define it so I fail to understand it fully… I can’t explain it to you as a result. I’m asking the result in SageMath, but I would be unable to explain it mathematically too. The isomorphism is between ec12 = EllipticCurve([F12(0),F12(3)]) and EllipticCurve([F2(0),F2(3/(9+z2))]). This is shown by the code I wrote : but how it works ? I don’t understand fully (again). 2A01:E0A:ACF:90B0:0:0:A03F:E788 (talk) 22:04, 17 July 2025 (UTC)

Explain, in your own words, what the groups involved are. Sesquilinear (talk) 22:10, 17 July 2025 (UTC)

Actually, I'll be more explicit on one point. The G1 and G2 they imply an isomorphism between aren't the full groups of elements of the elliptic curves over the given fields. Indeed, with Hasse's theorem on elliptic curves you can show that's impossible for a curve over $F_{p}$ and a curve over $F_{p^{2}}$ when $p$ is big enough. Sesquilinear (talk) 23:13, 17 July 2025 (UTC)

The curve over contains $\mathbb {F} _{p}^{12}$ contains 21888242871839275222246405745257275088548364400416034343698204186575808495617 has a subgroup/suborder. By reverse mapping, I’m also talking about points being in such correct subgroup. For the other finite fields, the specification the order is 21888242871839275222246405745257275088548364400416034343698204186575808495617 directly. I’m not wanting to map between $\mathbb {G} _{1}$ and $\mathbb {G} _{2}$. 2A01:E0A:ACF:90B0:0:0:A03F:E788 (talk) 06:44, 18 July 2025 (UTC)

Have all of your elliptic curve related questions been an attempt to solve the discrete logarithm problem for this zero knowledge proof system? Sesquilinear (talk) 06:33, 18 July 2025 (UTC)

In the case of this question, this is for solving a class of diffie Hellman. Although looking unlikely : Satoh’s latest Miller’s inversion algorithm seems to work for weil_pairing inversion (resulting root of unity of finite field elements in my case do satisfy the required unprobable criteria for the algorithm). The algorithm outputs a $\mathbb {G} _{2}$ point having the 21888242871839275222246405745257275088548364400416034343698204186575808495617 order that need to be mapped back from the ec12 curve to the ec curve of the SageMath code. 2A01:E0A:ACF:90B0:0:0:A03F:E788 (talk) 07:01, 18 July 2025 (UTC)

July 18

Tetration

Are there any web programs that can calculate last 100 or more digits of tetrational numbers, even power towers of 100 or more numbers? --40bus (talk) 04:31, 18 July 2025 (UTC)

July 26

Divisibility of units

Are there any SI units that divide evenly by 3 (and any of its multiples)? The only such metric units that come to mind are second, whose superunits are 60 times the previous unit (and thus divide evenly by 3), but second itself and its subunits do not, as well as degree of angle, but it is not an SI unit, the corresponding SI unit being radian, which does not divide evenly by 3. And of units that do not divide evenly, are thirds of these units ever used? --40bus (talk) 21:34, 26 July 2025 (UTC)

According to our article, minutes are non-SI units, so even the example you gave isn't valid. A minute is said to be "accepted for use" though, and if you look at the corresponding table it says minutes, hours and days as time units, and degrees and arcminutes as angle units, have subdivisions divisible by 3. That's 24 in the case of a day, and 60 for everything else. Astronomers also use light-hours, light-minutes, and light-seconds on occasion, but I gather these aren't strictly SI units. A parsec is based on an arcsecond, but it doesn't scale; 60 parsecs do not make an parmin, and if anything it would be the other way around. (Astronomy seems to be the outlier here since it still uses a plethora of non-metric units while the rest of the scientific community has (mostly) converted to SI units.) Really, the whole point of SI units is to use precise decimals instead of less precise fractions, and to reduce the number of units so there is less need for awkward conversion factors. --RDBury (talk) 23:03, 26 July 2025 (UTC)

If the list of (prefixless) SI units in International System of Units is complete, the answer is no. The only even non-negative divisors of (prefixed) SI-units are the numbers that can be written in the form $2^{m}5^{n}.$  ​‑‑Lambiam 13:59, 27 July 2025 (UTC)

Are thirds of metric units ever used? More numbers are divisible by 3 than 5, and every SI unit is divisible by 5. How common is to write fractions with metric units? --40bus (talk) 22:34, 27 July 2025 (UTC)

Of course.:

• Therefore, there was a loss of one third of a liter of electrolyte-free water in each liter of urine.^[1]
• Now, in vessel A there is one third of a liter of water and in vessel D there are two and thirds of a liter of wine ready for mixing.^[2]
• You administer a third of a liter of normal saline and obtain a right-sided 12-lead ECG.^[3]

For a use of fractions, just look here. Editors of scientific journals will ask authors to use decimal notation that complies with the rules in section 5.4 of The International System of Units.

BTW, this is not a maths question. In future, please post any similar questions either in the language section (if it is about actual use observed in the wild) or the science section (if it is about clarifying the rules).

 ​‑‑Lambiam 03:38, 28 July 2025 (UTC)

I think that's a bit of a nitpick; in my view this query can be thought of as mathematical as much as it can linguistic or scientific, particularly as the querant is not a native English speaker, and different cultures can view concepts differently. And surely most regular respondents on the Ref desks look at all of them (as I do)?

If it bothers you that much, why don't you transfer it to one of those other desks? Which would you pick? {The poster formerly known as 87.81.230.195} 90.193.253.201 (talk) 12:47, 28 July 2025 (UTC)

July 27

What’s the difference between commutative local fields and number fields ?

Everything is in the question… As far I understand number fields are Global Fields and thus a local field can’t be at the same time a local field isn’t it ? In addition to this, what are the differences ? 2A01:E0A:ACF:90B0:0:0:A03F:E788 (talk) 16:07, 27 July 2025 (UTC)

A local field is locally compact and complete with respect to a valuation. In characteristic zero, the real and complex numbers are the only archimedean local fields, with the non-archimedean local fields are finite extensions of the p-adic fields. A number field is a finite algebraic extension of the rationals. (In prime characteristic, the global fields are the function fields, and the local fields are completions of function fields with respect to a valuation.) Local fields are always uncountable, and global fields countable. Tito Omburo (talk) 16:21, 27 July 2025 (UTC)

In my case I was thinking about the case of lifting elliptic curve to local field since it preserve the discrete logarithm. This was in this way I was thinking to local fields. 2A01:E0A:ACF:90B0:0:0:A03F:E788 (talk) 12:28, 28 July 2025 (UTC)

For a lift to a local field which is unramified, there are basically two approaches. One approach, where you have the polynomial modulus f defining the finite field, you can just use that polynomial to define the quotient field $\mathbb {Q} _{q}=\mathbb {Q} _{p}[x]/(f(x)).$. In sage, you can do this with the class Qq(q, modulus=f) class (or for the valuation ring Zq(q, modulus=f).) If, on the other hand, multiplication in the field $\mathbb {F} _{q}$ is given as a black box, you can use the ring of Witt vectors to define the valuation ring $\mathbb {Z} _{q}$. I don't think sage implements the ring of Witt vectors, but it does implement the required symmetric functions. In the unramified case, there is usually a canonical way to lift an elliptic curve, but when the curve is supersingular, there is no canonical lift. Finally, I'm not sure about the case of lifts of elliptic curves in the case of ramified extensions. Tito Omburo (talk) 13:41, 28 July 2025 (UTC)