Quintic point group operation

Ordinary elliptic curves in the form
are used as resolvent equations to calculate the point group operation numerically over a quintic curve
by a limiting process with a point-averaging method, since the general quintic equation is insoluble in any closed form, by Abel’s famous proof. The iterative method of Doyle and McMullen is one such possibility, using a pair of bivariate “superresovlent” polynomials in the 9th and 12th degrees which have been derived for an iterative solution to the Briochi resolvent [1].
- ... We could also construct a single iteration that would find all five roots at once, but the formulas might be rather more complicated.
- (3) Remarkably, one can also derive the formulas for g and h by hand, without even knowing the basic invariants F12, H20 and T30 of the icosahedral group. ...
This is where much of the trouble with the monstrous moonshine “group theory” of modern abstract algebra lies. If it is so remarkable that these equations could have been derived by hand without it, then why was this not done so in the paper? It reads almost as if these algebraic methods are so abstract that they take place entirely “in one’s head” so to speak, or by “heart” as the paper suggests where the polynomials to be iterated are introduced without derivation or other explanation, and only “pretty pictures” of icosahedral supersymmetry to illustrate in a general sense of popular interest, which is exactly the same case with the so-called Calabi–Yau manifolds and ten-dimensional spacetime superstring “theories of everything” in physics which attempt to unify all fundamental forces of nature in one grand scheme.
The usefulness of these methods for performing cryptographic point group operations on quintic curves would depend on their rapidity of convergence and on the ease of finding rational points of bounded height at the limits of convergence.
There are some indications of quintic curves actually being used for cryptographic applications, a book going for [2] and a website [3] on the subject and some papers [4]. Without further reference, the idea seems preposterous at first glance, given the known insolubility of quintic equations in closed form. And then again, why not? We don’t want our crypto too easy to solve. Yet nevertheless, we have a sneaking suspicion none of the references on the topic are doing the sort of heavy-duty industrial curve-fitting math which civil engineers [5] for instance would use to solve these types of equations.
General quintic polynomials may be expressed in Bring radicals [6] with the help of Tschirnhausen transformations [7]:
- The irreducible quintic can be solved in terms of Jacobi theta functions, as was first done by Hermite in 1858. Once we have the 5 solutions, we must invert the Tschirnhaus-Bring transformation to obtain the solutions to the original quintic. Thus we shall in general obtain 20 candidates for five solutions. There is no way of knowing which ones are correct without numerical testing.
This journal article has some helpful information on deriving the said transformations but the summary conclusions are not consistent. Bring radicals or roots of quintic equations in Bring–Jerrard normal form can indeed be expressd in terms of Jacobi theta functions or possibly hypergeometric functions, etc., but that is not really a way of “solving” them without involving transcendental functions which are vastly more complicated to compute than simply estimating and finding polynomial roots numerically, if that is indeed the goal, and not a more abstract symbolic representation of the solutions to the general quintic and their algebraic group structure, which we would require here in order to perform any cryptographic operations on quintic curves over finite fields. Simple linear and quadratic transformations are applied, but “no way of knowing” is a way of saying “just don’t care,” because for every quadratic transformation applied the cases may be examined in particular which root should be taken to reverse the transformation.

Fitting elliptic curves over the quintic
When the degree of a solvend equation is odd and at least five, the degree of its resolvent is the least integer no less than half of it, and the resolvent is not squared as it is in the case of equations of high even degree. Therefore the first few leading terms of a resultant equation of odd degree are the negatives of the leading terms of the solvend.
A resolvent elliptic curve fitted through any one or two or three or four points on a quintic curve will intersect the quintic curve at a fifth point, uniquely determined, up to specified multiplicity and sign of the y-axis.
If P, Q, R, S and T are the five points of intersection of the resolvent elliptic curve with the quintic curve, permitting multiplicity, let
- ,
where O is the additional “point at infinity” considered to lie on the curve and serve as an identity for its additive point group operation. We will call this the “quincunx equation” and use direct sum notation for the point “sums” and additive inverses computed.
Curve intersections of multiplicity two and three are commonly referred to “in the business” as “tangency” and “osculation” [8] as described in the quartic point group operation. On quintic curves, without any more explicit terminology than that already introduced by Leibnitz, we will make use of intersections of multiplicity four as well, where the first, second and third derivatives with respect to x, as well as the y-coördinate, would be required to match, if the equations were put in the form of derivatives rather than polynomials with multiple roots at those points.
Curve fittings through one point
Choose a point P and solve , , and by curve-fitting. Now , , , and .
Curve fittings through two points
Choosing points P and Q it is possible to solve for and by curve-fitting: and , if scalar division is permitted. Also and so and , etc.
Goals and objectives
The goal is to reach the point from the point P alone, and to reach the point from the points P and Q if possible, and if so, by the shortest possible path of computation, i.e., using the least number of resolvent elliptic curve operations.
Point averaging and limit
Point averaging is possible.
- .
A closed form expression also exists for
although, apparently not for the point group “sum” itself. The only remaining option then is to approximate
using only those rational numbers n for which a closed form expression exists for the rational scalar multiple by n of the point group “sum” we are calculating.
Insolubility in closed form radical expressions
It has been known for over 500 years that unlike quartic equations, general quintic equations cannot be solved in closed form radical expressions, and this is indeed what we find here for the point group operation which we have defined by “summing” five points that intersect an elliptic curve to an additive identity “point at infinity.”
This is the difficulty of solving the “quincunx equation” which determines the “group operation” among the rational points on a quintic curve. Numbers which are not congruent to 1 modulo 5 are not “constructible” in the resolvent system of the quintic. Non-constructible scalar multiples of a point under the point group operation are not computable or expressible in any closed form. The scalar multiples and 2, which are needed for the “inverse” of one point and for point-doubling or calculating a “sum” of two points, since averaging is permitted without a scalar multiple, are not constructible because
- ; and
- .
Finding roots of equations which are soluble
Place the resolvent elliptic curve equation over [9] the equation for the quintic curve and subtract term by term:
- .
The fundamental theorem of algebra assures us that the roots (or radicals) which we seek do exist, and may be found when the equation is in this form. If a quintic equation with rational coefficients has four rational roots, then its fifth root must exist and be rational as well. Any known rational roots may be factored out by long division into the polynomial with rational coefficients, and in any case the degree of the equation reduced accordingly, and solved in closed form.
Numerical methods
Numerical or what we call “floating point” methods are often used in practice to solve equations in the fifth or higher degree. The point averaging method, which is expressible in closed form, certainly allows the solution to be approximated to any desired degree of accuracy.
Exact solutions from floating-point computations
If the point group operation as defined does indeed map pairs of rational points on the curve to rational points on the same curve not exceeding a given denominator or height, then floating point approximations of sufficient precision can potentially be proven adequate to rigorously determine the exact numerators and denominators of those rational points.
If we had anything else to offer in this regard, we would have succeeded in doing exactly what the classical Italian mathematicians already proved impossible centuries ago.
R code for example plot
#! /usr/bin/R -f
H <- function(x){
sqrt(1/9*(x+3)*(x+1)*(x+1/2)*(x-3/2)*(x-2))
}
xvals <- seq(0,3.1,0.0001)
plot(x=c(rev(xvals),-xvals,-rev(xvals),xvals),
y=c(H(rev(xvals)),H(-xvals),-H(-rev(xvals)),-H(xvals)),
type="l", lwd=3, col="purple",
xlab="x", ylab="y", asp="1",
main=expression(9*y^2 == (x+3)*(x+1)*(x+1/2)*(x-3/2)*(x-2))
)
abline(0,0)
- ↑ Peter Doyle and Curt McMullen. “Solving the quintic equation.” Acta Math., 163 (1989), 151-180 https://people.math.harvard.edu/~ctm/papers/home/text/papers/icos/icos.pdf
- ↑ Cohen, H., Frey, G., Avanzi, R., Doche, C., & Lange, T. (Eds.). (2005). Handbook of Elliptic and Hyperelliptic Curve Cryptography (1st ed.). Chapman and Hall/CRC. https://www.routledge.com/Handbook-of-Elliptic-and-Hyperelliptic-Curve-Cryptography/Cohen-Frey-Avanzi-Doche-Lange-Nguyen-Vercauteren/p/book/9781584885184
- ↑ https://hyperelliptic.org/
- ↑ Jasper Scholten and Frederik Vercauteren. “An Introduction to Elliptic and Hyperelliptic Curve Cryptography and the NTRU Cryptosystem.” K.U. Leuven, Dept. Elektrotechniek-ESAT/COSIC https://www.cse.iitk.ac.in/users/nitin/courses/WS2010-ref4.pdf
- ↑ City Hall is undoubtedly cracking hard codes and ciphers, given the politicization and weaponization of urban police departments.
- ↑ Wikipedia: “Bring radical.” https://en.wikipedia.org/wiki/Bring_radical
- ↑ Victor S. Adamchik and David J. Jeffrey. “Polynomial Transformations of Tschirnhaus, Bring and Jerrard.” ACM SIGSAM Bulletin, Vol 37, No. 3, September 2003 https://www.uwo.ca/apmaths/faculty/jeffrey/pdfs/Adamchik.pdf
- ↑ Latin for “touching” and “kissing.”
- ↑ Goals and objectives again!