Quintic point group operation - Revision history

Rational Point: TOE

2025-02-05T21:47:52Z

TOE

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	:<math>y^2=q_5x^5+q_4x^4+q_3x^3+q_2x^2+q_1x+q_0</math>		:<math>y^2=q_5x^5+q_4x^4+q_3x^3+q_2x^2+q_1x+q_0</math>

	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 which have been derived for an iterative solution to the Briochi resolvent <ref>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</ref>.		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 9<sup>th</sup> and 12<sup>th</sup> degrees which have been derived for an iterative solution to the Briochi resolvent <ref>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</ref>.

	:''... We could also construct a single iteration that would find all five roots at once, but the formulas might be rather more complicated.''		:''... 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 F''<sub>12</sub>, ''H''<sub>20</sub> ''and T''<sub>30</sub> ''of the icosahedral group. ...''		::(3) ''Remarkably, one can also derive the formulas for g and h by hand, without even knowing the basic invariants F''<sub>12</sub>, ''H''<sub>20</sub> ''and T''<sub>30</sub> ''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.		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.

Rational Point: Doyle and McMullen

2025-02-02T22:36:32Z

Doyle and McMullen

← Older revision		Revision as of 22:36, 2 February 2025
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	:<math>y^2=q_5x^5+q_4x^4+q_3x^3+q_2x^2+q_1x+q_0</math>		:<math>y^2=q_5x^5+q_4x^4+q_3x^3+q_2x^2+q_1x+q_0</math>

	by a limiting process with a point-averaging method, ~~because~~ the general quintic equation is insoluble in any closed form.		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 which have been derived for an iterative solution to the Briochi resolvent <ref>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</ref>.

			:''... 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 F''<sub>12</sub>, ''H''<sub>20</sub> ''and T''<sub>30</sub> ''of the icosahedral group. ...''

			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 <math>\frac14G</math> <ref>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</ref> and a website <ref>https://hyperelliptic.org/</ref> on the subject and some papers <ref>Jasper Scholten and Frederik Vercauteren. “An Introduction to Elliptic and Hyperelliptic Curve Cryptography and the NTRU Cryptosystem.” K.U. Leuven,		There are some indications of quintic curves actually being used for cryptographic applications, a book going for <math>\frac14G</math> <ref>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</ref> and a website <ref>https://hyperelliptic.org/</ref> on the subject and some papers <ref>Jasper Scholten and Frederik Vercauteren. “An Introduction to Elliptic and Hyperelliptic Curve Cryptography and the NTRU Cryptosystem.” K.U. Leuven,

Rational Point: Tschirnhausen

2025-02-02T17:23:08Z

Tschirnhausen

← Older revision		Revision as of 17:23, 2 February 2025
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	Dept. Elektrotechniek-ESAT/COSIC https://www.cse.iitk.ac.in/users/nitin/courses/WS2010-ref4.pdf</ref>. 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 [[bad curves and weak crypto\|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 <ref>City Hall is undoubtedly cracking hard codes and ciphers, given the politicization and weaponization of urban police departments.</ref> for instance would use to solve these types of equations.		Dept. Elektrotechniek-ESAT/COSIC https://www.cse.iitk.ac.in/users/nitin/courses/WS2010-ref4.pdf</ref>. 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 [[bad curves and weak crypto\|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 <ref>City Hall is undoubtedly cracking hard codes and ciphers, given the politicization and weaponization of urban police departments.</ref> for instance would use to solve these types of equations.

	General quintic polynomials may be expressed in '''Bring radicals''' <ref>Wikipedia: “Bring radical.” https://en.wikipedia.org/wiki/Bring_radical</ref> with the help of '''~~Tschirhausen~~ transformations''' <ref>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</ref>:		General quintic polynomials may be expressed in '''Bring radicals''' <ref>Wikipedia: “Bring radical.” https://en.wikipedia.org/wiki/Bring_radical</ref> with the help of '''Tschirnhausen transformations''' <ref>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</ref>:

	:''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.''		:''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.''

Rational Point at 17:18, 2 February 2025

2025-02-02T17:18:17Z

← Older revision		Revision as of 17:18, 2 February 2025
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	:''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.''		:''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 a 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.		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.

	[[Image:Gottfried-Wilhelm-Leibniz.jpg\|frame\|right\|Gottfried Wilhelm Leibniz, 1 July 1646 [O.S. 21 June] – 14 November 1716.]]		[[Image:Gottfried-Wilhelm-Leibniz.jpg\|frame\|right\|Gottfried Wilhelm Leibniz, 1 July 1646 [O.S. 21 June] – 14 November 1716.]]

Rational Point: journal article

2025-02-02T17:17:11Z

journal article

← Older revision		Revision as of 17:17, 2 February 2025
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	There are some indications of quintic curves actually being used for cryptographic applications, a book going for <math>\frac14G</math> <ref>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</ref> and a website <ref>https://hyperelliptic.org/</ref> on the subject and some papers <ref>Jasper Scholten and Frederik Vercauteren. “An Introduction to Elliptic and Hyperelliptic Curve Cryptography and the NTRU Cryptosystem.” K.U. Leuven,		There are some indications of quintic curves actually being used for cryptographic applications, a book going for <math>\frac14G</math> <ref>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</ref> and a website <ref>https://hyperelliptic.org/</ref> on the subject and some papers <ref>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</ref>. 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 [[bad curves and weak crypto\|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 <ref>City Hall is undoubtedly cracking hard codes and ciphers, given the politicization and weaponization of urban police departments.</ref> for instance would use to solve these types of equations.		Dept. Elektrotechniek-ESAT/COSIC https://www.cse.iitk.ac.in/users/nitin/courses/WS2010-ref4.pdf</ref>. 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 [[bad curves and weak crypto\|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 <ref>City Hall is undoubtedly cracking hard codes and ciphers, given the politicization and weaponization of urban police departments.</ref> for instance would use to solve these types of equations.

			General quintic polynomials may be expressed in '''Bring radicals''' <ref>Wikipedia: “Bring radical.” https://en.wikipedia.org/wiki/Bring_radical</ref> with the help of '''Tschirhausen transformations''' <ref>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</ref>:

			:''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 a 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.

	[[Image:Gottfried-Wilhelm-Leibniz.jpg\|frame\|right\|Gottfried Wilhelm Leibniz, 1 July 1646 [O.S. 21 June] – 14 November 1716.]]		[[Image:Gottfried-Wilhelm-Leibniz.jpg\|frame\|right\|Gottfried Wilhelm Leibniz, 1 July 1646 [O.S. 21 June] – 14 November 1716.]]
	== Fitting elliptic curves over the quintic ==		== Fitting elliptic curves over the quintic ==

Rational Point: /* Fitting elliptic curves over the quintic */ equations of high even degree

2025-01-27T23:09:07Z

Fitting elliptic curves over the quintic: equations of high even degree

← Older revision		Revision as of 23:09, 27 January 2025
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	== Fitting elliptic curves over the quintic ==		== 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.		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 [[quartic point group operation\|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.

	<math>\begin{array}{rcrrrrrrll}		<math>\begin{array}{rcrrrrrrll}

Rational Point: /* Fitting elliptic curves over the quintic */

2025-01-27T23:04:34Z

Fitting elliptic curves over the quintic

← Older revision		Revision as of 23:04, 27 January 2025
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	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.		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.


	~~<div style="clear:both">~~
	<math>\begin{array}{rcrrrrrrll}		<math>\begin{array}{rcrrrrrrll}
	y^2&=&&&r_3x^3&{}+r_2x^2&{}+r_1x&{}+r_0&&(\textrm{resolvent})\\		y^2&=&&&r_3x^3&{}+r_2x^2&{}+r_1x&{}+r_0&&(\textrm{resolvent})\\
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	&&-q_5x^5&{}-q_4x^4&{}+(r_3-q_3)x^3&{}+(r_2-q_2)x^2&{}(r_1-q_1)x+&{}r_0-q_0&=0&(\textrm{resultant})		&&-q_5x^5&{}-q_4x^4&{}+(r_3-q_3)x^3&{}+(r_2-q_2)x^2&{}(r_1-q_1)x+&{}r_0-q_0&=0&(\textrm{resultant})
	\end{array}</math>		\end{array}</math>
	~~</div>~~


	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.		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.

Rational Point: rearrange

2025-01-27T23:04:02Z

rearrange

← Older revision		Revision as of 23:04, 27 January 2025
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	by a limiting process with a point-averaging method, because the general quintic equation is insoluble in any closed form.		by a limiting process with a point-averaging method, because the general quintic equation is insoluble in any closed form.

			There are some indications of quintic curves actually being used for cryptographic applications, a book going for <math>\frac14G</math> <ref>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</ref> and a website <ref>https://hyperelliptic.org/</ref> on the subject and some papers <ref>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</ref>. 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 [[bad curves and weak crypto\|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 <ref>City Hall is undoubtedly cracking hard codes and ciphers, given the politicization and weaponization of urban police departments.</ref> for instance would use to solve these types of equations.
			[[Image:Gottfried-Wilhelm-Leibniz.jpg\|frame\|right\|Gottfried Wilhelm Leibniz, 1 July 1646 [O.S. 21 June] – 14 November 1716.]]
			== 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.		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.

	There are some indications of quintic curves actually being used for cryptographic applications, a book going for <math>\frac14G</math> <ref>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</ref> and a website <ref>https://hyperelliptic.org/</ref> on the subject and some papers <ref>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</ref>. 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 [[bad curves and weak crypto\|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 <ref>City Hall is undoubtedly cracking hard codes and ciphers, given the politicization and weaponization of urban police departments.</ref> for instance would use to solve these types of equations.

	<div style="clear:both">		<div style="clear:both">
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	</div>		</div>

	~~== Fitting elliptic curves over the quintic ==~~
	~~[[Image:Gottfried-Wilhelm-Leibniz.jpg\|frame\|right\|Gottfried Wilhelm Leibniz, 1 July 1646 [O.S. 21 June] – 14 November 1716.]]~~

	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.		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.

Rational Point: rearrange to clear float

2025-01-27T23:01:10Z

rearrange to clear float

← Older revision		Revision as of 23:01, 27 January 2025
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	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.		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.

			There are some indications of quintic curves actually being used for cryptographic applications, a book going for <math>\frac14G</math> <ref>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</ref> and a website <ref>https://hyperelliptic.org/</ref> on the subject and some papers <ref>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</ref>. 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 [[bad curves and weak crypto\|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 <ref>City Hall is undoubtedly cracking hard codes and ciphers, given the politicization and weaponization of urban police departments.</ref> for instance would use to solve these types of equations.

	<div style="clear:both">		<div style="clear:both">
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	\end{array}</math>		\end{array}</math>
	</div>		</div>

	There are some indications of quintic curves actually being used for cryptographic applications, a book going for <math>\frac14G</math> <ref>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</ref> and a website <ref>https://hyperelliptic.org/</ref> on the subject and some papers <ref>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</ref>. 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 [[bad curves and weak crypto\|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 <ref>City Hall is undoubtedly cracking hard codes and ciphers, given the politicization and weaponization of urban police departments.</ref> for instance would use to solve these types of equations.

	== Fitting elliptic curves over the quintic ==		== Fitting elliptic curves over the quintic ==

Rational Point: resultant

2025-01-27T23:00:21Z

resultant

← Older revision		Revision as of 23:00, 27 January 2025
Line 3:		Line 3:
	Ordinary '''elliptic curves''' in the form		Ordinary '''elliptic curves''' in the form

	:<math>y^2=ax^3+bx^2+cx+d</math>		:<math>y^2=r_3x^3+r_2x^2+r_1x+r_0</math>

	are used as '''resolvent equations''' to calculate the [[point group operation]] numerically over a '''quintic curve'''		are used as '''resolvent equations''' to calculate the [[point group operation]] numerically over a '''quintic curve'''

	:<math>y^2=px^5+qx^4+rx^3+sx^2+tx+u</math>		:<math>y^2=q_5x^5+q_4x^4+q_3x^3+q_2x^2+q_1x+q_0</math>

	by a limiting process with a point-averaging method, because the general quintic equation is insoluble in any closed form.		by a limiting process with a point-averaging method, because the general quintic equation is insoluble in any closed form.

			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.

			<div style="clear:both">
			<math>\begin{array}{rcrrrrrrll}
			y^2&=&&&r_3x^3&{}+r_2x^2&{}+r_1x&{}+r_0&&(\textrm{resolvent})\\
			y^2&=&q_5x^5&{}+q_4x^4&{}+q_3x^3&{}+q_2x^2&{}+q_1x&{}+q_0&&(\textrm{solvend})\\\hline
			&&-q_5x^5&{}-q_4x^4&{}+(r_3-q_3)x^3&{}+(r_2-q_2)x^2&{}(r_1-q_1)x+&{}r_0-q_0&=0&(\textrm{resultant})
			\end{array}</math>
			</div>

	There are some indications of quintic curves actually being used for cryptographic applications, a book going for <math>\frac14G</math> <ref>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</ref> and a website <ref>https://hyperelliptic.org/</ref> on the subject and some papers <ref>Jasper Scholten and Frederik Vercauteren. “An Introduction to Elliptic and Hyperelliptic Curve Cryptography and the NTRU Cryptosystem.” K.U. Leuven,		There are some indications of quintic curves actually being used for cryptographic applications, a book going for <math>\frac14G</math> <ref>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</ref> and a website <ref>https://hyperelliptic.org/</ref> on the subject and some papers <ref>Jasper Scholten and Frederik Vercauteren. “An Introduction to Elliptic and Hyperelliptic Curve Cryptography and the NTRU Cryptosystem.” K.U. Leuven,