Point group operation - Revision history

Rational Point: reword

2025-01-11T07:49:51Z

reword

← Older revision		Revision as of 07:49, 11 January 2025
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	:'' '''Caveat emptor:''' This article is incomplete, and more critical than others at this site. See any of various theses on the topic <ref>Noraldeen Alyounes. ''Elliptiska kurvor och kryptografi.'' Examensarbete i matematik, Uppsala Universitet, Februari 2020. https://uu.diva-portal.org/smash/get/diva2:1395121/FULLTEXT01.pdf</ref><ref>Gunendra Bikram Bidari. ''An algorithmic approach to elliptic curve cryptography.'' Thesis, ME in Computer Engineering, Kathmandu University, 2014. Lambert Academic Publishing, 2015.</ref>. This is proof that the unlearned and obstinate men, who have so far purported to implement “elliptic curve cryptography” and put their patented cryptographic schemes to widespread use under that name, had already been served with adequate instructions and examples, when they insisted on calling the law and getting all the details wrong in their official implementations and protocols.		:'' '''Caveat emptor:''' This article is incomplete, and more critical than others at this site. See any of various theses on the topic <ref>Noraldeen Alyounes. ''Elliptiska kurvor och kryptografi.'' Examensarbete i matematik, Uppsala Universitet, Februari 2020. https://uu.diva-portal.org/smash/get/diva2:1395121/FULLTEXT01.pdf</ref><ref>Gunendra Bikram Bidari. ''An algorithmic approach to elliptic curve cryptography.'' Thesis, ME in Computer Engineering, Kathmandu University, 2014. Lambert Academic Publishing, 2015.</ref>. This is proof that the unlearned and obstinate men, who have so far purported to implement “elliptic curve cryptography” and put their patented cryptographic schemes to widespread use under that name, had already been served with adequate instructions and examples, when they insisted on calling the law and getting all the details wrong in their official implementations and protocols.

	== ~~Euclidean plane geometry~~ ==		== Geometric derivation ==

	A '''point group operation''' on an elliptic curve is derived by geometric considerations on the curve in [[Weierstraß normal form]] <math>y^2 = x^3 + ax + b</math> over the real numbers.		A '''point group operation''' on an elliptic curve is derived by geometric considerations on the curve in [[Weierstraß normal form]] <math>y^2 = x^3 + ax + b</math> over the real numbers.
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	\end{array}</math>		\end{array}</math>

	However, we already know the slope ''m'' of the resolvent line equation, up to four choices of sign for ''y''<sub>''P''</sub> and ''y''<sub>''Q''</sub> respectively, and we have the ''x''-coördinate of the third point <math>R=(x,y)</math> in terms of it, from the '''quotient''' already calculated.		However, we already know the slope ''m'' of the resolvent line equation, up to four choices of sign for ''y''<sub>''P''</sub> and ''y''<sub>''Q''</sub> respectively, and we have the ''x''-coördinate of the third point <math>R=(x,y)</math> in terms of it, from the '''quotient''' already calculated. The special case of point-doubling (when ''Q''=''P'') is addressed by taking the derivative of the elliptic curve through that point to calculate the slope of the tangent line.

	:<math>m=\left\{\begin{array}{lr}		:<math>m=\left\{\begin{array}{lr}

Rational Point: special cases

2025-01-11T07:32:47Z

special cases

← Older revision		Revision as of 07:32, 11 January 2025
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	Because it is defined by an algebraic equation in the third degree, a straight line can intersect such a curve at at most three points in the Euclidean <math>(x,y)</math> plane.		Because it is defined by an algebraic equation in the third degree, a straight line can intersect such a curve at at most three points in the Euclidean <math>(x,y)</math> plane.

			=== Point addition ===

	The basic point group operation <math>\oplus</math> consists in finding the reflection across the ''x''-axis of the third point of intersection of the curve with a line through two given points on the curve.		The basic point group operation <math>\oplus</math> consists in finding the reflection across the ''x''-axis of the third point of intersection of the curve with a line through two given points on the curve.
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	An additional [[point at infinity]] is adjoined to the <math>(x,y)</math> plane, and considered to lie on the curve, although it is not given coördinates in the real numbers. This point is the [[group identity]]. The [[group inverse]] consists in taking the reflection of a point across the ''x''-axis.		An additional [[point at infinity]] is adjoined to the <math>(x,y)</math> plane, and considered to lie on the curve, although it is not given coördinates in the real numbers. This point is the [[group identity]]. The [[group inverse]] consists in taking the reflection of a point across the ''x''-axis.

	The operation of a point with itself, called '''point-doubling''', is defined similarly, but by considering the line tangent to the curve at that point rather than through two distinct points.		=== Point-doubling ===

			The special case of the addition operation of a point with itself, called '''point-doubling''', is defined similarly, but by considering the line tangent to the curve at that point rather than through two distinct points.

	[[Image:Pgo-2.svg\|right]]		[[Image:Pgo-2.svg\|right]]
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	\end{array}</math>		\end{array}</math>

	However, we already know the slope ''m'' of the resolvent line equation, and we have the ''x''-coördinate of the third point <math>R=(x,y)</math> in terms of it, from the '''quotient''' already calculated.		However, we already know the slope ''m'' of the resolvent line equation, up to four choices of sign for ''y''<sub>''P''</sub> and ''y''<sub>''Q''</sub> respectively, and we have the ''x''-coördinate of the third point <math>R=(x,y)</math> in terms of it, from the '''quotient''' already calculated.

			:<math>m=\left\{\begin{array}{lr}
			\dfrac{\pm\sqrt{x_Q^3+ax_Q+b}
			\mp\sqrt{x_P^3+ax_P+b}}{x_Q-x_P},&
			x_Q\ne x_P;\\
			\pm\dfrac{3x_P^2+a}{2\sqrt{x_P^3+ax_P+b}},&
			x_Q=x_P,\;y_Q=y_P;\\
			\infty,&
			x_Q=x_P,\;y_Q=-y_P.
			\end{array}\right.
			</math>

	:<math>~~m=\frac{\pm\sqrt{x_Q^3+ax_Q+b}~~		:<math></math>
	~~\mp\sqrt{x_P^3+ax_P+b}}{x_Q-x_P}~~</math>~~    (four possibilities!)~~

	:<math>x=m^2-x_P-x_Q</math>		:<math>x=m^2-x_P-x_Q</math>

	:<math>y=\pm\sqrt{x^3+ax+b}</math>~~    (choose sign correctly!)~~		:<math>y=\pm\sqrt{x^3+ax+b}</math>

	Now		Now

Rational Point: text flow

2025-01-11T06:57:20Z

text flow

← Older revision		Revision as of 06:57, 11 January 2025
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	The operation of a point with itself, called '''point-doubling''', is defined similarly, but by considering the line tangent to the curve at that point rather than through two distinct points.		The operation of a point with itself, called '''point-doubling''', is defined similarly, but by considering the line tangent to the curve at that point rather than through two distinct points.

	[[Image:~~Point~~-~~doubling.R~~.svg\|~~left~~]]		[[Image:Pgo-2.svg\|right]]
	[[Image:~~Pgo~~-2.svg]]		[[Image:Point-doubling.R.svg]]

	== Algebraic derivation ==		== Algebraic derivation ==

Rational Point: clear flow for text

2025-01-11T06:56:04Z

clear flow for text

← Older revision		Revision as of 06:56, 11 January 2025
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	The operation of a point with itself, called '''point-doubling''', is defined similarly, but by considering the line tangent to the curve at that point rather than through two distinct points.		The operation of a point with itself, called '''point-doubling''', is defined similarly, but by considering the line tangent to the curve at that point rather than through two distinct points.

	[[Image:~~Pgo~~-2.svg\|left]][[Image:~~Point~~-~~doubling.R~~.svg~~\|right~~]]		[[Image:Point-doubling.R.svg\|left]]
			[[Image:Pgo-2.svg]]

	== Algebraic derivation ==		== Algebraic derivation ==

Rational Point: /* Euclidean plane geometry */ another illustration

2025-01-11T06:50:38Z

Euclidean plane geometry: another illustration

← Older revision		Revision as of 06:50, 11 January 2025
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	The operation of a point with itself, called '''point-doubling''', is defined similarly, but by considering the line tangent to the curve at that point rather than through two distinct points.		The operation of a point with itself, called '''point-doubling''', is defined similarly, but by considering the line tangent to the curve at that point rather than through two distinct points.

	[[Image:Pgo-2.svg]]		[[Image:Pgo-2.svg\|left]][[Image:Point-doubling.R.svg\|right]]

	== Algebraic derivation ==		== Algebraic derivation ==

Rational Point: /* Algebraic derivation */ complete

2025-01-11T05:41:26Z

Algebraic derivation: complete

@@ Line 45: / Line 45: @@
 \end{array}</math>
-The '''remainder''' may be set equal to zero term by term (as we know it is having obtained three linear factors of a cubic equation) and then (with the variable ''x'' eliminated) a system of two quadratic equations <ref>Adelaide Denis. “A Discussion of the Cases when Two Quadratic Equations Involving Two Variables can be Solved by the Method of Quadratics.” ''The American Mathematical Monthly,''
+The '''remainder''' might be set equal to zero term by term (as we know it is having obtained three linear factors of a cubic equation) and then (with the variable ''x'' eliminated) a system of two quadratic equations <ref>Adelaide Denis. “A Discussion of the Cases when Two Quadratic Equations Involving Two Variables can be Solved by the Method of Quadratics.” ''The American Mathematical Monthly,''
 vol. 10, no. 8/9, Aug. - Sep., 1903. [[File:Denis-DiscussionCasesTwo-1903-9-jam8-join.pdf]] https://www.jstor.org/stable/2971350</ref> is left to sort out and solve for ''m'' and ''c'', the coefficients of the resolvent equation for the line passing through ''P'' and ''Q''.
@@ Line 52: / Line 52: @@
 -x_Px_Qm^2+c^2-b -x_P^2x_Q-x_Px_Q^2 &=& 0
 \end{array}</math>

Rational Point: file copy

2025-01-10T18:20:30Z

file copy

← Older revision		Revision as of 18:20, 10 January 2025
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	The '''remainder''' may be set equal to zero term by term (as we know it is having obtained three linear factors of a cubic equation) and then (with the variable ''x'' eliminated) a system of two quadratic equations <ref>Adelaide Denis. “A Discussion of the Cases when Two Quadratic Equations Involving Two Variables can be Solved by the Method of Quadratics.” ''The American Mathematical Monthly,''		The '''remainder''' may be set equal to zero term by term (as we know it is having obtained three linear factors of a cubic equation) and then (with the variable ''x'' eliminated) a system of two quadratic equations <ref>Adelaide Denis. “A Discussion of the Cases when Two Quadratic Equations Involving Two Variables can be Solved by the Method of Quadratics.” ''The American Mathematical Monthly,''
	vol. 10, no. 8/9, Aug. - Sep., 1903. https://www.jstor.org/stable/2971350</ref> is left to sort out and solve for ''m'' and ''c'', the coefficients of the resolvent equation for the line passing through ''P'' and ''Q''.		vol. 10, no. 8/9, Aug. - Sep., 1903. [[File:Denis-DiscussionCasesTwo-1903-9-jam8-join.pdf]] https://www.jstor.org/stable/2971350</ref> is left to sort out and solve for ''m'' and ''c'', the coefficients of the resolvent equation for the line passing through ''P'' and ''Q''.

	:<math>\begin{array}{rcl}		:<math>\begin{array}{rcl}

Rational Point: /* Algebraic derivation */ ref

2025-01-10T17:22:33Z

Algebraic derivation: ref

← Older revision		Revision as of 17:22, 10 January 2025
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	\end{array}</math>		\end{array}</math>

	The '''remainder''' may be set equal to zero term by term (as we know it is having obtained three linear factors of a cubic equation) and then (with the variable ''x'' eliminated) a system of two quadratic equations is left to sort out and solve for ''m'' and ''c'', the coefficients of the resolvent equation for the line passing through ''P'' and ''Q''.		The '''remainder''' may be set equal to zero term by term (as we know it is having obtained three linear factors of a cubic equation) and then (with the variable ''x'' eliminated) a system of two quadratic equations <ref>Adelaide Denis. “A Discussion of the Cases when Two Quadratic Equations Involving Two Variables can be Solved by the Method of Quadratics.” ''The American Mathematical Monthly,''
			vol. 10, no. 8/9, Aug. - Sep., 1903. https://www.jstor.org/stable/2971350</ref> is left to sort out and solve for ''m'' and ''c'', the coefficients of the resolvent equation for the line passing through ''P'' and ''Q''.

	:<math>\begin{array}{rcl}		:<math>\begin{array}{rcl}

Rational Point: /* Algebraic derivation */ typo

2025-01-10T13:39:16Z

Algebraic derivation: typo

← Older revision		Revision as of 13:39, 10 January 2025
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	== Algebraic derivation ==		== Algebraic derivation ==

	Let <math>y=mx+c</math> be the '''resolvent equation''' of the line passing through two given points ''P'' and ''Q'' of an elliptic curve <math>y^2=x^3+ax+b</math>. Square the resolvent on both sides, place it over the ~~euqation~~ for the elliptic curve, (here the [[Weierstraß normal form]],) and subtract to find the '''resultant equation'''.		Let <math>y=mx+c</math> be the '''resolvent equation''' of the line passing through two given points ''P'' and ''Q'' of an elliptic curve <math>y^2=x^3+ax+b</math>. Square the resolvent on both sides, place it over the equation for the elliptic curve, (here the [[Weierstraß normal form]],) and subtract to find the '''resultant equation'''.

	:<math>\begin{array}{rrrrrl}		:<math>\begin{array}{rrrrrl}

Rational Point: /* Algebraic derivation */ remainder

2025-01-10T13:32:51Z

Algebraic derivation: remainder

← Older revision		Revision as of 13:32, 10 January 2025
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	== Algebraic derivation ==		== Algebraic derivation ==

	Let <math>y=mx+c</math> be the '''resolvent equation''' of the line passing through two given points ''P'' and ''Q'' of an elliptic curve <math>y^2=x^3+ax+b</math>. Square the resolvent on both sides, place it over the elliptic curve ~~equation~~, and subtract.		Let <math>y=mx+c</math> be the '''resolvent equation''' of the line passing through two given points ''P'' and ''Q'' of an elliptic curve <math>y^2=x^3+ax+b</math>. Square the resolvent on both sides, place it over the euqation for the elliptic curve, (here the [[Weierstraß normal form]],) and subtract to find the '''resultant equation'''.

	:<math>\begin{array}{rrrrrl}		:<math>\begin{array}{rrrrrl}
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	[[long division\|Divide]] the factors for the two given points ''P'' and ''Q'' together with the leading coefficient		'''[[long division\|Divide]]''' the factors for the two given points ''P'' and ''Q'' together with the leading coefficient

	:<math>-(x-x_P)(x-x_Q)=-x^2+(x_P+x_Q)x-x_Px_Q</math>		:<math>-(x-x_P)(x-x_Q)=-x^2+(x_P+x_Q)x-x_Px_Q</math>

	into the resultant ~~equation,~~ to find the ''x''-coördinate of the third point ''R'' on the elliptic curve through which the same line passes.		into the resultant to find the ''x''-coördinate of the third point ''R'' on the elliptic curve through which the same line passes.

	:<math>\begin{array}{r\|rrrrl}		<math>\begin{array}{r\|rrrrl}
	&&&x&{}-{m^2-x_P\choose -x_Q}\\ \hline		&&&x&{}-{m^2-x_P\choose -x_Q}\\ \hline
	-x^2+(x_P+x_Q)x-x_Px_Q & -x^3&{}+m^2x^2&{}+(2mc-a)x&{}+c^2-b \\		-x^2+(x_P+x_Q)x-x_Px_Q & -x^3&{}+m^2x^2&{}+(2mc-a)x&{}+c^2-b \\
	&-x^3&{}+(x_P+x_Q)x^2&{}-x_Px_Qx\\ \hline		&-x^3&{}+(x_P+x_Q)x^2&{}-x_Px_Qx\\ \hline
	&&{m^2-x_P\choose -x_Q}x^2 &{}+{2mc-a\choose {}+x_Px_Q}x&{}+c^2-b\\		&&{m^2-x_P\choose -x_Q}x^2 &{}+{2mc-a\choose {}+x_Px_Q}x&{}+c^2-b\\
	&&{m^2-x_P\choose -x_Q}x^2&{}-{m^~~2x_P~~+m^~~2x_Q~~\choose -x_P^2-2x_Px_Q-x_Q^2}x&{}+{m^~~2x_Px_Q~~\choose -x_P^2x_Q-x_Px_Q^2} \\ \hline		&&{m^2-x_P\choose -x_Q}x^2&{}-{x_Pm^2+x_Qm^2\choose -x_P^2-2x_Px_Q-x_Q^2}x&{}+{x_Px_Qm^2\choose -x_P^2x_Q-x_Px_Q^2} \\ \hline
	R=		R=&&&{(x_P+x_Q)m^2 +2mc -a \choose -x_P^2-x_Px_Q-x_Q^2}x&{}+{-x_Px_Qm^2+c^2-b\choose -x_P^2x_Q-x_Px_Q^2}
	\end{array}</math>		\end{array}</math>

	The remainder may be set equal to zero term by term, (as we know it is having obtained three linear factors of a cubic equation) and then (with the variable ''x'' eliminated) a system of two quadratic equations is left to sort out and solve for ''m'' and ''c'', the coefficients of the resolvent equation for the line passing through ''P'' and ''Q''.		The '''remainder''' may be set equal to zero term by term (as we know it is having obtained three linear factors of a cubic equation) and then (with the variable ''x'' eliminated) a system of two quadratic equations is left to sort out and solve for ''m'' and ''c'', the coefficients of the resolvent equation for the line passing through ''P'' and ''Q''.

			:<math>\begin{array}{rcl}
			(x_P+x_Q)m^2 +2mc -a -x_P^2-x_Px_Q-x_Q^2 &=& 0\\
			-x_Px_Qm^2+c^2-b -x_P^2x_Q-x_Px_Q^2 &=& 0
			\end{array}</math>