Montgomery normal form - Revision history

Rational Point: spacing

2024-12-29T06:12:08Z

spacing

← Older revision		Revision as of 06:12, 29 December 2024
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	[[Category:Normal forms]]		[[Category:Normal forms]]
	[[Image:Elliptic-curve-2.svg\|frame\|left\|Elliptic curve in '''Montgomery normal form''' with <math>A=-2</math> and<math>B=\frac49</math>.]]		[[Image:Elliptic-curve-2.svg\|frame\|left\|Elliptic curve in '''Montgomery normal form''' with <math>A=-2</math> and <math>B=\frac49</math>.]]
	== The normal form ==		== The normal form ==

Rational Point: image

2024-12-29T06:09:32Z

image

← Older revision		Revision as of 06:09, 29 December 2024
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	[[Category:Normal forms]]		[[Category:Normal forms]]
			[[Image:Elliptic-curve-2.svg\|frame\|left\|Elliptic curve in '''Montgomery normal form''' with <math>A=-2</math> and<math>B=\frac49</math>.]]
	== The normal form ==		== The normal form ==

Rational Point: cateogory + mention efficiency

2024-12-19T22:35:17Z

cateogory + mention efficiency

← Older revision		Revision as of 22:35, 19 December 2024
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			[[Category:Normal forms]]

	== The normal form ==		== The normal form ==

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	:<math>By^2=x^3+Ax^2+x</math>		:<math>By^2=x^3+Ax^2+x</math>

	It has in common with the [[Weierstraß normal form]] the same important property of being symmetric about the ''x''-axis, and hence it is just as useful and possibly more convenient in some cases for computing the [[point group operation]] by using many if not most of the same techniques.		It has in common with the [[Weierstraß normal form]] the same important property of being symmetric about the ''x''-axis, and hence it is just as useful and possibly more convenient or efficient in some cases for computing the [[point group operation]] by using many if not most of the same techniques.

	== What is a Montgomery curve? ==		== What is a Montgomery curve? ==

Rational Point: /* What is a Montgomery curve? */ scaling and shift

2024-12-19T07:37:57Z

What is a Montgomery curve?: scaling and shift

← Older revision		Revision as of 07:37, 19 December 2024
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	:<math>By^2=(x+C)^3 + A (x+C)^2 + (x+C)</math>		:<math>By^2=(x+C)^3 + A (x+C)^2 + (x+C)</math>

	~~And that~~ preserves the exact same group structure among the rational points as the curve would have in [[Weierstraß normal form]], even in a finite field as the case may be, which is in all probability well understood, notwithstanding a certain presumptuousness or overexuberance in the nuts-and-bolts applications of so-called [[Montgomery arithmetic]] directly to modern-day [[strong cryptography]] together with a certain degree of ignorance of traditional or long-standing classical mathematical literature of the Victorian era.		This is a simple ''y''-axis scaling together with an ''x''-axis shift which preserves the exact same geometry and group structure among the rational points as the curve would have in [[Weierstraß normal form]], even in a finite field as the case may be, which is in all probability well understood, notwithstanding a certain presumptuousness or overexuberance in the nuts-and-bolts applications of so-called [[Montgomery arithmetic]] directly to modern-day [[strong cryptography]] together with a certain degree of ignorance of traditional or long-standing classical mathematical literature of the Victorian era.

Rational Point: Montgomery "curves"

2024-12-19T05:24:13Z

Montgomery "curves"

@@ Line 1: / Line 1: @@
 The '''Montgomery normal form'''<ref>Peter L. Montgomery. “Speeding the Pollard and elliptic curve methods of factorization.” ''Mathematics of Computation,'' vol. 48, no. 177, Jan 1987, pp. 243-264. https://www.ams.org/journals/mcom/1987-48-177/S0025-5718-1987-0866113-7/</ref> of an [[elliptic curve]] is a slightly different normalization from that of Weierstraß.
-:<math>by^2=x^3+ax^2+x</math>
+:<math>By^2=x^3+Ax^2+x</math>
 It has in common with the [[Weierstraß normal form]] the same important property of being symmetric about the ''x''-axis, and hence it is just as useful and possibly more convenient in some cases for computing the [[point group operation]] by using many if not most of the same techniques.

Rational Point: slightly different normalization

2024-12-19T03:20:35Z

slightly different normalization

New page

The '''Montgomery normal form'''<ref>Peter L. Montgomery. “Speeding the Pollard and elliptic curve methods of factorization.” ''Mathematics of Computation,'' vol. 48, no. 177, Jan 1987, pp. 243-264. https://www.ams.org/journals/mcom/1987-48-177/S0025-5718-1987-0866113-7/</ref> of an [[elliptic curve]] is a slightly different normalization from that of Weierstraß.

:<math>by^2=x^3+ax^2+x</math>

It has in common with the [[Weierstraß normal form]] the same important property of being symmetric about the ''x''-axis, and hence it is just as useful and possibly more convenient in some cases for computing the [[point group operation]] by using many if not most of the same techniques.