Elliptic Curve Cryptography - Revision history

NKirnbauer: expired link changed with link from wayback machine

2024-08-28T15:02:55Z

expired link changed with link from wayback machine

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	<ref name="vanet">https://www.researchgate.net/publication/216022102_Analysis_of_ECDSA_authentication_processing_in_VANETs</ref>		<ref name="vanet">https://www.researchgate.net/publication/216022102_Analysis_of_ECDSA_authentication_processing_in_VANETs</ref>

	<ref name="nfc">https://nfc-forum.org/wp-content/uploads/2013/12/Elliptic-Curve-Certificates-and-Signatures-for-NFC-Signature-Records-RIM.pdf</ref>		<ref name="nfc">http://web.archive.org/web/20170329062735/https://nfc-forum.org/wp-content/uploads/2013/12/Elliptic-Curve-Certificates-and-Signatures-for-NFC-Signature-Records-RIM.pdf</ref>
	<ref name="curve25519">Daniel Bernstein, Curve25519:new Diffie-Hellman speed records, Springer Berlin Heidelberg, 2006. https://cr.yp.to/ecdh/curve25519-20060209.pdf, [accessed 30.04.2020]</ref>		<ref name="curve25519">Daniel Bernstein, Curve25519:new Diffie-Hellman speed records, Springer Berlin Heidelberg, 2006. https://cr.yp.to/ecdh/curve25519-20060209.pdf, [accessed 30.04.2020]</ref>
	</references>		</references>
	[[Category:Basic]]		[[Category:Basic]]

NKirnbauer: /* References */

2024-08-23T07:01:09Z

References

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	<ref name="zigbee">https://zigbeealliance.org/wp-content/uploads/2019/11/docs-07-5356-19-0zse-zigbee-smart-energy-profile-specification.pdf</ref>		<ref name="zigbee">https://zigbeealliance.org/wp-content/uploads/2019/11/docs-07-5356-19-0zse-zigbee-smart-energy-profile-specification.pdf</ref>

	<ref name="vanet">~~hhttps~~://www.researchgate.net/publication/216022102_Analysis_of_ECDSA_authentication_processing_in_VANETs</ref>		<ref name="vanet">https://www.researchgate.net/publication/216022102_Analysis_of_ECDSA_authentication_processing_in_VANETs</ref>

	<ref name="nfc">https://nfc-forum.org/wp-content/uploads/2013/12/Elliptic-Curve-Certificates-and-Signatures-for-NFC-Signature-Records-RIM.pdf</ref>		<ref name="nfc">https://nfc-forum.org/wp-content/uploads/2013/12/Elliptic-Curve-Certificates-and-Signatures-for-NFC-Signature-Records-RIM.pdf</ref>

Ikramer at 09:33, 30 April 2020

2020-04-30T09:33:39Z

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	Curve25519 is a highly optimized curve proposed by Daniel J. Bernstein (djb) in 2005 <ref name="curve25519" />.		Curve25519 is a highly optimized curve proposed by Daniel J. Bernstein (djb) in 2005 <ref name="curve25519" />.
	The curve equation is		The curve equation is

	<math>y^2=x^3+486662x^2+x</math> over a prime field <math>2^{255}-19</math>		<math>y^2=x^3+486662x^2+x</math> over a prime field <math>2^{255}-19</math>
	=== Edward curves===		=== Edward curves===

Ikramer at 09:15, 30 April 2020

2020-04-30T09:15:29Z

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	== Curve25519==		== Curve25519==
	Curve25519 is a highly optimized curve proposed by Daniel J. Bernstein (djb) in 2005.		Curve25519 is a highly optimized curve proposed by Daniel J. Bernstein (djb) in 2005 <ref name="curve25519" />.
	The curve equation is		The curve equation is
	<math>y^2=x^3+486662x^2+x</math> over a prime field <math>2^{255}-19</math>		<math>y^2=x^3+486662x^2+x</math> over a prime field <math>2^{255}-19</math>

Ikramer at 09:13, 30 April 2020

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← Older revision		Revision as of 09:13, 30 April 2020
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	== Summary ==		== Summary ==

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	<ref name="nfc">https://nfc-forum.org/wp-content/uploads/2013/12/Elliptic-Curve-Certificates-and-Signatures-for-NFC-Signature-Records-RIM.pdf</ref>		<ref name="nfc">https://nfc-forum.org/wp-content/uploads/2013/12/Elliptic-Curve-Certificates-and-Signatures-for-NFC-Signature-Records-RIM.pdf</ref>
			<ref name="curve25519">Daniel Bernstein, Curve25519:new Diffie-Hellman speed records, Springer Berlin Heidelberg, 2006. https://cr.yp.to/ecdh/curve25519-20060209.pdf, [accessed 30.04.2020]</ref>
	</references>		</references>
	[[Category:Basic]]		[[Category:Basic]]

Ikramer at 13:25, 15 April 2020

2020-04-15T13:25:53Z

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	<math>p+1-2\sqrt{p} \leq \#E \leq p+1+2\sqrt{p}</math>		<math>p+1-2\sqrt{p} \leq \#E \leq p+1+2\sqrt{p}</math>
	</div>		</div>

			The next figure shows a elliptic cuve <math>E:y^2=x^3-2x+2</math> over a finite field, the prime field GF(199). On each y-axis are two points, the point and its inverse point, mapped into the positive space of the prime field.
	[[File:Curveover199.png\|400px\|center]]		[[File:Curveover199.png\|400px\|center]]

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	== Scalar Multiplication ==		== Scalar Multiplication ==
	Is the mainly used operation on elliptic curves in cryptography, it adds <math>n</math> times the point <math>P</math> of the elliptic curve over a finite field e.g. prime field.		Is the mainly used operation on elliptic curves in cryptography, it adds <math>n</math> times the point <math>P</math> of the elliptic curve over a finite field e.g. prime field <math>GF(p)</math>.
	<div align="center">		<div align="center">
	<math>nP= \underbrace{P+P+P+P+...+P}_\text{n times}</math>		<math>nP= \underbrace{P+P+P+P+...+P}_\text{n times} \ text{over} E</math>
	</div>		</div>

Ikramer at 12:46, 15 April 2020

2020-04-15T12:46:22Z

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	</div>		</div>

	<math>x_3=\bigg(\frac{y_2-y_1}{x_2-x_1}\bigg)^2- x_1 -x_2 \ \text{ and } \ y_3=\bigg(\frac{y_2-y_1}{x_2-x_1}\bigg)(x_1-x_3)-y_1.</math>		<math>x_3=\bigg(\frac{y_2-y_1}{x_2-x_1}\bigg)^2- x_1 -x_2 \ \text{ and } \ y_3=\bigg(\frac{y_2-y_1}{x_2-x_1}\bigg)(x_1-x_3)-y_1 \text{ over } E</math>

	This operation is also called the Point Addition in affine coordinates and the computational costs are 1 x inversion, 2 x multiplications and 1 x squaring.		This operation is also called the Point Addition in affine coordinates and the computational costs are 1 x inversion, 2 x multiplications and 1 x squaring.
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	</div>		</div>
	</div>		</div>

			<math>y_3=\bigg(\frac{3 x_1^2+a}{2 y_1}\bigg)(x_1-x_3)-y_1\ \text{ and } \ x_3=\bigg(\frac{3 x_1^2+a}{2 y_1}\bigg)^2-2 \cdot x_1 \ \text{ over } E</math>

			This operation is also called the Point Doubling in affine coordinates and the computational costs are 1 x inversion, 2 x multiplications and 2 x squarings.

	==== Example of elliptic point operations over an elliptic curve over a finite field ====		==== Example of elliptic point operations over an elliptic curve over a finite field ====

Ikramer at 12:12, 15 April 2020

2020-04-15T12:12:44Z

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	== Scalar Multiplication ==		== Scalar Multiplication ==
	Is ~~main~~ used operation in cryptography, it adds n times the point P of the elliptic curve over a prime field.		Is the mainly used operation on elliptic curves in cryptography, it adds <math>n</math> times the point <math>P</math> of the elliptic curve over a finite field e.g. prime field.
	<div align="center">		<div align="center">
	<math>nP= \underbrace{P+P+P+P+...+P}_\text{n times}</math>		<math>nP= \underbrace{P+P+P+P+...+P}_\text{n times}</math>
	</div>		</div>


	=== Double and Add algorithm ===		=== Double and Add algorithm ===
	=== Calculation in projective coordinates ===		=== Calculation in projective coordinates ===

Ikramer at 23:13, 31 March 2020

2020-03-31T23:13:04Z

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	* Case <math>Q=P</math> -> '''Point Doubling''': <math>2P=R</math>		* Case <math>Q=P</math> -> '''Point Doubling''': <math>2P=R</math>
	* Case <math>Q=-P</math> -> '''Inversion of a Point''': <math>P + (-P)=\mathcal{O}</math>		* Case <math>Q=-P</math> -> '''Inversion of a Point''': <math>P + (-P)=\mathcal{O}</math>
	~~TODO:Need to install link target extension: https://www.mediawiki.org/wiki/Extension:LinkTarget~~

	==== Point Addition ====		==== Point Addition ====
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	This operation is also called the Point Addition in affine coordinates and the computational costs are 1 x inversion, 2 x multiplications and 1 x squaring.		This operation is also called the Point Addition in affine coordinates and the computational costs are 1 x inversion, 2 x multiplications and 1 x squaring.
	Interactive tool to visualize ~~point addition online at~~ https://cdn.rawgit.com/andreacorbellini/ecc/920b29a/interactive/reals-add.html.		Interactive tool to visualize [https://cdn.rawgit.com/andreacorbellini/ecc/920b29a/interactive/reals-add.html point addition].

	==== Point Doubling ====		==== Point Doubling ====

Ikramer at 22:57, 31 March 2020

2020-03-31T22:57:18Z

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	Example usage of ECC in IoT:		Example usage of ECC in IoT:
	{\| class="wikitable"		{\| class="wikitable"
	\|ZigBee Smart Energy(1.x and 1.2) \|\| sect163k1(1.x), sect283k1(1.2)		\|ZigBee Smart Energy(1.x and 1.2)<ref name="zigbee"/> \|\| sect163k1(1.x), sect283k1(1.2)
	\|-		\|-
	\|Vehicular Ad-Hoc Networks (IEEE1609.2) \|\| secp224r1,secp256r1		\|Vehicular Ad-Hoc Networks (IEEE1609.2) <ref name="vanet"/> \|\| secp224r1,secp256r1
	\|-		\|-
	\|NFC Forum Signature Record Type Definition(RTD) \|\| sec192r1, secp224r1, sect233k1, sect233r1		\|NFC Forum Signature Record Type Definition(RTD) <ref name="nfc"/> \|\| sec192r1, secp224r1, sect233k1, sect233r1
	\|}		\|}

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	<ref name="NIST Suite B">Committee on National Security Systems. "National information assurance policy on the use of public standards for the secure sharing of information among national security systems." 1 October 2012.</ref>		<ref name="NIST Suite B">Committee on National Security Systems. "National information assurance policy on the use of public standards for the secure sharing of information among national security systems." 1 October 2012.</ref>
	~~</references>~~

			<ref name="zigbee">https://zigbeealliance.org/wp-content/uploads/2019/11/docs-07-5356-19-0zse-zigbee-smart-energy-profile-specification.pdf</ref>

			<ref name="vanet">hhttps://www.researchgate.net/publication/216022102_Analysis_of_ECDSA_authentication_processing_in_VANETs</ref>

			<ref name="nfc">https://nfc-forum.org/wp-content/uploads/2013/12/Elliptic-Curve-Certificates-and-Signatures-for-NFC-Signature-Records-RIM.pdf</ref>
			</references>
	[[Category:Basic]]		[[Category:Basic]]