Need help with this Elliptic curve theory (Edwards curve)

In summary, elliptic curve theory, specifically Edwards curves, is a mathematical concept used in cryptography and number theory. It involves the study of equations in the form of y^2 + x^2 = 1 + d x^2 y^2, where d is a non-square integer. This theory has many practical applications, including secure communication protocols and integer factorization. It is also used in the creation of digital signatures and key exchange algorithms. The main advantage of using Edwards curves is their efficient and secure implementation, making them a popular choice in modern cryptography.
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Firepanda
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That is the Edwards curve addition law, more info can be found here about the Edwards curve:

http://en.wikipedia.org/wiki/Edwards_curve

Everywhere I look it says the addition law takes 10M+1S+1D+7a, where M is field multiplication, S is field squaring, D is multiplying by a parameter d, and a is field addition. [Refer to the projective setting in the wiki page]

Please see the below and can anyone let me know if you can see any way to get my 11M down to 10M? I must have gone wrong somewhere in addition as I only have 4!

http://i.imgur.com/mrXov.jpg

Let me know if you can't make any sense of it..

Thanks
 
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  • #2
,Yes, there is a way to get the 11M down to 10M. You can use the fact that M = 2S - a, where M is field multiplication, S is field squaring and a is field addition. So, you can rewrite the equation as 10S + 1D + 7a.
 

1. What is an Edwards curve?

An Edwards curve is a type of elliptic curve that is defined by the equation x^2 + y^2 = c^2(1 + dx^2y^2), where c and d are constants. It was first introduced by Harold Edwards in 2007 and has gained popularity in cryptography due to its efficient and secure operations.

2. What is the significance of using Edwards curves in cryptography?

Edwards curves have several advantages over other types of elliptic curves, making them suitable for use in cryptography. They have a complete addition formula, which means that all points on the curve can be added together efficiently. They also have a large base point order, making them resistant to certain types of attacks.

3. How are points on an Edwards curve represented?

Points on an Edwards curve are represented by coordinates (x, y) where x and y are elements of a finite field. The x-coordinate is typically represented as a binary string, while the y-coordinate is represented as a sign bit and a binary string of the same length as the x-coordinate.

4. Can Edwards curves be used for both encryption and digital signatures?

Yes, Edwards curves can be used for both encryption and digital signatures. They are used in various cryptographic protocols such as the Edwards-curve Digital Signature Algorithm (EdDSA) and the Elliptic Curve Integrated Encryption Scheme (ECIES).

5. What is the current state of research on Edwards curves?

There is ongoing research on Edwards curves and their applications in cryptography. Some areas of focus include improving the efficiency of operations on Edwards curves, exploring their security properties, and finding new applications for them in other areas of computer science.

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