Dear Isaac Obiri,
You may want to look over the following data below:
What is so special about elliptic curves?
https://crypto.stackexchange.com/questions/11518/what-is-so-special-about-elliptic-curves
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A Study of Hyperelliptic Curves in Cryptography
Elliptic curves are some specific type of curves known as hyper elliptic curves. Compared to the integer factorization problem(IFP) based systems, using elliptic curve based cryptography will significantly decrease key size of the encryption. Therefore, application of this type of cryptography in systems that need high security and smaller key size has found great attention. Hyperelliptic curves help to make key length shorter. Many investigations are done with regard to improving computations, hardware and software implementation of these curves, their security and resistance against attacks. This paper studies and analyzes researches done about security and efficiency of hyperelliptic curves.
https://www.researchgate.net/publication/305878169_A_Study_of_Hyperelliptic_Curves_in_Cryptography
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Hyper-and-elliptic-curve cryptography
This paper introduces ‘hyper-and-elliptic-curve cryptography’, in which a single high-security group supports fast genus-2-hyperelliptic-curve formulas for variable-base-point single-scalar multiplication (for example, Diffie–Hellman shared-secret computation) and at the same time supports fast elliptic-curve formulas for fixed-base-point scalar multiplication (for example, key generation) and multi-scalar multiplication (for example, signature verification).
https://www.cambridge.org/core/services/aop-cambridge-core/content/view/4551593BD96904BCD48747DBB89F9615/S1461157014000394a.pdf/hyperandellipticcurve_cryptography.pdf
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Hyperelliptic curve cryptography
An (imaginary) hyperelliptic curve of genus over a field is given by the equation where is a polynomial of degree not larger than and is a monic polynomial of degree. From this definition it follows that elliptic curves are hyperelliptic curves of genus 1. In hyperelliptic curve cryptography is often a finite field. The Jacobian of, denoted, is a quotient group, thus the elements of the Jacobian are not points, they are equivalence classes of divisors of degree 0 under the relation of linear equivalence. This agrees with the elliptic curve case, because it can be shown that the Jacobian of an elliptic curve is isomorphic with the group of points on the elliptic curve.[1] The use of hyperelliptic curves in cryptography came about in 1989 from Neal Koblitz. Although introduced only 3 years after ECC, not many cryptosystems implement hyperelliptic curves because the implementation of the arithmetic isn't as efficient as with cryptosystems based on elliptic curves or factoring (RSA). The efficiency of implementing the arithmetic depends on the underlying finite field, in practice it turns out that finite fields of characteristic 2 are a good choice for hardware implementations while software is usually faster in odd characteristic.[2].
https://cryptography.fandom.com/wiki/Hyperelliptic_curve_cryptography
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The short answer is "it depends on the problem", like it always does. One size does not fit all, and one solution does not fit all. Think about what the validation criteria are for your problem set, and the underlying physical constraints. Understanding that will probably lead you to selection of the best tool to find the solution.
Some papers have shown that hyperelliptic curve cryptography is more efficient than elliptic curve cryptography for certain applications and scenarios. This is because hyperelliptic curve cryptography has several features that make it more efficient than elliptic curve cryptography in certain contexts.
One reason why hyperelliptic curve cryptography is more efficient is that it allows for the use of smaller key sizes. This means that it requires less computational power to encrypt and decrypt data using hyperelliptic curve cryptography, making it faster and more efficient than elliptic curve cryptography in some cases.
Another reason why hyperelliptic curve cryptography is more efficient is that it can provide higher security levels for the same key size. This is because the mathematical structure of hyperelliptic curves allows for stronger security guarantees than elliptic curves, making it more resistant to attacks.
Finally, hyperelliptic curve cryptography is more flexible than elliptic curve cryptography, as it can support a wider range of cryptographic functions and protocols. This allows it to be used in a wider range of applications and scenarios, making it more efficient in some cases.
Overall, the relative efficiency of hyperelliptic curve cryptography compared to elliptic curve cryptography depends on the specific context and requirements of the application. In some cases, hyperelliptic curve cryptography may be more efficient, while in other cases, elliptic curve cryptography may be more suitable.
The efficiency of HECC versus ECC depends on the specific application and parameters used. In some cases, HECC may be more efficient, while ECC may be more efficient in other cases.
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