## CryptoDB

### Laura Hitt

#### Publications

Year
Venue
Title
2007
EPRINT
Hyperelliptic curves of small genus have the advantage of providing a group of comparable size as that of elliptic curves, while working over a field of smaller size. Pairing-friendly hyperelliptic curves are those whose order of the Jacobian is divisible by a large prime, whose embedding degree is small enough for computations to be feasible, and whose minimal embedding field is large enough for the discrete logarithm problem in it to be difficult. We give a sequence of $\F_q$-isogeny classes for a family of Jacobians of genus two curves over $\F_{q}$, for $q=2^m$, and their corresponding small embedding degrees. We give examples of the parameters for such curves with embedding degree $k<(\log q)^2$, such as $k=8,13,16,23,26,37,46,52$. For secure and efficient implementation of pairing-based cryptography on genus g curves over $\F_q$, it is desirable that the ratio $\rho=\frac{g\log_2 q}{\log_2N}$ be approximately 1, where $N$ is the order of the subgroup with embedding degree $k$. We show that for our family of curves, $\rho$ is often near 1 and never more than 2. We also give a sequence of $\F_q$-isogeny classes for a family of Jacobians of genus 2 curves over $\F_{q}$ whose minimal embedding field is much smaller than the finite field indicated by the embedding degree $k$. That is, the extension degrees in this example differ by a factor of $m$, where $q=2^m$, demonstrating that the embedding degree can be a far from accurate measure of security. As a result, we use an indicator $k'=\frac{\ord_N2}{m}$ to examine the cryptographic security of our family of curves.
2007
EPRINT
We give a concise statement of a test for security of elliptic curves that should be inserted into the standards for elliptic curve cryptography. In particular, current validation for parameters related to the MOV condition that appears in the latest draft of the IEEE P1363 standard \cite[Section A.12.1, Section A.16.8]{P1363} should be replaced with our subfield-adjusted MOV condition. Similarly, the Standards for Efficient Cryptography Group's document SEC 1 \cite{sec_1} should make adjustments accordingly.
2006
EPRINT
We discuss the underlying mathematics that causes the embedding degree of a curve of any genus to not necessarily correspond to the minimal embedding field, and hence why it may fail to capture the security of a pairing-based cryptosystem. Let $C$ be a curve of genus $g$ defined over a finite field $\F_q$, where $q=p^m$ for a prime $p$. The Jacobian of the curve is an abelian variety, $J_C(\F_q)$, of dimension $g$ defined over $\F_q$. For some prime $N$, coprime to $p$, the embedding degree of $J_C(\F_q)[N]$ is defined to be the smallest positive integer $k$ such that $N$ divides $q^k-1$. Hence, $\F_{q^k}^*$ contains a subgroup of order $N$. To determine the security level of a pairing-based cryptosystem, it is important to know the minimal field containing the $N$th roots of unity, since the discrete logarithm problem can be transported from the curve to this field, where one can perform index calculus. We show that it is possible to have a dramatic (unbounded) difference between the size of the field given by the embedding degree, $\F_{p^{mk}}$, and the minimal embedding field that contains the $N$th roots of unity, $\F_{p^d}$, where $d\mid mk$. The embedding degree has utility as it indicates the field one must work over to compute the pairing, while a security parameter should indicate the minimal field containing the embedding. We discuss a way of measuring the difference between the size of the two fields and we advocate the use of two separate parameters. We offer a possible security parameter, $k'=\frac{\ord_Np}{g}$, and we present examples of elliptic curves and genus 2 curves which highlight the difference between them. While our observation provides a proper theoretical understanding of minimal embedding fields in pairing-based cryptography, it is unlikely to affect curves used in practice, as a discrepancy may only occur when $q$ is non-prime. Nevertheless, it is an important point to keep in mind and a motivation to recognize two separate parameters when describing a pairing-based cryptosystem.