Point compression for the trace zero subgroup over a small degree extension field

Using Semaev's summation polynomials, we derive a new equation for the $\mathbb{F}_q$-rational points of the trace zero variety of an elliptic curve defined over $\mathbb{F}_q$. Using this equation, we produce an optimal-size representation for such points. Our representation is compatible with scalar multiplication. We give a point compression algorithm to compute the representation and a decompression algorithm to recover the original point (up to some small ambiguity). The algorithms are efficient for trace zero varieties coming from small degree extension fields. We give explicit equations and discuss in detail the practically relevant cases of cubic and quintic field extensions.Comment: 23 pages, to appear in Designs, Codes and Cryptograph

Efficient Implementation on Low-Cost SoC-FPGAs of TLSv1.2 Protocol with ECC_AES Support for Secure IoT Coordinators

Security management for IoT applications is a critical research field, especially when taking into account the performance variation over the very different IoT devices. In this paper, we present high-performance client/server coordinators on low-cost SoC-FPGA devices for secure IoT data collection. Security is ensured by using the Transport Layer Security (TLS) protocol based on the TLS_ECDHE_ECDSA_WITH_AES_128_CBC_SHA256 cipher suite. The hardware architecture of the proposed coordinators is based on SW/HW co-design, implementing within the hardware accelerator core Elliptic Curve Scalar Multiplication (ECSM), which is the core operation of Elliptic Curve Cryptosystems (ECC). Meanwhile, the control of the overall TLS scheme is performed in software by an ARM Cortex-A9 microprocessor. In fact, the implementation of the ECC accelerator core around an ARM microprocessor allows not only the improvement of ECSM execution but also the performance enhancement of the overall cryptosystem. The integration of the ARM processor enables to exploit the possibility of embedded Linux features for high system flexibility. As a result, the proposed ECC accelerator requires limited area, with only 3395 LUTs on the Zynq device used to perform high-speed, 233-bit ECSMs in 413 µs, with a 50 MHz clock. Moreover, the generation of a 384-bit TLS handshake secret key between client and server coordinators requires 67.5 ms on a low cost Zynq 7Z007S device

Fast, uniform, and compact scalar multiplication for elliptic curves and genus 2 Jacobians with applications to signature schemes

We give a general framework for uniform, constant-time one-and two-dimensional scalar multiplication algorithms for elliptic curves and Jacobians of genus 2 curves that operate by projecting to the x-line or Kummer surface, where we can exploit faster and more uniform pseudomultiplication, before recovering the proper "signed" output back on the curve or Jacobian. This extends the work of L{\'o}pez and Dahab, Okeya and Sakurai, and Brier and Joye to genus 2, and also to two-dimensional scalar multiplication. Our results show that many existing fast pseudomultiplication implementations (hitherto limited to applications in Diffie--Hellman key exchange) can be wrapped with simple and efficient pre-and post-computations to yield competitive full scalar multiplication algorithms, ready for use in more general discrete logarithm-based cryptosystems, including signature schemes. This is especially interesting for genus 2, where Kummer surfaces can outperform comparable elliptic curve systems. As an example, we construct an instance of the Schnorr signature scheme driven by Kummer surface arithmetic

Generation, Verification, and Attacks on Elliptic Curves and their Applications in Signal Protocol

Elliptic curves (EC) are widely studied due to their mathematical and cryptographic properties. Cryptographers have used the properties of EC to construct elliptic curve cryptosystems (ECC). ECC are based on the assumption of hardness of special instances of the discrete logarithm problem in EC. One of the strong merits of ECC is providing the same cryptographic strength with smaller key size compared to other public key cryptosystems. A 256 bit ECC can provide similar cryptographic strength as a 3072 bit RSA cryptosystem. Due to smaller key sizes, elliptic curves are an attractive option in devices with limited storage capacity. It is therefore essential to understand how to generate these curves, verify their correctness and assure that they are resistant against attacks. The security of an EC cryptosystem is determined by the choice of the curve that is used in that cryptosystem. Over the years, a number of elliptic curves were introduced for cryptographic use. Elliptic curves such as FRP256V1, NIST P-256, Secp256k1 or SM2 curve are widely used in many applications like cryptocurrencies, transport layer protocol and Internet messaging applications. Another type of popular curves are Curve25519 introduced by Dan Bernstein and Curve448 introduced by Mike Hamburg, which are used in an end to end encryption protocol called Signal. This protocol is used in popular messaging applications like WhatsApp, Signal Messenger and Facebook Messenger. Recently, there has been a growing distrust among security researchers against the previously standardized curves. We have seen backdoors in the elliptic curve cryptosystems like the DUAL_EC_DRBG function that was standardized by NIST, and suspicious random seeds that were used in NIST P-curves. We can say that many of the previously standardized curves lack transparency in their generation and verification. We focus on transparent generation and verification of elliptic curves. We generate curves based on NIST standards and propose new standards to generate special types of elliptic curves. We test their resistance against the known attacks that target the ECC. Finally, we demonstrate ECDLP attacks on small curves with weak structure

Efficient algorithms for pairing-based cryptosystems

We describe fast new algorithms to implement recent cryptosystems based on the Tate pairing. In particular, our techniques improve pairing evaluation speed by a factor of about 55 compared to previously known methods in characteristic 3, and attain performance comparable to that of RSA in larger characteristics.We also propose faster algorithms for scalar multiplication in characteristic 3 and square root extraction over Fpm, the latter technique being also useful in contexts other than that of pairing-based cryptography

HLS-Based Methodology for Fast Iterative Development Applied to Elliptic Curve Arithmetic

International audienceHigh-Level Synthesis (HLS) is used by hardware developers to achieve higher abstraction in circuit descriptions. In order to shorten the hardware development time via HLS, we present an adjustment of the Iterative and Incremental Design (IID) methodology, frequently used in software development. In particular, our methodology is relevant for the development of applications with unusual complexity: the method was applied here to the development of large modular arithmetic, commonly used for cryptography applications (e.g., Elliptic Curves). Rapid feedback on circuit characteristics is used to evaluate deep architectural changes in short time, greatly reducing the time-to-market with respect to hand-made designs. In addition, our approach is highly flexible, since the same generic high-level description can be used to produce an entire set of circuits, each with different area/performance trade-offs. Thanks to the proposed approach, any change to the initial specification (e.g., the curve used) is also very fast, while it may require a large effort in the case of hand-made designs