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This work presents a new concept to implement the elliptic curve point multiplication (PM). This computation is based on a new modular arithmetic over Gaussian integer fields. Gaussian integers are a subset of the complex numbers such that the real and imaginary parts are integers. Since Gaussian integer fields are isomorphic to prime fields, this arithmetic is suitable for many elliptic curves. Representing the key by a Gaussian integer expansion is beneficial to reduce the computational complexity and the memory requirements of secure hardware implementations, which are robust against attacks. Furthermore, an area-efficient coprocessor design is proposed with an arithmetic unit that enables Montgomery modular arithmetic over Gaussian integers. The proposed architecture and the new arithmetic provide high flexibility, i.e., binary and non-binary key expansions as well as protected and unprotected PM calculations are supported. The proposed coprocessor is a competitive solution for a compact ECC processor suitable for applications in small embedded systems.
Large-scale quantum computers threaten the security of today's public-key cryptography. The McEliece cryptosystem is one of the most promising candidates for post-quantum cryptography. However, the McEliece system has the drawback of large key sizes for the public key. Similar to other public-key cryptosystems, the McEliece system has a comparably high computational complexity. Embedded devices often lack the required computational resources to compute those systems with sufficiently low latency. Hence, those systems require hardware acceleration. Lately, a generalized concatenated code construction was proposed together with a restrictive channel model, which allows for much smaller public keys for comparable security levels. In this work, we propose a hardware decoder suitable for a McEliece system based on these generalized concatenated codes. The results show that those systems are suitable for resource-constrained embedded devices.
The McEliece cryptosystem is a promising candidate for post-quantum public-key encryption. In this work, we propose q-ary codes over Gaussian integers for the McEliece system and a new channel model. With this one Mannheim error channel, errors are limited to weight one. We investigate the channel capacity of this channel and discuss its relation to the McEliece system. The proposed codes are based on a simple product code construction and have a low complexity decoding algorithm. For the one Mannheim error channel, these codes achieve a higher error correction capability than maximum distance separable codes with bounded minimum distance decoding. This improves the work factor regarding decoding attacks based on information-set decoding.
Many resource-constrained systems still rely on symmetric cryptography for verification and authentication. Asymmetric cryptographic systems provide higher security levels, but are very computational intensive. Hence, embedded systems can benefit from hardware assistance, i.e., coprocessors optimized for the required public key operations. In this work, we propose an elliptic curve cryptographic coprocessors design for resource-constrained systems. Many such coprocessor designs consider only special (Solinas) prime fields, which enable a low-complexity modulo arithmetic. Other implementations support arbitrary prime curves using the Montgomery reduction. These implementations typically require more time for the point multiplication. We present a coprocessor design that has low area requirements and enables a trade-off between performance and flexibility. The point multiplication can be performed either using a fast arithmetic based on Solinas primes or using a slower, but flexible Montgomery modular arithmetic.
Today, many resource-constrained systems, such as embedded systems, still rely on symmetric cryptography for authentication and digital signatures. Asymmetric cryptography provide a higher security level, but software implementations of public-key algorithms on small embedded systems are extremely slow. Hence, such embedded systems require hardware assistance, i.e. crypto coprocessors optimized for public key operations. Many such coprocessor designs aim on high computational performance. In this work, an area efficient elliptic curve cryptography (ECC) coprocessor is presented for applications in small embedded systems where high performance coprocessors are too costly. We propose a simple control unit with a small instruction set that supports different ECC point multiplication (PM) algorithms. The control unit reduces the logic and number of registers compared with other implementations of ECC point multiplications.
The reliability of flash memories suffers from various error causes. Program/erase cycles, read disturb, and cell to cell interference impact the threshold voltages and cause bit errors during the read process. Hence, error correction is required to ensure reliable data storage. In this work, we investigate the bit-labeling of triple level cell (TLC) memories. This labeling determines the page capacities and the latency of the read process. The page capacity defines the redundancy that is required for error correction coding. Typically, Gray codes are used to encode the cell state such that the codes of adjacent states differ in a single digit. These Gray codes minimize the latency for random access reads but cannot balance the page capacities. Based on measured voltage distributions, we investigate the page capacities and propose a labeling that provides a better rate balancing than Gray labeling.
Code-based cryptosystems are promising candidates for post-quantum cryptography. Recently, generalized concatenated codes over Gaussian and Eisenstein integers were proposed for those systems. For a channel model with errors of restricted weight, those q-ary codes lead to high error correction capabilities. Hence, these codes achieve high work factors for information set decoding attacks. In this work, we adapt this concept to codes for the weight-one error channel, i.e., a binary channel model where at most one bit-error occurs in each block of m bits. We also propose a low complexity decoding algorithm for the proposed codes. Compared to codes over Gaussian and Eisenstein integers, these codes achieve higher minimum Hamming distances for the dual codes of the inner component codes. This property increases the work factor for a structural attack on concatenated codes leading to higher overall security. For comparable security, the key size for the proposed code construction is significantly smaller than for the classic McEliece scheme based on Goppa codes.
Large-scale quantum computers threaten today's public-key cryptosystems. The code-based McEliece and Niederreiter cryptosystems are among the most promising candidates for post-quantum cryptography. Recently, a new class of q-ary product codes over Gaussian integers together with an efficient decoding algorithm were proposed for the McEliece cryptosystems. It was shown that these codes achieve a higher work factor for information-set decoding attacks than maximum distance separable (MDS) codes with comparable length and dimension. In this work, we adapt this q-ary product code construction to codes over Eisenstein integers. We propose a new syndrome decoding method which is applicable for Niederreiter cryptosystems. The code parameters and work factors for information-set decoding are comparable to codes over Gaussian integers. Hence, the new construction is not favorable for the McEliece system. Nevertheless, it is beneficial for the Niederreiter system, where it achieves larger message lengths. While the Niederreiter and McEliece systems have the same level of security, the Niederreiter system can be advantageous for some applications, e.g., it enables digital signatures. The proposed coding scheme is interesting for lightweight Niederreiter cryptosystems and embedded security due to the short code lengths and low decoding complexity.
Reed-Muller (RM) codes have recently regained some interest in the context of low latency communications and due to their relation to polar codes. RM codes can be constructed based on the Plotkin construction. In this work, we consider concatenated codes based on the Plotkin construction, where extended Bose-Chaudhuri-Hocquenghem (BCH) codes are used as component codes. This leads to improved code parameters compared to RM codes. Moreover, this construction is more flexible concerning the attainable code rates. Additionally, new soft-input decoding algorithms are proposed that exploit the recursive structure of the concatenation and the cyclic structure of the component codes. First, we consider the decoding of the cyclic component codes and propose a low complexity hybrid ordered statistics decoding algorithm. Next, this algorithm is applied to list decoding of the Plotkin construction. The proposed list decoding approach achieves near-maximum-likelihood performance for codes with medium lengths. The performance is comparable to state-of-the-art decoders, whereas the complexity is reduced.
The code-based McEliece cryptosystem is a promising candidate for post-quantum cryptography. The sender encodes a message, using a public scrambled generator matrix, and adds a random error vector. In this work, we consider q-ary codes and restrict the Lee weight of the added error symbols. This leads to an increased error correction capability and a larger work factor for information-set decoding attacks. In particular, we consider codes over an extension field and use the one-Lee error channel, which restricts the error values to Lee weight one. For this channel model, generalized concatenated codes can achieve high error correction capabilities. We discuss the decoding of those codes and the possible gain for decoding beyond the guaranteed error correction capability.