Radix-2² Algorithm for the Odd New Mersenne Number Transform (ONMNT)

\ua9 2023 by the authors. This paper introduces a new derivation of the radix- (Formula presented.) fast algorithm for the forward odd new Mersenne number transform (ONMNT) and the inverse odd new Mersenne number transform (IONMNT). This involves introducing new equations and functions in finite fields, bringing particular challenges unlike those in other fields. The radix- (Formula presented.) algorithm combines the benefits of the reduced number of operations of the radix-4 algorithm and the simple butterfly structure of the radix-2 algorithm, making it suitable for various applications such as lightweight ciphers, authenticated encryption, hash functions, signal processing, and convolution calculations. The multidimensional linear index mapping technique is the conventional method used to derive the radix- (Formula presented.) algorithm. However, this method does not provide clear insights into the underlying structure and flexibility of the radix- (Formula presented.) approach. This paper addresses this limitation and proposes a derivation based on bit-unscrambling techniques, which reverse the ordering of the output sequence, resulting in efficient calculations with fewer operations. Butterfly and signal flow diagrams are also presented to illustrate the structure of the fast algorithm for both ONMNT and IONMNT. The proposed method should pave the way for efficient and flexible implementation of ONMNT and IONMNT in applications such as lightweight ciphers and signal processing. The algorithm has been implemented in C and is validated with an example

Secure and Efficient RNS Approach for Elliptic Curve Cryptography

Scalar multiplication, the main operation in elliptic curve cryptographic protocols, is vulnerable to side-channel (SCA) and fault injection (FA) attacks. An efficient countermeasure for scalar multiplication can be provided by using alternative number systems like the Residue Number System (RNS). In RNS, a number is represented as a set of smaller numbers, where each one is the result of the modular reduction with a given moduli basis. Under certain requirements, a number can be uniquely transformed from the integers to the RNS domain (and vice versa) and all arithmetic operations can be performed in RNS. This representation provides an inherent SCA and FA resistance to many attacks and can be further enhanced by RNS arithmetic manipulation or more traditional algorithmic countermeasures. In this paper, extending our previous work, we explore the potentials of RNS as an SCA and FA countermeasure and provide an description of RNS based SCA and FA resistance means. We propose a secure and efficient Montgomery Power Ladder based scalar multiplication algorithm on RNS and discuss its SCAFA resistance. The proposed algorithm is implemented on an ARM Cortex A7 processor and its SCA-FA resistance is evaluated by collecting preliminary leakage trace results that validate our initial assumptions

Orthogonal transforms and their application to image coding

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On the Cryptanalysis of Public-Key Cryptography

Nowadays, the most popular public-key cryptosystems are based on either the integer factorization or the discrete logarithm problem. The feasibility of solving these mathematical problems in practice is studied and techniques are presented to speed-up the underlying arithmetic on parallel architectures. The fastest known approach to solve the discrete logarithm problem in groups of elliptic curves over finite fields is the Pollard rho method. The negation map can be used to speed up this calculation by a factor √2. It is well known that the random walks used by Pollard rho when combined with the negation map get trapped in fruitless cycles. We show that previously published approaches to deal with this problem are plagued by recurring cycles, and we propose effective alternative countermeasures. Furthermore, fast modular arithmetic is introduced which can take advantage of prime moduli of a special form using efficient "sloppy reduction." The effectiveness of these techniques is demonstrated by solving a 112-bit elliptic curve discrete logarithm problem using a cluster of PlayStation 3 game consoles: breaking a public-key standard and setting a new world record. The elliptic curve method (ECM) for integer factorization is the asymptotically fastest method to find relatively small factors of large integers. From a cryptanalytic point of view the performance of ECM gives information about secure parameter choices of some cryptographic protocols. We optimize ECM by proposing carry-free arithmetic modulo Mersenne numbers (numbers of the form 2M – 1) especially suitable for parallel architectures. Our implementation of these techniques on a cluster of PlayStation 3 game consoles set a new record by finding a 241-bit prime factor of 21181 – 1. A normal form for elliptic curves introduced by Edwards results in the fastest elliptic curve arithmetic in practice. Techniques to reduce the temporary storage and enhance the performance even further in the setting of ECM are presented. Our results enable one to run ECM efficiently on resource-constrained platforms such as graphics processing units

Advanced cryptographic system : design, architecture and FPGA implementation

PhD ThesisThe field programmable gate array (FPGA) is a powerful technology, and since its introduction broad prospects have opened up for engineers to creatively design and implement complete systems in various fields. One such area that has a long history in information and network security is cryptography, which is considered in this thesis. The challenge for engineers is to design secure cryptographic systems, which should work efficiently on different platforms with the levels of security required. In addition, cryptographic functionalities have to be implemented with acceptable degrees of complexity while demanding lower power consumption. The present work is devoted to the design of an efficient block cipher that meets contemporary security requirements, and to implement the proposed design in a configurable hardware platform. The cipher has been designed according to Shannon’s principles and modern cryptographic theorems. It is an iterated symmetric-key block cipher based on the substitution permutation network and number theoretic transform with variable key length, block size and word length. These parameters can be undisclosed when determined by the system, making cryptanalysis almost impossible. The aim is to design a more secure, reliable and flexible system that can run as a ratified standard, with reasonable computational complexity for a sufficient service time. Analyses are carried out on the transforms concerned, which belong to the number theoretic transforms family, to evaluate their diffusion power, and the results confirm good performance in this respect mostly of a minimum of 50%. The new Mersenne number transform and Fermat number transform were included in the design because their characteristics meet the basic requirements of modern cryptographic systems. A new 7×7 substitution box (S-box) is designed and its non-linear properties are evaluated, resulting in values of 2-6 for maximum difference propagation probability and 2-2.678 for maximum input-output correlation. In addition, these parameters are calculated for all S-boxes belonging to the previous and current standard algorithms. Moreover, three extra S-boxes are derived from the new S-box and another three from the current standard, preserving the same non-linear properties by reordering the output elements. The robustness of the proposed cipher in terms of differential and linear cryptanalysis is then considered, and it is proven that the algorithm is secure against such well-known attacks from round three onwards regardless of block or key length. A number of test vectors are run to verify the correctness of the algorithm’s implementation in terms of any possible error, and all results were promising. Tests included the known answer test, the multi-block message test, and the Monte Carlo test. Finally, efficient hardware architectures for the proposed cipher have been designed and implemented using the FPGA system generator platform. The implementations are run on the target device, Xilinx Virtex 6 (XC6VLX130T-2FF484). Using parallel loop-unrolling architecture, a high throughput of 44.9 Gbits/sec is achieved with a power consumption of 1.83W and 8030 slices for implementing the encryption module with key and block lengths of 16×7 bits. There are a variety of outcomes when the cipher is implemented on different FPGA devices as well as for different block and key lengths.Ministry of Higher Education and Scientific Research in Ira