Efficient modular arithmetic units for low power cryptographic applications

The demand for high security in energy constrained devices such as mobiles and PDAs is growing rapidly. This leads to the need for efficient design of cryptographic algorithms which offer data integrity, authentication, non-repudiation and confidentiality of the encrypted data and communication channels. The public key cryptography is an ideal choice for data integrity, authentication and non-repudiation whereas the private key cryptography ensures the confidentiality of the data transmitted. The latter has an extremely high encryption speed but it has certain limitations which make it unsuitable for use in certain applications. Numerous public key cryptographic algorithms are available in the literature which comprise modular arithmetic modules such as modular addition, multiplication, inversion and exponentiation. Recently, numerous cryptographic algorithms have been proposed based on modular arithmetic which are scalable, do word based operations and efficient in various aspects. The modular arithmetic modules play a crucial role in the overall performance of the cryptographic processor. Hence, better results can be obtained by designing efficient arithmetic modules such as modular addition, multiplication, exponentiation and squaring. This thesis is organized into three papers, describes the efficient implementation of modular arithmetic units, application of these modules in International Data Encryption Algorithm (IDEA). Second paper describes the IDEA algorithm implementation using the existing techniques and using the proposed efficient modular units. The third paper describes the fault tolerant design of a modular unit which has online self-checking capability --Abstract, page iv

Efficient Computation and FPGA implementation of Fully Homomorphic Encryption with Cloud Computing Significance

Homomorphic Encryption provides unique security solution for cloud computing. It ensures not only that data in cloud have confidentiality but also that data processing by cloud server does not compromise data privacy. The Fully Homomorphic Encryption (FHE) scheme proposed by Lopez-Alt, Tromer, and Vaikuntanathan (LTV), also known as NTRU(Nth degree truncated polynomial ring) based method, is considered one of the most important FHE methods suitable for practical implementation. In this thesis, an efficient algorithm and architecture for LTV Fully Homomorphic Encryption is proposed. Conventional linear feedback shift register (LFSR) structure is expanded and modified for performing the truncated polynomial ring multiplication in LTV scheme in parallel. Novel and efficient modular multiplier, modular adder and modular subtractor are proposed to support high speed processing of LFSR operations. In addition, a family of special moduli are selected for high speed computation of modular operations. Though the area keeps the complexity of O(Nn^2) with no advantage in circuit level. The proposed architecture effectively reduces the time complexity from O(N log N) to linear time, O(N), compared to the best existing works. An FPGA implementation of the proposed architecture for LTV FHE is achieved and demonstrated. An elaborate comparison of the existing methods and the proposed work is presented, which shows the proposed work gains significant speed up over existing works

A Low Complexity Algorithm and Architecture for Systematic Encoding of Hermitian Codes

We present an algorithm for systematic encoding of Hermitian codes. For a Hermitian code defined over GF(q^2), the proposed algorithm achieves a run time complexity of O(q^2) and is suitable for VLSI implementation. The encoder architecture uses as main blocks q varying-rate Reed-Solomon encoders and achieves a space complexity of O(q^2) in terms of finite field multipliers and memory elements.Comment: 5 Pages, Accepted in IEEE International Symposium on Information Theory ISIT 200

Area Efficient Modular Reduction in Hardware for Arbitrary Static Moduli

Modular reduction is a crucial operation in many post-quantum cryptographic schemes, including the Kyber key exchange method or Dilithium signature scheme. However, it can be computationally expensive and pose a performance bottleneck in hardware implementations. To address this issue, we propose a novel approach for computing modular reduction efficiently in hardware for arbitrary static moduli. Unlike other commonly used methods such as Barrett or Montgomery reduction, the method does not require any multiplications. It is not dependent on properties of any particular choice of modulus for good performance and low area consumption. Its major strength lies in its low area consumption, which was reduced by 60% for optimized and up to 90% for generic Barrett implementations for Kyber and Dilithium. Additionally, it is well suited for parallelization and pipelining and scales linearly in hardware resource consumption with increasing operation width. All operations can be performed in the bit-width of the modulus, rather than the size of the number being reduced. This shortens carry chains and allows for faster clocking. Moreover, our method can be executed in constant time, which is essential for cryptography applications where timing attacks can be used to obtain information about the secret key.Comment: 7 pages, 2 figure

Communication over Finite-Chain-Ring Matrix Channels

Though network coding is traditionally performed over finite fields, recent work on nested-lattice-based network coding suggests that, by allowing network coding over certain finite rings, more efficient physical-layer network coding schemes can be constructed. This paper considers the problem of communication over a finite-ring matrix channel $Y = AX + BE$, where $X$ is the channel input, $Y$ is the channel output, $E$ is random error, and $A$ and $B$ are random transfer matrices. Tight capacity results are obtained and simple polynomial-complexity capacity-achieving coding schemes are provided under the assumption that $A$ is uniform over all full-rank matrices and $BE$ is uniform over all rank-$t$ matrices, extending the work of Silva, Kschischang and K\"{o}tter (2010), who handled the case of finite fields. This extension is based on several new results, which may be of independent interest, that generalize concepts and methods from matrices over finite fields to matrices over finite chain rings.Comment: Submitted to IEEE Transactions on Information Theory, April 2013. Revised version submitted in Feb. 2014. Final version submitted in June 201