Shannon Perfect Secrecy in a Discrete Hilbert Space

The One-time-pad (OTP) was mathematically proven to be perfectly secure by Shannon in 1949. We propose to extend the classical OTP from an n-bit finite field to the entire symmetric group over the finite field. Within this context the symmetric group can be represented by a discrete Hilbert sphere (DHS) over an n-bit computational basis. Unlike the continuous Hilbert space defined over a complex field in quantum computing, a DHS is defined over the finite field GF(2). Within this DHS, the entire symmetric group can be completely described by the complete set of n-bit binary permutation matrices. Encoding of a plaintext can be done by randomly selecting a permutation matrix from the symmetric group to multiply with the computational basis vector associated with the state corresponding to the data to be encoded. Then, the resulting vector is converted to an output state as the ciphertext. The decoding is the same procedure but with the transpose of the pre-shared permutation matrix. We demonstrate that under this extension, the 1-to-1 mapping in the classical OTP is equally likely decoupled in Discrete Hilbert Space. The uncertainty relationship between permutation matrices protects the selected pad, consisting of M permutation matrices (also called Quantum permutation pad, or QPP). QPP not only maintains the perfect secrecy feature of the classical formulation but is also reusable without invalidating the perfect secrecy property. The extended Shannon perfect secrecy is then stated such that the ciphertext C gives absolutely no information about the plaintext P and the pad.Comment: 7 pages, 1 figure, presented and published by QCE202

Secure Key Encapsulation Mechanism with Compact Ciphertext and Public Key from Generalized Srivastava code

Code-based public key cryptosystems have been found to be an interesting option in the area of Post-Quantum Cryptography. In this work, we present a key encapsulation mechanism (KEM) using a parity check matrix of the Generalized Srivastava code as the public key matrix. Generalized Srivastava codes are privileged with the decoding technique of Alternant codes as they belong to the family of Alternant codes. We exploit the dyadic structure of the parity check matrix to reduce the storage of the public key. Our encapsulation leads to a shorter ciphertext as compared to DAGS proposed by Banegas et al. in Journal of Mathematical Cryptology which also uses Generalized Srivastava code. Our KEM provides IND-CCA security in the random oracle model. Also, our scheme can be shown to achieve post-quantum security in the quantum random oracle model

Chaves mais pequenas para criptossistemas de McEliece usando codificadores convolucionais

The arrival of the quantum computing era is a real threat to the confidentiality and integrity of digital communications. So, it is urgent to develop alternative cryptographic techniques that are resilient to quantum computing. This is the goal of pos-quantum cryptography. The code-based cryptosystem called Classical McEliece Cryptosystem remains one of the most promising postquantum alternatives. However, the main drawback of this system is that the public key is much larger than in the other alternatives. In this thesis we study the algebraic properties of this type of cryptosystems and present a new variant that uses a convolutional encoder to mask the so-called Generalized Reed- Solomon code. We conduct a cryptanalysis of this new variant to show that high levels of security can be achieved using significant smaller keys than in the existing variants of the McEliece scheme. We illustrate the advantages of the proposed cryptosystem by presenting several practical examples.A chegada da era da computação quântica é uma ameaça real à confidencialidade e integridade das comunicações digitais. É, por isso, urgente desenvolver técnicas criptográficas alternativas que sejam resilientes à computação quântica. Este é o objetivo da criptografia pós-quântica. O Criptossistema de McEliece continua a ser uma das alternativas pós-quânticas mais promissora, contudo, a sua principal desvantagem é o tamanho da chave pública, uma vez que é muito maior do que o das outras alternativas. Nesta tese estudamos as propriedades algébricas deste tipo de criptossistemas e apresentamos uma nova variante que usa um codificador convolucional para mascarar o código de Generalized Reed-Solomon. Conduzimos uma criptoanálise dessa nova variante para mostrar que altos níveis de segurança podem ser alcançados usando uma chave significativamente menor do que as variantes existentes do esquema de McEliece. Ilustramos, assim, as vantagens do criptossistema proposto apresentando vários exemplos práticos.Programa Doutoral em Matemátic

A Modified Symmetric Key Fully Homomorphic Encryption Scheme Based on Read-Muller Code

Homomorphic encryption became popular and powerful cryptographic primitive for various cloud computing applications. In the recent decades several developments has been made. Few schemes based on coding theory have been proposed but none of them support unlimited operations with security.   We propose a modified Reed-Muller Code based symmetric key fully homomorphic encryption to improve its security by using message expansion technique. Message expansion with prepended random fixed length string provides one-to-many mapping between message and codeword, thus one-to many mapping between plaintext and ciphertext. The proposed scheme supports both (MOD 2) additive and multiplication operations unlimitedly.   We make an effort to prove the security of the scheme under indistinguishability under chosen-plaintext attack (IND-CPA) through a game-based security proof. The security proof gives a mathematical analysis and its complexity of hardness. Also, it presents security analysis against all the known attacks with respect to the message expansion and homomorphic operations

Reinforcing Security and Usability of Crypto-Wallet with Post-Quantum Cryptography and Zero-Knowledge Proof

Crypto-wallets or digital asset wallets are a crucial aspect of managing cryptocurrencies and other digital assets such as NFTs. However, these wallets are not immune to security threats, particularly from the growing risk of quantum computing. The use of traditional public-key cryptography systems in digital asset wallets makes them vulnerable to attacks from quantum computers, which may increase in the future. Moreover, current digital wallets require users to keep track of seed-phrases, which can be challenging and lead to additional security risks. To overcome these challenges, a new algorithm is proposed that uses post-quantum cryptography (PQC) and zero-knowledge proof (ZKP) to enhance the security of digital asset wallets. The research focuses on the use of the Lattice-based Threshold Secret Sharing Scheme (LTSSS), Kyber Algorithm for key generation and ZKP for wallet unlocking, providing a more secure and user-friendly alternative to seed-phrase, brain and multi-sig protocol wallets. This algorithm also includes several innovative security features such as recovery of wallets in case of downtime of the server, and the ability to rekey the private key associated with a specific username-password combination, offering improved security and usability. The incorporation of PQC and ZKP provides a robust and comprehensive framework for securing digital assets in the present and future. This research aims to address the security challenges faced by digital asset wallets and proposes practical solutions to ensure their safety in the era of quantum computing

Energy efficiency analysis of selected public key cryptoschemes

Public key cryptosystems in both classical and post-quantum settings usually involve a lot of computations. The amount as well as the type of computations involved vary among these cryptosystems. As a result, when the computations are performed on processors or devices, they can lead to a wide range of energy consumption. Since a lot of devices implementing these cryptosystems might have a limited source of power or energy, energy consumption by such schemes is an important aspect to be considered. The Diffie-Hellman key exchange is one of the most commonly used technique in the classical setting of public key cryptographic shceme, and elliptic curve based Diffie-Hellman (ECDH) has been in existence for more than three decades. An elliptic curve based post-quantum version of Diffie-Hellman, called supersingular isogeny based Diffie-Hellman (SIDH) was developed in 2011. For computations involved in ECDH and SIDH, elliptic curve points can be represented in various coordinate systems. In this thesis, a comparative analysis of energy consumption is carried out for the affine and projective coordinate based elliptic curve point addition and doubling used in ECDH and SIDH. We also compare the energy consumption of the entire ECDH and SIDH schemes. SIDH is one of the more than sixty algorithms currently being considered by NIST to develop and standardize quantum-resistant public key cryptographic algorithms. In this thesis, we use a holistic approach to provide a comprehensive report on the energy consumption and power usage of the candidate algorithms executed on a 64-bit processor

Theoretical analysis of decoding failure rate of non-binary QC-MDPC codes

In this paper, we study the decoding failure rate (DFR) of non-binary QC-MDPC codes using theoretical tools, extending the results of previous binary QC-MDPC code studies. The theoretical estimates of the DFR are particularly significant for cryptographic applications of QC-MDPC codes. Specifically, in the binary case, it is established that exploiting decoding failures makes it possible to recover the secret key of a QC-MDPC cryptosystem. This implies that to attain the desired security level against adversaries in the CCA2 model, the decoding failure rate must be strictly upper-bounded to be negligibly small. In this paper, we observe that this attack can also be extended to the non--binary case as well, which underscores the importance of DFR estimation. Consequently, we study the guaranteed error-correction capability of non-binary QC-MDPC codes under one-step majority logic (OSML) decoder and provide a theoretical analysis of the 1-iteration parallel symbol flipping decoder and its combination with OSML decoder. Utilizing these results, we estimate the potential public-key sizes for QC-MDPC cryptosystems over $\mathbb{F}_4$ for various security levels. We find that there is no advantage in reducing key sizes when compared to the binary case

Cryptanalysis of Ivanov-Krouk-Zyablov cryptosystem

Recently, F.Ivanov, E.Krouk and V.Zyablov proposed new cryptosystem based of Generalized Reed--Solomon (GRS) codes over field extensions. In their approach, the subfield images of GRS codes are masked by a special transform, so that the resulting public codes are not equivalent to subfield images of GRS code but burst errors still can be decoded. In this paper, we show that the complexity of message-recovery attack on this cryptosystem can be reduced due to using burst errors, and the secret key of Ivanov-Krouk-Zyablov cryptosystem can successfully recovered in polynomial time with a linear-algebra based attack and a square-based attack