Solving the Discrete Logarithm Problem for Ephemeral Keys in Chang and Chang Password Key Exchange Protocol

The first published solution to key distribution problem is due to Diffie–Hellman, which allows two parties that have never communicated earlier, to jointly establish a shared secret key over an insecure channel. In this paper, we propose a new key exchange protocol in a non-commutative semigroup over group ring whose security relies on the hardness of Factorization with Discrete Logarithm Problem (FDLP). We have also provided its security and complexity analysis. We then propose a ElGamal cryptosystem based on FDLP using the group of invertible matrices over group rings.

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Non-Commutative Key Exchange Protocol

10.20944/preprints202105.0174.v1 ◽

2021 ◽

Author(s):

Luis Adrián Lizama-Pérez ◽

José Mauricio López Romero

Keyword(s):

Discrete Logarithm ◽

Matrix Multiplication ◽

Key Exchange ◽

Security Level ◽

Search Problem ◽

Key Exchange Protocol ◽

Secret Communication ◽

Perfect Forward Secrecy ◽

Iot Devices ◽

Commutative Matrix

We introduce a novel key exchange protocol based on non-commutative matrix multiplication defined in $\mathbb{Z}_p^{n \times n}$. The security of our method does not rely on computational problems as integer factorization or discrete logarithm whose difficulty is conjectured. We claim that the unique eavesdropper's opportunity to get the secret/private key is by means of an exhaustive search which is equivalent to the unsorted database search problem. Furthermore, we show that the secret/private keys become indistinguishable to the eavesdropper. Remarkably, to achieve a 512-bit security level, the keys (public/private) are of the same size when matrix multiplication is done over a reduced 8-bit size modulo. Also, we discuss how to achieve key certification and Perfect Forward Secrecy (PFS). Therefore, Lizama's algorithm becomes a promising candidate to establish shared keys and secret communication between (IoT) devices in the quantum era.

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A Novel Provably Secure Key-Agreement Using Secret Subgroup Generator

10.21203/rs.3.rs-474430/v2 ◽

2021 ◽

Author(s):

Abdelhaliem Babiker

Keyword(s):

Key Agreement ◽

Cyclic Subgroup ◽

Discrete Logarithm ◽

Key Exchange ◽

Discrete Logarithm Problem ◽

Secret Key ◽

Shared Secret ◽

Shared Secret Key ◽

Provably Secure

Abstract In this paper, a new key-agreement scheme is proposed and analyzed. In addition to being provably secure in shared secret key indistinguishability model, the scheme has an interesting feature: while using exponentiation over a cyclic subgroup to establish the key-agreement, the generator of that subgroup is hidden to secure the scheme against adversaries that are capable of solving the Discrete Logarithm Problem, which means that the scheme might be candidate as a post-quantum key exchange scheme.

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A Summary of Recent and Old Results on the Security of the Diffie-Hellman Key Exchange Protocol in Finite Groups

Handbook of Research on Secure Multimedia Distribution ◽

10.4018/978-1-60566-262-6.ch010 ◽

2011 ◽

pp. 181-200

Author(s):

Ionut Florescu

Keyword(s):

Discrete Logarithm ◽

Key Exchange ◽

Underlying Structure ◽

Key Exchange Protocol ◽

Statistical Concept ◽

Dimensional Group ◽

Diffie Hellman ◽

Recent Developments ◽

Diffie Hellman Key Exchange ◽

The Relationship

Regarding fundamental protocols in cryptography, the Diffie-Hellman (Diffie and Hellman, 1976) public key exchange protocol is one of the oldest and most widely used in today’s applications. Consequently, many specific cryptographic implementations depend on its security. Typically, an underlying (finite dimensional) group is selected to provide candidates for the key. The study of the security of the exchange as depending on the structure of the underlying group is even today poorly understood, with the most common approaches relying on the security of the Discrete Logarithm problem or on the size of the group. Recent developments bring to attention that the relationship is not necessarily valid and that more research is needed that will relate the underlying structure of the group and the security of the Diffie- Hellman exchange. In this chapter, we describe the problem in detail, we present the relationship with the previously studied Discrete Logarithm and Computational Diffie-Hellman problems, we expose the various concepts of security, and we introduce a new statistical concept specifically designed to serve the assessment of the security of the exchange.

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A UC-Secure Authenticated Contributory Group Key Exchange Protocol Based on Discrete Logarithm

Information Systems Security - Lecture Notes in Computer Science ◽

10.1007/978-3-642-45204-8_29 ◽

2013 ◽

pp. 390-401 ◽

Cited By ~ 2

Author(s):

Yasser Sobhdel ◽

Rasool Jalili

Keyword(s):

Discrete Logarithm ◽

Key Exchange ◽

Key Exchange Protocol ◽

Group Key ◽

Group Key Exchange

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The Diffie–Hellman Key Exchange and the Discrete Logarithm Problem

Springer Undergraduate Mathematics Series - Cryptography ◽

10.1007/978-3-319-94818-8_11 ◽

2018 ◽

pp. 99-112

Author(s):

Simon Rubinstein-Salzedo

Keyword(s):

Discrete Logarithm ◽

Key Exchange ◽

Discrete Logarithm Problem ◽

Diffie Hellman ◽

Diffie Hellman Key Exchange

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Non-Commutative Key Exchange Protocol

10.20944/preprints202103.0716.v1 ◽

2021 ◽

Author(s):

Luis Adrián Lizama-Pérez ◽

José Mauricio López Romero

Keyword(s):

Discrete Logarithm ◽

Matrix Multiplication ◽

Key Exchange ◽

Secret Key ◽

Key Exchange Protocol ◽

Promising Alternative ◽

Private Key ◽

Secret Communication ◽

Perfect Forward Secrecy ◽

Commutative Matrix

We introduce a novel key exchange protocol based on non-commutative matrix multiplication. The security of our method does not rely on computational problems as integer factorization or discrete logarithm whose difficulty is conjectured. We claim that the unique opportunity for the eavesdropper to get the private key is by means of an exhaustive search which is equivalent to searching an unsorted database problem. Therefore, the algorithm becomes a promising candidate to be used in the quantum era to establish shared keys and achieve secret communication. Furthermore, to establish a 256-bit secret key the size of the public key only requires 256 bits while the private key occupies just 384 bits. Matrix multiplications can be done over a reduced 4-bit size modulo. Also, we show that in a generalized method, private numbers become indistinguishable and we discuss how to achieve Perfect Forward Secrecy (PFS). As a consequence, Lizama's protocol becomes a promising alternative for Internet-of-Things (IoT) computational devices in the quantum era.

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