[Paper Review] Restructuring of Discrete Logarithm Problem and ElGamal Cryptosystem by Using the Power Fibonacci Sequence Module M

Çağla Özyılmaz, Ayşe Nallı|arXiv (Cornell University)|Jul 16, 2018

Coding theory and cryptography2 references3 citations

TL;DR

This paper proposes a novel variant of the ElGamal cryptosystem based on the Discrete Logarithm Problem over power Fibonacci sequences modulo a composite number $ m $. By leveraging the cyclic and multiplicative group structure of these sequences, the authors reformulate the discrete logarithm problem to work over composite moduli, enabling a new cryptosystem that enhances security and expands the plaintext space beyond traditional prime-based ElGamal. The new system demonstrates superior cryptographic advantages due to increased computational difficulty for adversaries.

ABSTRACT

In this paper, we have studied on adapting to asymmetric cryptography power Fibonacci sequence module m . To do this, We have restructed Discreate Logarithm Problem which is one of mathematical difficult problems by using power Fibonacci sequence module m and by means of this sequences, we have made the mathematical difficult problem which is used only in prime modules is also useful for composite modules. Then we have constructed cryptographic system based on this more difficult problem which we have rearranged. Hence, we have obtained a new cryptosystem as ElGamal Cryptosystem. Lastly, we have compared that ElGamal Cryptosystem and a new cryptosystem which we constitute in terms of cryptography and we have obtained that a new cryptosystem is more advantageuos than ElGamal Cryptosystem.

Motivation & Objective

• To extend the applicability of the discrete logarithm problem from prime moduli to composite moduli using power Fibonacci sequences.
• To construct a new public-key cryptosystem based on the reformulated discrete logarithm problem over power Fibonacci sequences modulo $ m $.
• To improve upon the ElGamal cryptosystem by increasing the difficulty of cryptanalysis and expanding the plaintext space.
• To demonstrate that the new system offers stronger cryptographic security compared to standard ElGamal, especially under composite modulus conditions.

Proposed method

• Define power Fibonacci sequences modulo $ m $, showing they form cyclic multiplicative subgroups of $ \mathbb{Z}_m^* $.
• Reformulate the discrete logarithm problem as finding $ a $ such that $ \alpha^a \equiv \beta \pmod{m} $, where $ \alpha $ is a generator of a subgroup derived from the power Fibonacci sequence modulo $ m $.
• Construct a new ElGamal-like cryptosystem using $ \alpha $, $ \beta = \alpha^a \mod m $, and a random $ k $ to compute $ y_1 = \alpha^k \mod m $, $ y_2 = x \cdot \beta^k \mod m $.
• Enable decryption via $ x = y_2 \cdot (y_1^a)^{-1} \mod m $, using the private key $ a $.
• Use composite $ m $, such as $ m = 1045 $, with $ \alpha $ as a primitive element in a subgroup of $ \mathbb{Z}_m^* $, to increase the size of the plaintext space.
• Demonstrate the system with a concrete example using $ m = 1045 $, $ \alpha = 338 $, $ a = 547 $, $ x = 1001 $, and $ k = 162 $, resulting in $ (y_1, y_2) = (229, 374) $, which correctly decrypts back to 1001.

Experimental results

Research questions

• RQ1 Can the discrete logarithm problem be reformulated to work over composite moduli using power Fibonacci sequences?
• RQ2 Does constructing a cryptosystem based on power Fibonacci sequences modulo $ m $ yield a more secure alternative to standard ElGamal?
• RQ3 How does the plaintext space of the new system compare to that of classical ElGamal in terms of size and security?
• RQ4 What is the impact of using composite $ m $, especially with $ m = 5p_1^{e_1}p_2^{e_2}\cdots $, on the security and efficiency of the new cryptosystem?
• RQ5 Can the power Fibonacci sequence modulo $ m $ generate cyclic multiplicative subgroups of $ \mathbb{Z}_m^* $ with order dividing $ \varphi(m) $?

Key findings

• The power Fibonacci sequence modulo $ m $ forms a cyclic multiplicative subgroup of $ \mathbb{Z}_m^* $, with order dividing $ \varphi(m) $, enabling algebraic structure for cryptographic use.
• The discrete logarithm problem over power Fibonacci sequences modulo $ m $ is reformulated to allow composite moduli, extending the applicability of the problem beyond prime fields.
• The new cryptosystem, named ElGamal with Power Fibonacci Sequence modulo $ m $, supports plaintexts up to $ m-1 $, whereas standard ElGamal is limited to $ p-1 $, significantly increasing the plaintext space.
• For the same security level, the new system increases the number of possible values an attacker must try, enhancing resistance to brute-force attacks.
• The example with $ m = 1045 $, $ \alpha = 338 $, $ a = 547 $, $ x = 1001 $, and $ k = 162 $ successfully encrypts to $ (229, 374) $, which decrypts correctly to 1001, validating the scheme.
• The new system is more advantageous than standard ElGamal due to the increased complexity of solving the discrete logarithm problem in the composite modulus setting using power Fibonacci sequences.