Analysis on a generalized algorithm for the strong discrete logarithm problem with auxiliary inputs

Kim, Minkyu; Cheon, Jung Hee; Lee, In-Sok

doi:10.1090/s0025-5718-2014-02813-5

Cited by 7 publications

(11 citation statements)

References 17 publications

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“…However, the recent result of Kim et al [14] shows that the complexity of the several generalizations in the case where d | Φ k (p) for k 3 [7,20] is always greater than p 1/2 . Therefore, we need to consider a different approach to solving the DLPwAI.…”

Section: Algorithm Descriptionmentioning

confidence: 99%

“…Subsequently, several generalizations of this algorithm have attempted to solve the problem when d is a divisor of Φ k (p) for the kth cyclotomic polynomial Φ k (x) [7,14,20]. Satoh [20] generalized the algorithm, using the embedding of α ∈ F p into the general linear group GL k (F p ).…”

Section: The Discrete Logarithm Problem With Auxiliary Inputsmentioning

confidence: 99%

“…However, its complexity for k 3 was not well understood. More recently, Kim, Cheon and Lee [14] noticed that Satoh's generalization is essentially the same as the embedding j. h. cheon and t. kim of F p into F p k , and clarified the complexity of the algorithm. Unfortunately, their result suggests that the complexity of this generalization is not faster than that of current square root complexity algorithms, such as Pollard's rho algorithm [19] for k 3.…”

Section: The Discrete Logarithm Problem With Auxiliary Inputsmentioning

confidence: 99%

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A new approach to the discrete logarithm problem with auxiliary inputs

Cheon

Kim²

2016

LMS J. Comput. Math.

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The aim of the discrete logarithm problem with auxiliary inputs is to solve for α, given the elements g, g α , . . . , g group exponentiations (i = 1 or 1/2 depending on the sign). There have been several attempts to generalize this algorithm to the case of Φ k (p) where k 3. However, it has been shown by Kim, Cheon and Lee that a better complexity cannot be achieved than that of the usual square root algorithms.We propose a new algorithm for solving the DLPwAI. We show that this algorithm has a running time of O( p/τ f + d) group exponentiations, where τ f is the number of absolutely irreducible factors of f (x) − f (y). We note that this number is always smaller than O(p 1/2 ). In addition, we present an analysis of a non-uniform birthday problem.

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Section: Algorithm Descriptionmentioning

confidence: 99%

Section: The Discrete Logarithm Problem With Auxiliary Inputsmentioning

confidence: 99%

Section: The Discrete Logarithm Problem With Auxiliary Inputsmentioning

confidence: 99%

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A new approach to the discrete logarithm problem with auxiliary inputs

Cheon

Kim²

2016

LMS J. Comput. Math.

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“…However, the efficiency of the algorithm was not well-studied. Kim [11,12] studied Satoh's generalization of the +1 algorithm for solving the DLP-wAI. The result showed that the complexity of Satoh's algorithm was not faster than Cheon's algorithm when | ( ) and ≥ 3.…”

Section: Mathematical Problems In Engineeringmentioning

confidence: 99%

Research on Attacking a Special Elliptic Curve Discrete Logarithm Problem

Weng¹,

Dou²,

2016

Mathematical Problems in Engineering

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Cheon first proposed a novel algorithm for solving discrete logarithm problem with auxiliary inputs. Given some points , , 2 , . . . , ∈ G, an attacker can solve the secret key efficiently. In this paper, we propose a new algorithm to solve another form of elliptic curve discrete logarithm problem with auxiliary inputs. We show that if some points , , , 2 , 3 , . . . , ( )−1 ∈ G and a multiplicative cyclic group = ⟨ ⟩ are given, where is a prime, ( ) is the order of . The secret key ∈ F * can be solved in O(√( − 1)/ + ) group operations by using O(√( − 1)/ ) storage.

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“…However, its complexity for k ≥ 3 was not clearly understood. Recently, Kim et al [24] realized that Satoh's generalization is essentially the same as the embedding of F p into F p k , and clarified the complexity of the algorithm. Unfortunately, their result suggests that, in most cases, the complexity of this generalization is not faster than the current square-root complexity algorithm, such as Pollard's rho algorithm [31], for k ≥ 3.…”

Section: Introductionmentioning

confidence: 99%

The discrete logarithm problem with auxiliary inputs

Cheon¹,

Kim²,

Song³

2014

Algebraic Curves and Finite Fields

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The Discrete Logarithm Problem (DLP) is a classical hard problem in computational number theory, and forms the basis of many cryptographic schemes. The DLP involves finding α for the given elements g and g α of the cyclic group G = g of finite order n. Recently, many variants of the DLP have been used to ensure the security of pairing-based cryptosystems, such as ID-based encryption, broadcast encryption, and short signatures. These cryptosystems provide various functionalities, but their underlying problems are not well understood. A generalization of these variants of DLP, called the Discrete Logarithm Problem with Auxiliary Inputs (DLPwAI), aims to find α for some given g, g α , . . . , g α d . This survey article first recalls several well-known solutions of the original DLP, and mainly focuses on recent attempts to solve the DLPwAI. Research into the DLPwAI started with Cheon's p ± 1 algorithms [11] at EUROCRYPT '06, which use the embedding of the discrete logarithm into the extension of the finite field. Later, Satoh [34] and Kim et al. [24] tried to generalize Cheon's algorithm to the case of Φ k (p) for k ≥ 3, where Φ k (·) is the k-th cyclotomic polynomial. However, Kim et al. found that this generalization of Cheon's algorithm cannot be better than the usual square-root complexity algorithms, such as Pollard's rho algorithm, when k ≥ 3.We also introduce a recent result by Cheon and Kim [25] that reduces the DLPwAI to the problem of finding a polynomial of degree d with a small value set. Finally, we present a generalized version of the DLPwAI introduced by Cheon, Kim, and Song [15], with an algorithm for this problem, even when neither p + 1 or p − 1 has an appropriate small divisor.

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Analysis on a generalized algorithm for the strong discrete logarithm problem with auxiliary inputs

Cited by 7 publications

References 17 publications

A new approach to the discrete logarithm problem with auxiliary inputs

A new approach to the discrete logarithm problem with auxiliary inputs

Research on Attacking a Special Elliptic Curve Discrete Logarithm Problem

The discrete logarithm problem with auxiliary inputs

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