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Pollard rho algorithm for elliptic curves over GF(2n) with negation map, frobenius map and normal basis
Abstract
One of the strongest attacks on ECC is the attack using the Pollard rho method. In this paper, we explain how to speed up the Pollard rho method for elliptic curves over binary fields by using negation and Frobenius maps. We also use basis normal to speed up the squaring in Frobenius map. Experimental result with GF(213) yields a modified Pollard rho which needs far less iterations.
... This study is a research development by [14] and [21]. Research [14] used the Pollard Rho algorithm without negation mapping, while research [21] used negation and Frobenius mapping but applied it to different fields, namely ...
... This study is a research development by [14] and [21]. Research [14] used the Pollard Rho algorithm without negation mapping, while research [21] used negation and Frobenius mapping but applied it to different fields, namely ...
El Gamal encryption was introduced in 1985 and is still commonly used today. Its hardness is based on a discrete logarithm problem defined over the finite abelian cyclic group group chosen in the original paper was but later it was proven that using the group of Elliptic Curve points could significantly reduce the key size required. The modified El Gamal encryption is dubbed its analog version. This analog encryption bases its hardness on Elliptic Curve Discrete Logarithm Problem (ECDLP). One of the fastest attacks in cracking ECDLP is the Pollard Rho algorithm, with the expected number of iterations where is the number of points in the curve. This paper proposes a modification of the Pollard Rho algorithm using a negation map. The experiment was done in El Gamal analog encryption of elliptic curve defined over the field with different values of small digit . The modification was expected to speed up the algorithm by times. The average of speed up in the experiment was 1.9 times.
... Previously we have conducted research related to problems of implementing elliptic curve cryptography based on composite fields [Paryasto et al, 2012] and the implementation of Elliptic Curve Integrated Encryption Scheme [Susantio and Muchtadi, 2016]. In addition, we also studied implementation of modified algorithm Pollard Rho which is basically attack on Elliptic Curve Cryptography [Muchtadi et al, 2013], [Muchtadi et al, 2014], [Muchtadi and Utomo, 2016]). ...
Maritime has special needs for information security, including the protection of classified information. As written in “Cryptography’s Role in Securing the Information Societyâ€: ‘Cryptography provides important capabilities that can help deal with the vulnerabilities of electronic information. Cryptography can help to assure the integrity of data, to authenticate the identity of specific parties, to prevent individuals from plausibly denying that they have signed something, and to preserve the confidentiality of information that may have improperly come into the possession of unauthorized parties [Dam and Lin, 1996]. Elliptic Curve Cryptography (ECC) was first introduced by Neal Koblitz and Victor Miller ([Koblitz1987],[Miller1985]). They independently introduced the elliptic curve to design a public-key cryptography. Compared to other cryptography method, it has several advantages: its arithmetic operations are spesific and can not be predicted, it offers smaller key length for the same security level compared to other method and its operations have many layers and combinations. ECC relies on the security level of the discrete logarithm problem called Elliptic Curve Discrete Logarithm Problem (ECDLP) [Hankerson et al, 2004].
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