On binary signed digit representations of integers

Elliptic curve cryptosystems have become increasingly popular due to their efficiency and the small size of the keys they use. Particularly, the anomalous curves introduced by Koblitz allow a complex representation of the keys, denoted τ NAF, that make the computations over these curves more efficient. In this report, we propose an efficient method for randomizing a τ NAF to produce different equivalent representations of the same key to the same complex base τ . We prove that the average Hamming density of the resulting representations is 0.5. We identify the pattern of the τ NAFs yielding the maximum number of representations and the formula governing this number. We also present deterministic methods to compute the average and the exact number of possible representations of a τ NAF.

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“…It is interesting to note the similarity of the results obtained here to those obtained for the BSD representation of integers [3].…”

Section: Discussionsupporting

confidence: 81%

On τ-adic representations of integers

Ebeid

Hasan

2007

Des. Codes Cryptogr.

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“…In 2007, Ebeid and Hasan [5] proposed an algorithm to generate all possible signed digit representation of any integer k.…”

Section: Preliminariesmentioning

confidence: 99%

High Performance Methods of Elliptic Curve Scalar Multiplication

Saffar¹,

Said²

2014

IJCA

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Elliptic curve scalar multiplication is the operation of successively adding a point along an elliptic curve to itself k times. It is used in elliptic curve cryptography (ECC) as a means of producing a trapdoor function. In this paper, algorithms to compute the elliptic curve scalar multiplication using a special form for integers will introduce, and then two types of signed digit representation will use. The signed digit form of the scalar is calculated by many types of algorithms such as binary , non adjacent form and direct recoding. The results indicate that the proposed methods perform better to compute the scalar multiplication on elliptic curves and it is more efficient than the existing methods.

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“…For example, this technique is used in elliptic curve arithmetic because the computation of the additive inverse (therefore the computation of a subtraction) is costless. The reader could refer to [10,16] for details.…”

Section: Examplementioning

confidence: 99%

A matrix approach for FCSR automata

2011

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LFSRs are primitives widely used in information theory, coding theory and cryptography. However since 2002, they have faced algebraic attacks. To avoid this kind of attacks, FCSRs have been proposed as an alternative in [2][3][4]. In this paper, we first give a general representation of 2-adic automata using a traditional matrix representation. We then explore the special case of binary and ternary automata. We also study the complexity in terms of memory to implement such automata. Finally, we expose some proposed FCSR constructions for hardware and software oriented stream ciphers.

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On binary signed digit representations of integers

Cited by 19 publications

References 11 publications

On τ-adic representations of integers

On τ-adic representations of integers

High Performance Methods of Elliptic Curve Scalar Multiplication

A matrix approach for FCSR automata

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