A Continued Fraction-Hyperbola based Attack on RSA cryptosystem

04/08/2023

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by   Gilda Rech Bansimba, et al.

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In this paper we present new arithmetical and algebraic results following the work of Babindamana and al. on hyperbolas and describe from the new results an approach to attacking a RSA-type modulus based on continued fractions, independent and not bounded by the size of the private key d nor public exponent e compared to Wiener's attack. When successful, this attack is bounded by 𝒪( blogα_j4log(α_i3+α_j3)) with b=10^y, α_i3+α_j3 a non trivial factor of n and α_j4 such that (n+1)/(n-1)=α_i4/α_j4. The primary goal of this attack is to find a point X_α=(-α_3, α_3+1 ) ∈ℤ^2_⋆ that satisfies ⟨ X_α_3, P_3⟩ =0 from a convergent of α_i4/α_j4+δ, with P_3∈ℬ_n(x, y)_|_x≥ 4n. We finally present some experimental examples. We believe these results constitute a new direction in RSA Cryptanalysis using continued fractions.