scholarly journals Quantum Algorithm for Attacking RSA Based on Fourier Transform and Fixed-Point

2021 ◽  
Vol 26 (6) ◽  
pp. 489-494
Author(s):  
Yahui WANG ◽  
Huanguo ZHANG
Keyword(s):  
Fixed Point ◽  
Polynomial Time ◽  
Multiplicative Group ◽  
Success Probability ◽  
Quantum Algorithm ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Rsa Cryptosystem ◽  
Shor's Algorithm ◽  
Quantum Polynomial

Shor in 1994 proposed a quantum polynomial-time algorithm for finding the order r of an element a in the multiplicative group Zn*, which can be used to factor the integer n by computing [see formula in PDF]and hence break the famous RSA cryptosystem. However, the order r must be even. This restriction can be removed. So in this paper, we propose a quantum polynomial-time fixed-point attack for directly recovering the RSA plaintext M from the ciphertext C, without explicitly factoring the modulus n. Compared to Shor’s algorithm, the order r of the fixed-point C for RSA(e, n) satisfying [see formula in PDF] does not need to be even. Moreover, the success probability of the new algorithm is at least [see formula in PDF] and higher than that of Shor’s algorithm, though the time complexity for both algorithms is about the same.

An O(m^2)-depth quantum algorithm for the elliptic curve discrete logarithm problem over GF(2^m)

2009 ◽  
Vol 9 (7&8) ◽  
pp. 610-621
Author(s):  
D. Maslov ◽  
J. Mathew ◽  
D. Cheung ◽  
D.K. Pradhan
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Nearest Neighbor ◽  
Discrete Logarithm ◽  
Quantum Algorithm ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Discrete Logarithm Problem ◽  
Quantum Polynomial

We consider a quantum polynomial-time algorithm which solves the discrete logarithm problem for points on elliptic curves over $GF(2^m)$. We improve over earlier algorithms by constructing an efficient circuit for multiplying elements of binary finite fields and by representing elliptic curve points using a technique based on projective coordinates. The depth of our proposed implementation, executable in the Linear Nearest Neighbor (LNN) architecture, is $O(m^2)$, which is an improvement over the previous bound of $O(m^3)$ derived assuming no architectural restrictions.

Learning Importance of Preferences

10.29007/v68w ◽  
2018 ◽  
Author(s):  
Ying Zhu ◽  
Mirek Truszczynski
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Scoring Rules ◽  
Np Hard ◽  
Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

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