Optimization and implementation of the number theoretic transform butterfly unit for large integer multiplication

2021 ◽  
Vol 59 ◽  
pp. 102857
Author(s):  
Siliang Hua ◽  
Huiguo Zhang ◽  
Jingya Zhang ◽  
Shuchang Wang
Keyword(s):  
Large Integer ◽  
Integer Multiplication

Karatsuba-ZOT Multiplication Algorithm and its Application in Cryptography

2012 ◽  
Vol 241-244 ◽  
pp. 2417-2423 ◽  
Author(s):  
Shahram Jahani ◽  
Azman Samsudin
Keyword(s):  
Public Key Cryptography ◽  
Public Key ◽  
Large Integer ◽  
Compact Representation ◽  
Number Representation ◽  
The Public ◽  
Public Key Cryptosystems ◽  
Multiplication Algorithm ◽  
Multiplication Operation ◽  
Integer Multiplication

The number theory based cryptography algorithms are the most commonly used public-key cryptosystems. One of the fundamental arithmetic operations for such systems is the large integer multiplication. The efficiency of these cryptosystems is directly related to the efficiency of this large integer multiplication operation. Classical multiplication algorithm and Karatsuba multiplication algorithm, and their hybrid, are among the most popular multiplication algorithms used for this purpose. In this paper, we propose a hybrid of Karatsuba and a classical-based multiplication algorithm, enhanced by a new number representation system. The new number representation, known as "Big-Digits”, is used to carry out the sub-multiplication operation in the new multiplication algorithm. Big-Digits has a compact representation with lower Hamming weight. As the result, the number of sub-multiplication operations for the multiplication algorithm that is based on the Big-Digits representation is significantly reduced. Our results show that the proposed multiplication algorithm is significantly faster than the classical, Karasuba and the hybrid of Karatsuba-Classical multiplication algorithms within the implementation domain of the public-key cryptography.

Computing Interval Discrete Logarithm Problem with Restricted Jump Method

2020 ◽  
Vol 177 (2) ◽  
pp. 189-201
Author(s):  
Bin Qi ◽  
Jie Ma ◽  
Kewei Lv
Keyword(s):  
Cyclic Group ◽  
Discrete Logarithm ◽  
Discrete Logarithm Problem ◽  
Large Integer ◽  
Finite Cyclic Group ◽  
Speed Up ◽  
Integer Multiplication ◽  
Intermediate Value ◽  
Improved Algorithm

The interval discrete logarithm problem(IDLP) is to find a solution n such that gn = h in a finite cyclic group G = 〈g〉, where h ∈ G and n belongs to a given interval. To accelerate solving IDLP, a restricted jump method is given to speed up Pollard’s kangaroo algorithm in this paper. Since the Pollard’ kangaroo-like method need to compute the intermediate value during every iteration, the restricted jump method gives another way to reuse the intermediate value so that each iteration is speeded up at least 10 times. Actually, there are some variants of kangaroo method pre-compute the intermediate value and reuse the pre-computed value in each iteration. Different from the pre-compute method that reuse the pre-computed value, the restricted jump method reuse the value naturally arised in pervious iteration, so that the improved algorithm not only avoids precomputation, but also speeds up the efficiency of each iteration. So only two or three large integer multiplications are needed in each iteration of the restricted jump method. And the average large integer multiplication times is (1:633 + o(1)) N in restricted jump method, which is verified in the experiment.

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