• Title/Summary/Keyword: baby-step giant-step algorithm

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AN IMPROVED BABY-STEP-GIANT-STEP METHOD FOR CERTAIN ELLIPTIC CURVES

  • OH BYEONG-KWEON;HA KIL-CHAN;OH JANGHEON
    • Journal of applied mathematics & informatics
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    • v.20 no.1_2
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    • pp.485-489
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    • 2006
  • In this paper, we slightly improve the Baby-step Giant-step for certain elliptic curves. This method gives the running time improvement of $200\%$ in precomputation (Baby-step) and requires half as much storage as the original Baby-step Giant-step method.

Modified Baby-Step Giant-Step Algorithm for Discrete Logarithm (최단 보폭-최장 보폭 이산대수 알고리즘의 변형)

  • Lee, Sang-Un
    • Journal of the Korea Society of Computer and Information
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    • v.18 no.8
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    • pp.87-93
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    • 2013
  • A baby-step giant-step algorithm divides n by n blocks that possess $m={\lceil}\sqrt{n}{\rceil}$ elements, and subsequently computes and stores $a^x$ (mod n) for m elements in the 1st block. It then calculates mod n for m blocks and identifies each of them with those in the 1st block of an identical elemental value. This paper firstly proposes a modified baby-step giant-step algorithm that divides ${\lceil}m/2{\rceil}$ blocks with m elements applying $a^{{\phi}(n)/2}{\equiv}1(mod\;n)$ and $a^x(mod\;n){\equiv}a^{{\phi}(n)+x}$ (mod n) principles. This results in a 50% decrease in the process of the giant-step. It then suggests a reverse baby-step giant step algorithm that performs and saves ${\lceil}m/2{\rceil}$ blocks firstly and computes $a^x$ (mod n) for m elements. The proposed algorithm is found to successfully halve the memory and search time of the baby-step giant step algorithm.

Baby-Step 2k-ary Adult-Step Algorithm for Symmetric-Key Decryption (대칭키 해독을 위한 아기걸음 2k-ary 성인걸음 알고리즘)

  • Lee, Sang-Un
    • The Journal of the Institute of Internet, Broadcasting and Communication
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    • v.15 no.2
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    • pp.23-29
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    • 2015
  • The baby-step giant-step algorithm seeks b in a discrete logarithm problem when a,c,p of $a^b{\equiv}c$(mod p) are already given. It does so by dividing p by m block of $m={\lceil}{\sqrt{p}}{\rceil}$ length and letting one giant walk straight toward $a^0$ with constant m strides in search for b. In this paper, I basically reduce $m={\lceil}{\sqrt{p}}{\rceil}$ to p/l, $a^l$ > p and replace a giant with an adult who is designed to walk straight with constant l strides. I also extend the algorithm to allow $2^k$ adults to walk simultaneously. As a consequence, the proposed algorithm quarters the execution time of the basic adult-walk method when applied to $2^k$, (k=2) in the range of $1{\leq}b{\leq}p-1$. In conclusion, the proposed algorithm greatly shorten the step number of baby-step giant-step.

Reverse Baby-step 2k-ary Adult-step Method for 𝜙((n) Decryption of Asymmetric-key RSA (비대칭키 RSA의 𝜙(n) 해독을 위한 역 아기걸음- 2k-ary 성인걸음법)

  • Lee, Sang-Un
    • The Journal of the Institute of Internet, Broadcasting and Communication
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    • v.14 no.6
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    • pp.25-31
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    • 2014
  • When the public key e and the composite number n=pq are disclosed but not the private key d in an asymmetric-key RSA, message decryption is carried out by obtaining ${\phi}(n)=(p-1)(q-1)=n+1-(p+q)$ and subsequently computing $d=e^{-1}(mod{\phi}(n))$. The most commonly used decryption algorithm is integer factorization of n/p=q or $a^2{\equiv}b^2$(mod n), a=(p+q)/2, b=(q-p)/2. But many of the RSA numbers remain unfactorable. This paper therefore applies baby-step giant-step discrete logarithm and $2^k$-ary modular exponentiation to directly obtain ${\phi}(n)$. The proposed algorithm performs a reverse baby-step and $2^k$-ary adult-step. As a results, it reduces the execution time of basic adult-step to $1/2^k$ times and the memory $m={\lceil}\sqrt{n}{\rceil}$ to l, $a^l$ > n, hence obtaining ${\phi}(n)$ by executing within l times.

Optimization Algorithm for Economic Load Dispatch Problem Using Balance and Swap Method (균형-교환방법을 적용한 경제급전문제 최적화 알고리즘)

  • Lee, Sang-Un
    • The Journal of the Institute of Internet, Broadcasting and Communication
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    • v.15 no.2
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    • pp.255-262
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    • 2015
  • In the absence of a deterministic algorithm for economic load dispatch optimization problem (ELDOP), existing algorithms proposed as solutions are inevitably non-deterministic heuristic algorithms. This paper, therefore, proposes a balance-and-swap algorithm to solve an ELDOP. Firstly, it balances the initial value to ${\Sigma}P_i=P_d$ by subsequently reducing power generation for each adult-step and baby-step and selects the minimum cost-generating method. Subsequently, it selects afresh the minimum cost-generating method after an optimization of the previously selected value with adult-step baby-step swap and giant-step swap methods. Finally, we perform the $P_i{\pm}{\beta}$, (${\beta}=0.1,0.01,0.001,0.0001$) swap. When applied to the 3 most prevalently used economic load dispatch problem data, the proposed algorithm has obtained improved results for two and a result identical to the existing one for the rest. This algorithm thus could be applied to ELDOP for it has proven to consistently yield identical results and to be applicable to all types of data.