Addition-chain Exponentiation Algorithm
M r med L r μ r the arithmetric average of numbers which have a shortest addition chain of length r. Addition chain-based exponentiation methods are used to find a sequence of elements in a given multiplicative group G such that the first number is 1 G and the last is s for some arbitrary s2G and 2Z.
The largest number which has an addition chain of length r.
Addition-chain exponentiation algorithm. For an RSA implemented with the Chinese Remainder Theorem method one faulty execution suffices to factorize the public modulus and. Where I set the initial upper limit at the number of multiplications it would take the normal binary squaremultiply algorithm and it does give the right answers but Im not satisfied with the running time it can take several seconds for numbers around 200. The question of finding the Minimal Addition Chain MAC for needed for Addition chain exponentiation seems to be NP-complete.
If aiai-1 ak is called a star step. A Brauer chain aka star-chain is an addition chain where each new element is formed as the sum of the previous element and some element possibly the same. Ask Question Asked 9 years 3 months ago.
1 the first number is one 2 every element is the sum of two earlier elements and 3 the given number occurs at the end of the sequence. It works by creating a shortest addition chain that generates the desired exponent. A satisfying bk-.
M r the median of numbers which have a shortest addition chain of length r. An addition-chain of length 1 for an integer n is a sequence of integers ao a1. It is extremely popular.
In mathematics and computer science optimal addition-chain exponentiation is a method of exponentiation by positive integer powers that requires a minimal number of multiplications. Fault Analysis is a powerful cryptanalytic technique that enables to break cryptographic implementations embedded in portable devices more efficiently than any other technique. More generally addition-chain exponentiation may also refer to exponentiation by non-minimal addition chains constructed by a variety of algorithms since a shortest addition chain is very difficult to find.
An addition-chain is used to represent the sequence which is defined as follows. C r min L r 2r. Genetic Algorithm Modular Multiplication Cipher Text Modular Exponentiation Addition Chain These keywords were added by machine and not by the authors.
The Euclidean method was first introduced in Efficient exponentiation using precomputation and vector addition chains by PD Rooij. Of the modular exponentiation in advance because E is known previously. So finding an optimal addition chain amounts to finding a chain with minimal small steps.
Brauers algorithm is often called the left-to-right 2k-ary method or simply 2k-ary method. It uses the fact that this exponentiation algorithm works with a pair of intermediate variables a 0a 1 storing values of the form mαmα1. For any given target the number of large steps is fixed by n.
The shortest addition-chain algorithm requires no more multiplications than binary exponentiation and usually less. An addition chain exponentiation may require an important amoun t of memory to store the intermediate powers required for the computation of subsequen t powers. CiteSeerX - Document Details Isaac Councill Lee Giles Pradeep Teregowda.
2r max L r holds trivally. Again for step i we say that. If a fault is injected during the computation the coherence between a.
At the end of the exponentiation the pair a 0a 1 equals md1md and the consistency of the computation can be verified by checking wether a 0 m equals a 1. The algorithm uses wh-2 multiplications and w1 elements must be stored to compute x n see. Chain found by the algorithm is an optimal addition chain and it does not ne ed too much space.
Each exponentiation in the chain can be evaluated by multiplying two of the earlier exponentiation results. This process is experimental and the keywords may be updated as the learning algorithm improves. Given a natural number e an addition chain AC for e is a finite sequence of numbers having three properties.
In practice therefore shortest addition-chain exponentiation is primarily used for small fixed exponents for which a shortest chain can be precomputed and is not too large. If ai2ai-1 is called a doubling. The variable y is multiplied h-1 times with the initial u h-2 times with the next highest powers etc.
As such it would be nice to have a list for the small powers already computed. The best method for exponentiation depends strongly on the group being used the hardware the. λ ai λ ai-1 and we call this a small step.
Even given a shortest chain addition-chain exponentiation requires more memory than the binary method because it must potentially store many previous exponents from the chain simultaneously. Brauer chain is a sac for n 12509.
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