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One of their last mechanical calculators is the HL-21, an Odhner type machine. [5] Their first calculator with a microprocessor is the Busicom 141-PF. Their entry based calculators, the Busicom LE-120A (Handy-LE) and LE-120S (Handy), [6] were the first to fit in a pocket and also the first calculators to use an LED display.
The size of the input to the algorithm is log 2 n or the number of bits in the binary representation of n. Any element of the order n c for a constant c is exponential in log n. The running time of the number field sieve is super-polynomial but sub-exponential in the size of the input.
The most significant digit is an exception to this: for an n-bit Gray code, the most significant digit follows the pattern 2 n-1 on, 2 n-1 off, which is the same (cyclic) sequence of values as for the second-most significant digit, but shifted forwards 2 n-2 places. The four-bit version of this is shown below:
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In number theory, a branch of mathematics, the special number field sieve (SNFS) is a special-purpose integer factorization algorithm. The general number field sieve (GNFS) was derived from it. The special number field sieve is efficient for integers of the form r e ± s, where r and s are small (for instance Mersenne numbers).
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An important aspect in the study of elliptic curves is devising effective ways of counting points on the curve.There have been several approaches to do so, and the algorithms devised have proved to be useful tools in the study of various fields such as number theory, and more recently in cryptography and Digital Signature Authentication (See elliptic curve cryptography and elliptic curve DSA).