The idea of RSA challenges [5] has been set before with lower-strength ciphers to encourage researcher with computational number theory and practical difficulty of factoring

large integers. The RSA challenges were held by the RSA laboratories.

For every sufficiently

large integer n and for every constant k [greater than or equal to] 3 there exists a c = c(n, k) such that 1/5 < c < 1/4 and the remainder r of the division of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]is positive and satisfies r = [omicron]n).

There are the three possibilities: random generation of values 0 and 1 when the probability for the appearance of each value is 50% in every bit of test pattern; random generation of

large integer values, which then are split into the sequences of 0's and 1 's according to the rules of the conversion of the decimal number to the binary number; anti-random generation when the presence of the earlier generated patterns is taken into account.

The author wrote function multOneDigit to multiply a positive,

large integer by a single digit and function shiftLeft to left-shift all digits in a

large integer by an arbitrary number of places in the array Digit.

In particular, we propose a DPF-based

large integer representation and adapt the Montgomery multiplication algorithm to it.

During definition, we choose a

large integer N as in the previous section.

Given a very

large integer (say, several hundred bits), how would you test it for primality?

Suppose the contrary, then [[gamma].sub.n] = p/q for all sufficiently

large integers n.

For instance, Lehmer's algorithm reduces the

large integers to common base (b) with fast convergence rate at complexity of O (n/log (n)) which make it faster than other algorithms.

One can always fit the constraint using very

large integers! This possibility is eliminated when the total angular momentum is known.

Finally, the framework proposed reduced the pitfalls by demonstating each established rule with the help of their recursive applications on

large integers.

This point is illustrated by the progression from the RSA algorithm that relies on factoring very

large integers. Next came work by Diffie, Heilman, and Merkle that involves computing discrete logarithms.

Shor (1994) developed a polynomial time algorithm for factoring

large integers. According to Williams and Clearwater (1998), it is not known if there is a classical algorithm for factoring

large integers efficiently, but the best algorithms published thus far are super-polynomial.

This led to many lively discussions regarding the generation of arbitrarily

large integers, and we considered both practical and theoretical aspects of this problem.

The RSA cryptosystem - introduced by Rivest, Shamir, and Adlement in 1977 - relies for its security on the difficulty of working out the factors dividing

large integers. Though it has performed well until now, the level of protection it provides has been eroded by constant efforts to develop more efficient methods for breaking it.