I have read some papers relating to the Number Field Sieve, but I could not figure out how this algorithm helps in Logjam, or even what is meant by the number field. What is this? What is meant by factorization?

I am new to this topic and need a very simple description for the above questions...


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Factorization is in mathematics the process by which an algebraic expression is converted from a sum to a factor. The terms involved in the product are called factors and when multiplied together they give the original expression. The reverse process is called analysis. Usually factorization is used to find the greatest common divisor (gcd). and of the Lowest Common Multiple (lcm). polynomials, for simplifying fractional expressions, for adding and subtracting fractional expressions and for solving quadratic or higher equations. There are two basic kinds of factorization, the factorization of natural numbers (into prime factors) and the factorization of polynomials.

Many public key cryptographic schemes are based on the fact that it is computationally difficult to factor large integers. The fastest, and at the same time the most complex, classical algorithm known to date for factoring integers longer than 110 decimal places is the General Number Field Sieve (GNFS). This algorithm is based of many years of research, during which increasingly faster algorithms were produced to arrive at the GNFS algorithm so far.

The trust problem is fundamental to cryptography but it is not the only vulnerability of the DH algorithm. Applying of the General Number Field Sieve (GNFS) algorithm, to the discrete logarithm problem which uses four (4) computational steps, an attacker need only to perform the last step, which is much less computationally expensive from the first three steps. Logjam attack took the advantages of this problem to compromise a variety of Internet services that allowed the use of groups whose order was a 512-bit prime, called the extraction degree.


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