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It is customary to omit a monomial ai x i in the notation of a polynomial whenever ai = 0. Verifying the equality of two polynomials consists of checking whether they have the same coefficients standing at corresponding powers of their independent variables. Let n f (x) = m ai x i , g(x) = i=0 bi x i , m ≤ n. i=0 If n = m, then the condition on the equivalence of two polynomials can be written as follows: f (x) = g(x) ⇔ ai = bi , 0 ≤ i ≤ n. 46 2 Mathematical Foundations of Cryptography We define the sum and the product of two polynomials in the following way: n (ai + bi )x i , f (x) + g(x) = i=0 n+m f (x)g(x) = ck = ck x k , ai bj , 0 ≤ i ≤ n, 0 ≤ j ≤ m.
Before Enigma reached its final form, various models of the machine were produced and deployed. This section presents a description of the origin of these machines and their principles of operation. 1 History of the Enigma There is no single father of Enigma (Fig. 9). Several teams of cryptographers from the Netherlands, Germany, Sweden and the United States worked to develop its ancestors. Enigma’s history dates back to the beginning of World War I, when in 1915 two Dutch officers Theo van Hengel and R.
In this way, the machine performs the afore mentioned permutation of the alphabet. The most famous machine of this type was Enigma, developed in Germany and used before and during the World War II by the Axis powers. 5 Enigma The name Enigma comes from Greek and literally means secret. The designation Enigma was used to name a number of encryption machines with the same root 24 1 Basic Concepts and Historical Overview and use. Before Enigma reached its final form, various models of the machine were produced and deployed.