Some properties of the output sequences of combined generator over finite fields.pdf Introduction The randomness is an important property in the cryptographic scheme. One of the components that ensure this property is the random sequence that is built by generators. The elements of random sequences can be used as initialization vectors, in cyclic codes, as keys in block cipher, and in stream cipher. The combined generator presents one class of generators that are used to obtain pseudorandom sequence. Examples of its use are stream ciphers: A.1 of standard GSM [1], Grain, Trivium [2]. The most results belong to generators over the field GF(2) [1, 3, 4]. In this work, we analyze the following characteristics for combined generator: period of the output sequence and distribution elements in output sequence over finite field. Let P = GF(q) be a finite field with q elements, F1(x),..., Fk(x) be polynomials with coefficients in P of degrees m1,... , mk respectively. We assume that F1(x),..., Fk(x) are primitive polynomials [5], also gcd(mi, mj) = 1 for each i = j. For each function ^ • Pk^P, we consider the combined generator [1, 6, 7] with the output sequence v(i) = ^(ui(i),u2(i),... ,uk(i)), i ^ 0, where Uj is a linear recurring sequence over P with minimal polynomial Fj (x). 1. Period In [6] the general bounds for the period of combined generator and the exact equality in the case GF(2) is presented. In this work, we give bounds for the period of a given generator for one class of function over any finite field and show, how this period can be calculated. Theorem 1. If ^ has the form k ^(x1, . . . , xk) /U /U ciii2...isxiixi2 . . . xis , s=1 1^ii

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