J. Rifa and J. Borrell. Improving the Time Complexity of the Computation of Irreducible and Primitive Polynomials in Finite Fields. Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... In particular, if the characteristic polynomial of an LFSR is primitive 4 , then the LFSR has maximum possible period 2 k Gamma 1 (assuming the initial state of the LFSR is not the zero vector) 20, 19] Primitive polynomials can be generated efficiently using a probabilistic algorithm [22]. It is thus possible to efficiently construct an LFSR which counts from 0 to n Gamma 1 using the minimum possible dlog 2 ne registers (each representing a single bit) Given a state s of an LFSR (and assuming knowledge of the initial state) it is easy to decode the state and determine the ....

....leaking any information about the transition. Below is a complete description of the protocol (here, dlog 2 ne) Key Generation Algorithm G 0 (1 k ) 1. Run G(1 k ) to generate public key pk 0 and secret key sk 0 . 2. Generate a primitive polynomial g 2 Z 2 [x] of degree using [22]. 3. Set r 1 = E pk 0 (1) and r 2 = E pk 0 (0) r = E pk 0 (0) 4. Set s 0 = r 1 ; r ) sk = sk 0 ; g) and pk = pk 0 ; g) Output pk; sk, and s 0 . Transition Algorithm [defined for i 2 Z n ] T ( pk 0 ; g) r 1 ; r ) i) 1. Polynomial g defines (nonzero) f(r ....

J. Rifa and J. Borrell. Improving the Time Complexity of the Computation of Irreducible and Primitive Polynomials in Finite Fields. Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, 1991.

....of f t (x) In fact d is equal to the smallest positive integer k such that tq k # t mod e. If t is relatively prime to e, then d is equal to n. Thus if gcd(t, e) 1 then f t (x) C t (x) Several methods of computing C t (x) are given by Alanen and Knuth [6] Daykin [39] Rifa and Borrell [113] and Thiong Ly [136] We will not discuss these methods here. Instead we will show that the coe#cients of f t (x) are a unique solution of a system of linear equations whose coe#cients are from the coe#cients of the quotient polynomial (x e 1) f(x) Though Theorem 3.3.3 seems not provide any ....

J. Rif a and J. Borrell, "Improving the time complexity of computation of irreducible and primitive polynomials in finite fields", Proc. AAECC-9, Lecture Notes in Computer Science 539, Springer-Verlag, Berlin, 1991, 352-359.

....1 with O(m ( Gamma1) 2 n m 1=2 nL(n) operations in K. By Fact 3, step 2 can be executed using O(mL(m) log m) operations in K. This proves the theorem. 2 Remarks. The idea of using linearly generated sequences over K to compute minimum polynomials for elements in K[ already appears in Rif a Borrell (1991) and Thiong Ly (1989) Our technical contribution is the improved asymptotic running time that results from applying Brent Kung s algorithm together with the transposition principle. We will freely make use of a slightly more general version of Theorem 4 in which the residue class field K[ is ....

J. Rif`a and J. Borrell. Improving the time complexity of the computation of irreducible and primitive polynomials in finite fields. In Proc. AAECC-9, Lecture Notes in Computer Science 539, vol. 12, pp. 352--359, 1991.

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J. Rifa and J. Borrell. Improving the Time Complexity of the Computation of Irreducible and Primitive Polynomials in Finite Fields. Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, 1991.

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