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10 results for “Primitive Polynomials”
BRAIN Journal-A Synoptic of Software Implementation for Shift Registers Based on 16th Degree Primitive Polynomials-Figure 4. Scheme for the polynomial with [16, 14, 13, 11] tap sequence
<p>A simulation program for the functioning on LFSR of the 16th degree for the Galois implementation was developed. In the following example an analysis for the 14 selected primitive polynomials will be presented. A list with the positions which will influence the future state is called tap sequence.</p> <p>This sequence can be represented by a polynomial mod 2, only with coefficients 1 and 0, called Feedback Polynomial or Characteristic Polynomial. For the above scheme this polynomial is: P(X)= X^16+X^14+X^13+X^11+1</p>
BRAIN Journal-A Synoptic of Software Implementation for Shift Registers Based on 16th Degree Primitive Polynomials-Figure 3. Galois implementation
<p>In Galois implementation there is a Shift Register, whose content is modified each step at a binary value sent to the output. In Galois configuration the single shifted out bit is XOR ed with several bits in the shift register and in conventional configuration each new bit input to the shift register is the XOR of several bits in the register.</p>
BRAIN Journal-A Synoptic of Software Implementation for Shift Registers Based on 16th Degree Primitive Polynomials-Figure 2. Fibonacci implementation
<p>A LFSR can be represented as a polynomial of variable x referred to as the generator polynomial or the characteristic polynomial. The input bit is given from a linear function of the initial status for a special shift register called Linear Feedback Shift Register (LFSR). The initial value of the register is called seed and the produced sequence is completely determined by the initial status. Because the register has a finite number of possible statuses, after a period the sequence will be repeated. If the feedback function is very well chosen, the produced sequence will be random and the cycle will be very long, called by Golomb (1967) maximum lengths shift register sequences. Goresky and Klapper (2004) show two possibilities to implement a LFSR: • Fibonacci Form • Galois. </p>
BRAIN Journal-A Synoptic of Software Implementation for Shift Registers Based on 16th Degree Primitive Polynomials-Figure 1. Basis scheme for a Feedback Shift Register
<p>Every LFSR works by taking the XOR of the selected bits in its internal state and any LFSR containing all zero bits will never move to any other state, so one possible state must be excluded from any cycle. A LFSR is composed of memory cells connected together as a shift register with linear feedback. In digital circuits a shift register is formed by flip-flops and EXOR gates chained together with a synchronous clock. Shift registers are a form of sequential logic like counters. Always the shift registers produce a discrete delay of a digital signal or waveform. Considering that a shift register has n stages, the waveform is delayed by n discrete clock times. Usually the naming of the shift register follows a type of convention shown normally in digital logic, with the least significant bit on the left. According to the communication protocol, the signals will be addressed, not the registers. There are n+1 signals for each n-bit register. Always the next state of an LFSR is uniquely determined from the previous one by the feedback network. Any LFSR will generate a sequence of different states starting with the initial one, called seed. A feedback shift register is composed of: - a shift register - a feedback function. </p>
BRAIN Journal-A Synoptic of Software Implementation for Shift Registers Based on 16th Degree Primitive Polynomials-Figure 10. Graphic containing the results for 1000 bits
<p>The distribution obtained depending on the length of the input string shows that time depends on the input length, but for lengths even closer together, the times are also close (this can be seen in Figure 8 for 20 bits inputs). Time does not change so much depending on which of the 14 different 16th degree primitive polynomials has been used. </p>
BRAIN Journal-A Synoptic of Software Implementation for Shift Registers Based on 16th Degree Primitive Polynomials-Figure 9. Graphic containing the results for 1000 bits
<p>The next two graphics show the obtained results from the execution of the main program for each of the 14 degrees, 16th primitive polynomials for three different situations depending on the lengths of the entrance data polynomial. The lengths of the input polynomials were 20. 30. 40, 50, 100 and 1000 bits. The maximum number of sequences is 216-1(Solomon, 1967). </p>
BRAIN Journal-A Synoptic of Software Implementation for Shift Registers Based on 16th Degree Primitive Polynomials-Figure 8. Graphic containing the results for 20 bits
<p>The next two graphics show the obtained results from the execution of the main program for each of the 14 degrees, 16th primitive polynomials for three different situations depending on the lengths of the entrance data polynomial. The lengths of the input polynomials were 20. 30. 40, 50, 100 and 1000 bits. The maximum number of sequences is 216-1(Solomon, 1967). </p>
BRAIN Journal-A Synoptic of Software Implementation for Shift Registers Based on 16th Degree Primitive Polynomials-Figure 7. Ring Implementation for the Polynomial X6+X^12+X^3+X+1
<p>VlăduŃiu and Crişan (1989) show three types of schemes for a 4th degree polynomial. Similar to it, there were developed the three different implementations for the Primitive Polynomial X^16+X^12+X^3+X+1. It can be specified that these schemes are according to the well-known Galois Form, Fibonacci representation and some other Forms that are rarely used, called Ring Implementation. All of these Implementations have the same function, because they describe a linear feedback shift register. In the experimental work there has been analyzed the behavior of 14 Primitive Polynomial degrees, 16 randomly selected. For each of these polynomials there has been developed the simulation of the specific functioning with a program. Because the goal of this experimental work was to compare the different obtained results, a few rows of input data of a different length have been selected. </p>
BRAIN Journal-A Synoptic of Software Implementation for Shift Registers Based on 16th Degree Primitive Polynomials-Figure 5. Galois Implementation for the Polynomial X^16+X^12+X^3+X+1
<p>VlăduŃiu and Crişan (1989) show three types of schemes for a 4th degree polynomial. Similar to it, there were developed the three different implementations for the Primitive Polynomial X^16+X^12+X^3+X+1. It can be specified that these schemes are according to the well-known Galois Form, Fibonacci representation and some other Forms that are rarely used, called Ring Implementation. All of these Implementations have the same function, because they describe a linear feedback shift register. In the experimental work there has been analyzed the behavior of 14 Primitive Polynomial degrees, 16 randomly selected. For each of these polynomials there has been developed the simulation of the specific functioning with a program. Because the goal of this experimental work was to compare the different obtained results, a few rows of input data of a different length have been selected</p>
RAIN Journal-A Synoptic of Software Implementation for Shift Registers Based on 16th Degree Primitive Polynomials-Figure 6. Fibonacci Implementation for the Polynomial X^16+X^12+X^3+X+1
<p>VlăduŃiu and Crişan (1989) show three types of schemes for a 4th degree polynomial. Similar to it, there were developed the three different implementations for the Primitive Polynomial X^16+X^12+X^3+X+1. It can be specified that these schemes are according to the well-known Galois Form, Fibonacci representation and some other Forms that are rarely used, called Ring Implementation. All of these Implementations have the same function, because they describe a linear feedback shift register. In the experimental work there has been analyzed the behavior of 14 Primitive Polynomial degrees, 16 randomly selected. For each of these polynomials there has been developed the simulation of the specific functioning with a program. Because the goal of this experimental work was to compare the different obtained results, a few rows of input data of a different length have been selected</p>
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