This is due to the fact that a 4-bit LFSR can only represent 16 unique values, which means that there is a significant probability that multiple errors in the data stream could result in the two checksum values being identical. In the real world, a 4-bit CRC calculator would not be considered to provide sufficient confidence in the integrity of the transmitted data. This form of error detection is very efficient in terms of the small number of bits that have to be transmitted in addition to the data. Once all of the data bits have arrived, the receiver compares its internally generated checksum value with the checksum sent by the transmitter to determine whether any corruption occurred during the course of the transmission. The receiver contains an identical CRC calculator and generates its own checksum value from the incoming data. After all of the data bits have been transmitted, the transmitter sends its checksum value to the receiver.
The final CRC value stored in the LFSR is known as a checksum, and is dependent on every bit in the data stream. Cyclic redundancy check (CRC) calculations. Schenk, "Electronic circuits: Design and applications", Springer (1991) Rueppel, "Analysis and design of stream ciphers", Springer (1986) Niederreiter, "Introduction to finite fields and their applications", Cambridge Univ. Jungnickel, "Finite fields: Structure and arithmetics", Bibliographisches Inst. Golomb, "Shift register sequences", Aegean Park Press (1982)ĭ. While shift register sequences per se are too weak for use in cryptography, suitable (non-linear) combinations of such sequences have been studied in this context, see, e.g. \begin$ are periodic with least period $r_0$ equal to the order of $\alpha$ (that is, the least positive integer $e$ such that $\alpha^e=1$) and split into $(q^n-1)/r_0$ equivalence classes of $r_0$ sequences each. While the LFSR discards (or outputs) the rightmost entry $b_0$ (and replaces it by $b_1$), it computes the linear function The classical reference on shift register sequences is see also or for expositions.Ī linear feedback shift register of length $n$ (LFSR) is a time-dependent device (running on a clock) of $n$ cells each capable of holding a value from some field $F$, such that with each clock cycle the contents of the cells are shifted cyclically by one position (to the right, say). The term "shift register sequence" stems from the engineering literature in mathematics, the terms recursive sequence or recurrent sequence are more common. A sequence which can be obtained as the output of a linear feedback shift register.
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