# Overview

prim_lfsr is a parameterized linear feedback shift register (LFSR) implementation that supports Galois (XOR form) and Fibonacci (XNOR form) polynomials. The main difference between Galois and Fibonacci is that the former has a shorter critical timing path since the XOR Gates are interleaved with the shift register, whereas the latter combines several shift register taps and reduces them with an XNOR tree. For more information, refer to this page. Both LFSR flavors have maximal period (2^LfsrDw - 1). The recommendation is to use the Galois type and fall back to the Fibonacci type depending on the polynomial width availability in the lookup table (see below).

## Parameters

Name type Description
LfsrType string LFSR form, can be "GAL_XOR" or "FIB_XNOR"
LfsrDw int Width of the LFSR
EntropyDw int Width of the entropy input
StateOutDw int Width of the LFSR state to be output (lfsr_q[StateOutDw-1:0])
DefaultSeed logic Initial state of the LFSR, must be nonzero for XOR and non-all-ones for XNOR forms.
CustomCoeffs logic Custom polynomial coefficients of length LfsrDw.
MaxLenSVA bit Enables maximum length assertions, use only in sim and FPV.

## Signal Interfaces

Name In/Out Description
seed_en_i input External seed input enable
seed_i[LfsrDw] input External seed input
lfsr_en_i input Lfsr enable
entropy_i[EntropyDw] input Entropy input
state_o[StateOutDw] output LFSR state output.

# Theory of Operations

             /----------------\
seed_en_i    |                |
------------>|      lfsr      |
seed_i       |                |
=====/======>|     LfsrDw     |
[LfsrDw]    |    LfsrType    |
lfsr_en_i    |   EntropyDw    |
------------>|   StateOutDw   |
entropy_i    |   DefaultSeed  |  state_o
=====/======>|  CustomCoeffs  |=====/=======>
[EntropyDw] |   MaxLenSVA    |  [StateOutDw]
|                |
\----------------/


The LFSR module has an enable input and an additional entropy input that is XOR'ed into the LFSR state (connect to zero if this feature is unused). The state output contains the lower bits of the LFSR state from StateOutDw-1 downto 0. As the entropy input may cause the LFSR to jump into its parasitic state (all-zero for XOR, all-ones for XNOR), the LFSR state transition function contains a lockup protection which re-seeds the state with DefaultSeed once this condition is detected.

The LFSR contains an external seed input seed_i which can be used to load a custom seed into the LFSR by asserting seed_en_i. This operation takes precedence over internal state updates. If the external seed happens to be a parasitic state, the lockup protection feature explained above will reseed the LFSR with the DefaultSeed in the next cycle.

The LFSR coefficients are taken from an internal set of lookup tables with precomputed coefficients. Alternatively, a custom polynomial can be provided using the Custom parameter. The lookup tables contain polynomials for both LFSR forms and range from 4bit to 64bit for the Galois form and 3bit to 168bit for the Fibonacci form. The polynomial coefficients have been obtained from this page and Xilinx application note 52. The script ./script/get-lfsr-coeffs.py can be used to download, parse and dump these coefficients in SV format as follows:

\$ script/get-lfsr-coeffs.py -o <output_file>


The default is to get the Galois coefficients. If the Fibonacci coefficients are needed, add the --fib switch to the above command.

The implementation of the state transition function of both polynomials have been formally verified. Further, all polynomials up to 34bit in length have been swept through in simulation in order to ensure that they are of maximal-length.