mirror of
https://github.com/restic/restic.git
synced 2024-12-25 17:27:25 +00:00
82 lines
2.9 KiB
Go
82 lines
2.9 KiB
Go
// Copyright 2014 Alexander Neumann. All rights reserved.
|
|
// Use of this source code is governed by a BSD-style
|
|
// license that can be found in the LICENSE file.
|
|
|
|
/*
|
|
Package chunker implements Content Defined Chunking (CDC) based on a rolling
|
|
Rabin Checksum.
|
|
|
|
Choosing a Random Irreducible Polynomial
|
|
|
|
The function RandomPolynomial() returns a new random polynomial of degree 53
|
|
for use with the chunker. The degree 53 is chosen because it is the largest
|
|
prime below 64-8 = 56, so that the top 8 bits of an uint64 can be used for
|
|
optimising calculations in the chunker.
|
|
|
|
A random polynomial is chosen selecting 64 random bits, masking away bits
|
|
64..54 and setting bit 53 to one (otherwise the polynomial is not of the
|
|
desired degree) and bit 0 to one (otherwise the polynomial is trivially
|
|
reducible), so that 51 bits are chosen at random.
|
|
|
|
This process is repeated until Irreducible() returns true, then this
|
|
polynomials is returned. If this doesn't happen after 1 million tries, the
|
|
function returns an error. The probability for selecting an irreducible
|
|
polynomial at random is about 7.5% ( (2^53-2)/53 / 2^51), so the probability
|
|
that no irreducible polynomial has been found after 100 tries is lower than
|
|
0.04%.
|
|
|
|
Verifying Irreducible Polynomials
|
|
|
|
During development the results have been verified using the computational
|
|
discrete algebra system GAP, which can be obtained from the website at
|
|
http://www.gap-system.org/.
|
|
|
|
For filtering a given list of polynomials in hexadecimal coefficient notation,
|
|
the following script can be used:
|
|
|
|
# create x over F_2 = GF(2)
|
|
x := Indeterminate(GF(2), "x");
|
|
|
|
# test if polynomial is irreducible, i.e. the number of factors is one
|
|
IrredPoly := function (poly)
|
|
return (Length(Factors(poly)) = 1);
|
|
end;;
|
|
|
|
# create a polynomial in x from the hexadecimal representation of the
|
|
# coefficients
|
|
Hex2Poly := function (s)
|
|
return ValuePol(CoefficientsQadic(IntHexString(s), 2), x);
|
|
end;;
|
|
|
|
# list of candidates, in hex
|
|
candidates := [ "3DA3358B4DC173" ];
|
|
|
|
# create real polynomials
|
|
L := List(candidates, Hex2Poly);
|
|
|
|
# filter and display the list of irreducible polynomials contained in L
|
|
Display(Filtered(L, x -> (IrredPoly(x))));
|
|
|
|
All irreducible polynomials from the list are written to the output.
|
|
|
|
Background Literature
|
|
|
|
An introduction to Rabin Fingerprints/Checksums can be found in the following articles:
|
|
|
|
Michael O. Rabin (1981): "Fingerprinting by Random Polynomials"
|
|
http://www.xmailserver.org/rabin.pdf
|
|
|
|
Ross N. Williams (1993): "A Painless Guide to CRC Error Detection Algorithms"
|
|
http://www.zlib.net/crc_v3.txt
|
|
|
|
Andrei Z. Broder (1993): "Some Applications of Rabin's Fingerprinting Method"
|
|
http://www.xmailserver.org/rabin_apps.pdf
|
|
|
|
Shuhong Gao and Daniel Panario (1997): "Tests and Constructions of Irreducible Polynomials over Finite Fields"
|
|
http://www.math.clemson.edu/~sgao/papers/GP97a.pdf
|
|
|
|
Andrew Kadatch, Bob Jenkins (2007): "Everything we know about CRC but afraid to forget"
|
|
http://crcutil.googlecode.com/files/crc-doc.1.0.pdf
|
|
|
|
*/
|
|
package chunker
|