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