chunker

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Published: Aug 21, 2015 License: BSD-2-Clause Imports: 8 Imported by: 0

Documentation

Overview

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

Index

Constants

View Source
const (
	KiB = 1024
	MiB = 1024 * KiB

	// MinSize is the minimal size of a chunk.
	MinSize = 512 * KiB
	// MaxSize is the maximal size of a chunk.
	MaxSize = 8 * MiB
)

Variables

This section is empty.

Functions

This section is empty.

Types

type Chunk

type Chunk struct {
	Start  uint
	Length uint
	Cut    uint64
	Digest []byte
}

Chunk is one content-dependent chunk of bytes whose end was cut when the Rabin Fingerprint had the value stored in Cut.

func (Chunk) Reader

func (c Chunk) Reader(r io.ReaderAt) io.Reader

type Chunker

type Chunker struct {
	// contains filtered or unexported fields
}

Chunker splits content with Rabin Fingerprints.

func New

func New(rd io.Reader, pol Pol, h hash.Hash) *Chunker

New returns a new Chunker based on polynomial p that reads from rd with bufsize and pass all data to hash along the way.

func (*Chunker) Next

func (c *Chunker) Next() (*Chunk, error)

Next returns the position and length of the next chunk of data. If an error occurs while reading, the error is returned with a nil chunk. The state of the current chunk is undefined. When the last chunk has been returned, all subsequent calls yield a nil chunk and an io.EOF error.

type Pol

type Pol uint64

Pol is a polynomial from F_2[X].

func RandomPolynomial

func RandomPolynomial() (Pol, error)

RandomPolynomial returns a new random irreducible polynomial of degree 53 (largest prime number below 64-8). There are (2^53-2/53) irreducible polynomials of degree 53 in F_2[X], c.f. Michael O. Rabin (1981): "Fingerprinting by Random Polynomials", page 4. If no polynomial could be found in one million tries, an error is returned.

func (Pol) Add

func (x Pol) Add(y Pol) Pol

Add returns x+y.

func (Pol) Deg

func (x Pol) Deg() int

Deg returns the degree of the polynomial x. If x is zero, -1 is returned.

func (Pol) Div

func (x Pol) Div(d Pol) Pol

Div returns the integer division result x / d.

func (Pol) DivMod

func (x Pol) DivMod(d Pol) (Pol, Pol)

DivMod returns x / d = q, and remainder r, see https://en.wikipedia.org/wiki/Division_algorithm

func (Pol) Expand

func (x Pol) Expand() string

Expand returns the string representation of the polynomial x.

func (Pol) GCD

func (x Pol) GCD(f Pol) Pol

GCD computes the Greatest Common Divisor x and f.

func (Pol) Irreducible

func (x Pol) Irreducible() bool

Irreducible returns true iff x is irreducible over F_2. This function uses Ben Or's reducibility test.

For details see "Tests and Constructions of Irreducible Polynomials over Finite Fields".

func (Pol) MarshalJSON

func (p Pol) MarshalJSON() ([]byte, error)

func (Pol) Mod

func (x Pol) Mod(d Pol) Pol

Mod returns the remainder of x / d

func (Pol) Mul

func (x Pol) Mul(y Pol) Pol

Mul returns x*y. When an overflow occurs, Mul panics.

func (Pol) MulMod

func (x Pol) MulMod(f, g Pol) Pol

MulMod computes x*f mod g

func (Pol) String

func (x Pol) String() string

String returns the coefficients in hex.

func (*Pol) UnmarshalJSON

func (p *Pol) UnmarshalJSON(data []byte) error

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