CA2632857A1 - Closed galois field combination - Google Patents

Closed galois field combination Download PDF

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Publication number
CA2632857A1
CA2632857A1 CA002632857A CA2632857A CA2632857A1 CA 2632857 A1 CA2632857 A1 CA 2632857A1 CA 002632857 A CA002632857 A CA 002632857A CA 2632857 A CA2632857 A CA 2632857A CA 2632857 A1 CA2632857 A1 CA 2632857A1
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Canada
Prior art keywords
number sequence
galois
field
sequence
mapping
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Granted
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CA002632857A
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French (fr)
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CA2632857C (en
Inventor
Alan J. Michaels
David B. Chester
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Harris Corp
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Harris Corp
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Publication of CA2632857A1 publication Critical patent/CA2632857A1/en
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Publication of CA2632857C publication Critical patent/CA2632857C/en
Expired - Fee Related legal-status Critical Current
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F7/00—Methods or arrangements for processing data by operating upon the order or content of the data handled
    • G06F7/60—Methods or arrangements for performing computations using a digital non-denominational number representation, i.e. number representation without radix; Computing devices using combinations of denominational and non-denominational quantity representations, e.g. using difunction pulse trains, STEELE computers, phase computers
    • G06F7/72—Methods or arrangements for performing computations using a digital non-denominational number representation, i.e. number representation without radix; Computing devices using combinations of denominational and non-denominational quantity representations, e.g. using difunction pulse trains, STEELE computers, phase computers using residue arithmetic
    • G06F7/724—Finite field arithmetic
    • G06F7/725—Finite field arithmetic over elliptic curves

Abstract

A method is provided for combining two or more input sequences in a communications system to increase a repetition period of the input sequences in a resource-efficient manner. The method includes a receiving step, a mapping step, and a generating step. The receiving step involves receiving a first number sequence and a second number sequence, each expressed in a Galois field GF[p k]. The mapping step involves mapping the first and second number sequences to a Galois extension field GF[p k+1]. The generating step involves generating an output sequence by combining the first number sequence with the second number sequence utilizin g a Galois field multiplication operation in the Galois extension field GF[p k+1 ]. p is a prime number. k is an integer. p k+1 defines a finite field size of the Galo is extension field GF[p k+1].

Claims (10)

1. A method for combining at least two input sequences in a communications system to increase a repetition period of the at least two input sequences in a resource efficient manner, comprising:
receiving a first number sequence and a second number sequence, each expressed in a Galois field GF[p k];
mapping said first number sequence and said second number sequence to a Galois extension field GF[p k+1]; and generating an output sequence by combining said first number sequence with said second number sequence utilizing a Galois field multiplication operation in said Galois extension field GF[p k+1], where p is a prime number, k is an integer value, and p k+1 defines a finite field size of said Galois extension field GF [p k+1]
2. The method according to claim 1, further comprising selecting p k to be a prime number defining a finite field size of said Galois field GF[p k]
3. The method according to claim 1, further comprising expressing said first number sequence and said second number sequence in a p-adic number system representation prior to said receiving step.
4. The method according to claim 1, further comprising expressing said first number sequence and said second number sequence in a mixed-radix number system representation prior to said receiving step.
5. The method according to claim 1, further comprising selecting each of said first number sequence and said second number sequence to be comprised of at least two digits expressed in a weighted number system.
6. The method according to claim 1, wherein said generating step further comprising combining said first number sequence and said second number sequence by selectively performing a plurality of Galois field multiplication operations using a distinct subset of digits from each of said first number sequence and said second number sequence.
7. The method according to claim 1, further comprising selecting said first number sequence to be payload data or a random number sequence.
8. The method according to claim 1, further comprising selecting said second number sequence to be a pseudo random number sequence or a pseudo-chaotic random number sequence.
9. The method according to claim 1, wherein said mapping step further comprises performing an Affine transformation of said first number sequence and said second number sequence.
10. A closed Galois field cryptographic (CGFC) system, comprising:
a receiving means configured for receiving a first number sequence and a second number sequence, each expressed in a Galois field GF[p k];
a mapping means configured for mapping said first number sequence and said second number sequence to a Galois extension field GF[p k+1]; and a generating means configured for generating an output sequence by combining said first number sequence with said second number sequence utilizing a Galois field multiplication operation in said Galois extension field GF[p k+1], where p is a prime number, k is an integer value, and p k+1 defines a finite field size of said Galois extension field GF[p k+1].
CA2632857A 2007-05-31 2008-05-30 Closed galois field combination Expired - Fee Related CA2632857C (en)

Applications Claiming Priority (2)

Application Number Priority Date Filing Date Title
US11/756,086 US7995757B2 (en) 2007-05-31 2007-05-31 Closed galois field combination
US11/756,086 2007-05-31

Publications (2)

Publication Number Publication Date
CA2632857A1 true CA2632857A1 (en) 2008-11-30
CA2632857C CA2632857C (en) 2012-11-27

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CA2632857A Expired - Fee Related CA2632857C (en) 2007-05-31 2008-05-30 Closed galois field combination

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US (1) US7995757B2 (en)
EP (1) EP2003546B1 (en)
JP (1) JP4866389B2 (en)
CA (1) CA2632857C (en)
DE (1) DE602008006296D1 (en)

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Publication number Publication date
CA2632857C (en) 2012-11-27
DE602008006296D1 (en) 2011-06-01
JP4866389B2 (en) 2012-02-01
EP2003546B1 (en) 2011-04-20
JP2008299330A (en) 2008-12-11
EP2003546A2 (en) 2008-12-17
US7995757B2 (en) 2011-08-09
EP2003546A3 (en) 2010-04-07
US20090044080A1 (en) 2009-02-12

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