Punctured bent function sequences for watermarked DS-CDMA
Transcript of Punctured bent function sequences for watermarked DS-CDMA
Punctured bent function sequences for
watermarked DS-CDMAHong-Yeop Song
Yonsei University, Seoul, [email protected]
Based on the paper with the same title,IEEE Communications Letters, 2019, to appear soon.
July 5, 2019Aachen, Germany
Table of Contents
Introduction to Watermarked DS-CDMA
Proposed Model of W-DS-CDMA
Analysis and Design Criteria
Proposed (optimal) watermarked sequences punctured bent function sequences
various properties including optimality
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Introduction: Watermarked DS-CDMA
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Watermarking chips
Spreading code
Watermarked spreading code
β’ Insert some watermarking chips into spreading codeβ’ Any two watermarks at different time are different
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periodic patternswith non-periodic chips
periodic sequence
Introduction: advantage/disadvantage
Watermarked DS-CDMA have been considered to provide security at the signal level
SteganographyWatermark conveys some βsecretβ information which can
be extracted after synchronized.
Authentication of GNSS open signalsWatermark is used to provide where a signal comes from Protect from spoofing attacks
at the price of degrading the correlation performance (communication performance) of spreading sequences for multiple access, for example.
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Introduction: Summary
Investigate the effect of inserting some randomly generated watermarking chips into known ( set of ) spreading sequences
β’ In terms of periodic correlations
Propose two design criteria for βgoodβ watermarked sequences in the sense of
1) Reducing the average correlation value2) Minimizing the variance of correlations
for the best performance of multiple-access
Specifically, we propose, for ππ = 2ππ with even ππ, an optimal set of ππππβππ punctured bent function sequences of length ππππ β ππ in the sense of the above two criteria such that
β’ all of which are punctured by the single pattern obtained by the Singer difference set, (Criteria 2) Hence, half the bits are punctured in one period of the sequence
β’ the max non-trivial correlation magnitude maintains 2ππ + 1, (Criteria 1) which is the same value as those for un-punctured bent function sequences but is in fact twice of the Welch bound
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Introduction: (selected) References
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S. W. Golomb and G. Gong, Signal design for good correlation: for wirelesscommunications, cryptography, and radar, New York, NY, USA: Cambridge UniversityPress, 2005.
Global Positioning Systems Directorate Systems Engineering & Integration InterfaceSpecification, document IS-GPS-200H, Mar. 2014.
G. Caparra and J. T. Curran, ``On the achievable equivalent security of GNSS ranging codeencryption,'' in Proc. 2018 IEEE/ION Positions, Location and Navigation Symposium(PLANS), Monterey, USA, pp. 956-966, Apr. 2018.
X. Li, C. Yu, M. Hizlan, W.-T. Kim, and S. Park, ``Physical layer watermarking of directsequence spread spectrum signals,'' in Proc. IEEE MILCOM 2013, San Diego, USA. pp. 476-481, Nov. 2013.
C. Yang, ``FFT acquisition of periodic, aperiodic, puncture, and overlaid code sequencesin GPS,'' in Proc. ION GPS 2001, Salt Lake City, USA, pp. 137-147, Sep. 2001.
M. Villanti, M. Iubatti, A. Vanelli-Coralli, and G. E. Corazza, ``Design of distributed uniquewords for enhanced frame synchronization,'' IEEE Trans. Commun., vol. 57, no. 8, pp. 2430-2440, Aug. 2009.
J. D. Olsen, R. A. Scholtz, and L. R. Welch, ``Bent-function sequences,'' IEEE Trans. Inf. Theory, vol. 28, no. 6, pp.858-864, Nov. 1982.
L. R. Welch, βLower bounds on the maximum cross correlation of signals (Corresp.),β IEEE Trans. Inf. Theory, vol. 20, no. 3, pp. 397-399, May 1974.
L. D. Baumert, Cyclic difference sets, New York, NY, USA: Springer-Verlag, 1972.
J. Singer, βA theorem in finite projective geometry and some applications to number theory,β Trans. Amer. Math. Soc., vol. 43, pp. 377-385, 1938.
DS-CDMA for communications
DS-CDMA for navigations
W-DS-CDMA for authentication
W-DS-CDMA for steganography
W-DS-CDMA for fast acquisition
Effect of watermarking on single spreading sequence only in terms of aperiodic autocorrelation
Bent function sequences
Welch Bound
Cyclic difference sets
Singer difference sets
Proposed model ofW-DS-CDMA
Previous results are focused on how to use watermarks for security.
Usually assume the aggregated insertion
The watermark insertion affects on auto- and cross-correlation of spreading code
Question:
What insertion is better in the sense of acquisition performance?
How to insert watermark?
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watermark
Case 2. spread
watermark
Case 1. aggregated
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Why not
Proposed model
Equivalent model
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Watermark
Spreading code
Watermarked spreading code
length πΏπΏ length πΏπΏ
Nulled(No signal here)
Nulled(No signal here)
Watermarkcode
Punctured spreading
code
insertion
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Proposed model
Some properties
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watermark
Punctured spreading code
length πΏπΏ
Nulled(No signal here)
Nulled(No signal here)
Ternary0, +1,β1
Ternary0, +1,β1
Binary+1,β1
Alphabet? Repeated?
Not repeated
Repeated periodically
Partially repeated
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Watermarked spreading code
Proposed model
Acquisition for watermarked DS-CDMA
During the acquisition process, the receiver knows which chips are watermarked (only the position information)
but has no information about what each value is. (no idea on its value)
Therefore, the receiver can only use the punctured spreading code, which is repeated, periodically.
Watermark chips will be extracted after the signal is obtained/acquired the receiver will use these chips for some other purpose (steganography/authentication/extra security, etc)
Our goal is to find BEST watermarking chips (position) PLUS spreading codes so that the multiple-access performance is NOT MUCH degraded compared with the conventional DS-CDMA systems without watermarks.
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Received signal(watermarked)
Reference signal:punctured
spreading code
spreading code of length πΏπΏfor the ππ-th transmit symbol
spreading code of length πΏπΏfor the (ππ+ ππ)-th transmit symbol
movingfrom ππ = 0 to πΏπΏ β 1
time delay ππ
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Proposed model
Watermarked DS-CDMA system
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.
.
Transmitted signal for user1
Transmitted signal for user2
Transmitted signal for userππ
User1 Reference signal of user1
User2
User ππ
Reference signal of user2...
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Reference signal of userππ
Transmitted signal for user ππ
Puntured spreading sequence for user ππ
Watermark for user ππ
Watermarked sequence for user ππ
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Want acquireDonβt want acquire
Proposed model
Analysis on Watermarksand
Design Criteria
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ππππππ, π π ππ ππ = πππ π ππ, π π ππ(ππ) + πππ€π€ππ, π π ππ(ππ)
Crosscorrelationof watermarked sequence and punctured sequence
β» ππππ : watermarked sequence for user ππ.π π ππ: punctured sequence for user ππ.π€π€ππ: watermark for user ππ
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movingfrom ππ = 0 to πΏπΏ β 1
time delay ππ ππππ: punctured sequence for user ππ.
ππππ: punctured sequence for user ππ.
movingfrom ππ = 0 to πΏπΏ β 1
time delay ππ ππππ: punctured sequence for user ππ.
ππππ: watermark for user ππ
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For desired signal (when ππ = ππ)
desired signal ππππ(watermarked)
reference signal:punctured
spreading code for user ππ
spreading code of length πΏπΏfor the ππ-th transmit symbol
spreading code of length πΏπΏfor the (ππ + 1)-th transmit symbol
movingfrom ππ = 0 to πΏπΏ β 1
time delay ππ
movingfrom ππ = 0 to πΏπΏ β 1
time delay ππ
movingfrom ππ = 0 to πΏπΏ β 1
time delay ππ
autocorrelation of punctured
code
crosscorrelation of punctured
code and watermark code
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ππππππ, π π ππ ππ = πππ π ππ, π π ππ(ππ) + π½π½ππππ, ππππ(ππ)
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For undesired signal (when ππ β ππ)
undesired signal ππππ(watermarked)
reference signal:punctured
spreading code for user ππ (β ππ)
spreading code of length πΏπΏfor the ππ-th transmit symbol
spreading code of length πΏπΏfor the (ππ + 1)-th transmit symbol
movingfrom ππ = 0 to πΏπΏ β 1
time delay ππ
movingfrom ππ = 0 to πΏπΏ β 1
time delay ππ
movingfrom ππ = 0 to πΏπΏ β 1
time delay ππ
crosscorrelation of two punctured
codes
crosscorrelation of punctured
code and watermark code
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ππππππ, π π ππ ππ = πππ π ππ, π π ππ(ππ) + π½π½ππππ, ππππ(ππ)
What is required
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ππππππ, π π ππ ππ = πππ π ππ, π π ππ(ππ) + π½π½ππππ, ππππ(ππ)
ππππππ, π π ππ ππ = πππ π ππ, π π ππ(ππ) + π½π½ππππ, ππππ(ππ)
as small as possiblefor all ππ
as small as possiblefor all ππ
as small as possiblefor all ππ β ππ
deterministic random (?)
undesired signal
desired signal
β’ ππππ is a watermark, which have values Β±1 at positions indicated by the puncturing pattern π π , which is a ππ-subset of β€πΏπΏ to be OPTIMIZED
β’ It turned out that it is enough to assume that all the users have the same π π .
β’ Assume that the watermarking chips are i.i.d. random variables with Β±1 equally likely
Crosscorrelation π½π½ππ, ππππ(ππ)of watermarking chips and punctured sequence
π½π½ππ, ππππ ππ = οΏ½ππ
ππ ππ + ππ ππππ(ππ)
β’ Watermarking chip sequence ππ has a non-zero value ONLY at index ππ + ππ β π π or at index ππ β π π β ππ.
β’ Punctured sequence ππππ has a non-zero value ONLY at index ππ β π π or at ππ β πππΏπΏ\π π β’ Therefore, ππ ππ + ππ ππππ(ππ) has a non-zero value ONLY at
ππ β π π β ππ β© πππΏπΏ\π π = π π β ππ \π π β’ Therefore, the number of non-zeros will be
= | π π β ππ \π π |= |π π | β |π π β© π π β ππ |= ππ β π·π·π π (ππ)
This must be the variance of π½π½ππ, ππππ ππ .The mean of π½π½ππ, ππππ ππ becomes 0 since πΈπΈ[π€π€] = 0
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the non-zero values of ππ are i.i.d. random with Β±1 equally likely,hence, mean-zero
DESIGN CRITERIA
ππππππ, π π ππ ππ = πππ π ππ, π π ππ ππ + π½π½ππππ,ππππ ππ
This is a random variable with mean = πππ π ππ, π π ππ ππ
variance = ππ β π·π·π π (ππ) = |π π | β |π π β© π π β ππ |
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It is a random variable with mean-zero and variance ππ β π·π·π π (ππ) = |π π | β |π π β© π π β ππ |where π π is a puncturing pattern of size ππ.
πͺπͺππ: Minimize the mean of π½π½ππππ,ππππ ππ
β‘ Minimize πππ π ππ, π π ππ ππ the non-trivial correlation magnitude of
punctured sequences π π ππ , π π ππ for all possible ππ, ππ. πͺπͺππ: Minimize the variance of π½π½ππππ,ππππ ππ
β‘ Maximize minππβ 0
π·π·π π (ππ) = minππβ 0
|π π β© (π π β ππ)| β π«π«ππππππ(πΉπΉ)
Upper bound of min of |π π β© (π π β ππ)|
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min1β€ππβ€πΏπΏβ1
π π β© (π π β ππ) β€ππ2 β πππΏπΏ β 1
.
Lemma (πͺπͺππ). Assume that ππ watermarking chips are inserted in a watermarked spreading code of length πΏπΏ, according to a puncturing pattern π π . Then,
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Proof: Recall that π π is a ππ-subset of β€πΏπΏ. Therefore, for any such π π of size ππ, we have
οΏ½ππ=0
πΏπΏβ1
π·π·π π (ππ) = ππ2
since each member in π π will match every member of π π (including itself) exactly once as ππ runs from 0 to πΏπΏ β 1.
Since π·π·π π 0 = ππ, we have
1πΏπΏ β 1οΏ½
ππ=1
πΏπΏβ1
π·π·π π (ππ) =ππ2 β πππΏπΏ β 1
x x xx
x xx x
x
Proposed Optimal Watermarked Spreading Sequences Set
We consider πͺπͺππ first, and then consider πͺπͺππ.
puncturing pattern
optimization
which spreading sequence is best with
the selected puncturing?
Does there any spreading sequence that is good with this
puncturing?
or that can be proved to be good with this puncturing?
(Almost) Cyclic Difference Sets
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Definition. Let π π be a ππ-subset of β€πΏπΏ. Then,
β π π is called a (πΏπΏ,ππ, ππ, π‘π‘)-almost cyclic difference set if, for ππ = 1, 2, β¦ , πΏπΏ β 1,
|π π β© (π π β ππ)| = οΏ½ ππ π‘π‘ timesππ + 1 πΏπΏ β 1 β π‘π‘ times.
β‘ π π is called a (πΏπΏ,ππ, ππ)-cyclic difference set if, for ππ =1, 2, β¦ , πΏπΏ β 1,
|π π β© (π π β ππ)| = ππ.
This is equivalent to almost cyclic difference set with π‘π‘ = πΏπΏ β 1.
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Optimal Puncturing Pattern
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Theorem. (ACDS β πΆπΆ2 optimal)
Let π π be a ππ-subset of β€πΏπΏ. Then π π is an optimal puncturing pattern if it is an (πΏπΏ, ππ, ππ, π‘π‘)-ACDS in the sense of
min1β€ππβ€πΏπΏβ1
π π β© (π π β ππ) attains its maximum value ππ = ππ2βπππΏπΏβ1
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Well-known Lemma on the existence:β If an (πΏπΏ, ππ, ππ, π‘π‘)-almost cyclic difference set π π exists, then we have
ππ ππ β 1 = L β 1 ππ + πΏπΏ β 1 β π‘π‘β‘ If an (πΏπΏ, ππ, ππ)-cyclic difference set π π exists, then we have
ππ ππ β 1 = (L β 1)ππ
For both cases, we haveππ2 β πππΏπΏ β 1
= ππ
Singer Difference Sets (J. Singer 1938)
β’ πΏπΏ = 2ππ β 1 with ππ = 0 (mod 4)
β’ ππ = 2ππ/2 β 1 and ππ = 2ππ/4 β 1
β’ πΌπΌ β π½π½2ππ be a primitive element
β’ tr1ππ π₯π₯ = βππ=0ππβ1 π₯π₯2ππ is the trace of π₯π₯ β π½π½2ππ to π½π½2
Then, a ππ-subset π π of β€πΏπΏ is an (πΏπΏ, ππ, ππ)-CDS if, for each ππ β β€πΏπΏ,
ππ β π π iff tr1ππ πΌπΌππ = 0
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We will use the puncturing pattern π π from the Singer difference set constructed above.β’ This is optimal (πͺπͺππ)β’ It punctures about half the bits in one
period of the sequence of length πΏπΏ = 2ππ β 1Is it too much?
Bent Function Sequences
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β’ ππ = 2ππ be a positive integer with even ππ.β’ ππ be a bent function over π½π½2ππ.β’ πΌπΌ β π½π½2ππ be a primitive element and a constant ππ β π½π½2ππ\π½π½2ππ .
The set ππ of 2ππ binary sequences of length 2ππ β 1 for each constant ππ β π½π½2ππ given as, for ππ = 0,1, β¦ , 2ππ β 2,
ππππ ππ = β1 ππ trππππ πΌπΌππ +tr1ππ ππ+ππ πΌπΌππ
is called bent function sequence family and
ππππππππ ππ β€ 2ππ + 1.Hence, it is optimal in terms of the Welch bound.
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Original Contribution: J. D. Olsen, R. A. Scholtz, and L. R. Welch (1982)Above formulation by traces: Golomb and Gong (2005) Chapter 10
MAIN ContributionPunctured bent function sequences
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β’ ππ = 2ππ be a positive integer with evenππ.β’ ππ be a bent function over π½π½2ππ .β’ πΌπΌ β π½π½2ππ be a primitive element and a constant ππ β π½π½2ππ\π½π½2ππ .
β’ ππππ ππ = β1 ππ trππππ πΌπΌππ +tr1ππ ππ+ππ πΌπΌππ be the bent function sequences of length 2ππ β 1 for each ππ β π½π½2ππ , constructed earlier in previous page.
β’ ππ be a subset of π½π½ππππ such that ππ + ππ β ππ for any ππ,ππ β ππ. β’ πΉπΉ is the puncturing pattern from the Singer difference set, i.e.,
ππ β πΉπΉ iff ππππππππ πΆπΆππ = ππConsider the set of punctured bent function sequences ππ = π π ππ:ππ β Ξwhere
π π ππ ππ = οΏ½ππππ ππ if ππ β π π β tr1ππ πΌπΌππ = 10 otherwise
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Main Theorem
Let ππ = π π ππ:ππ β Ξ be the set of punctured bent function sequences in previous page, with puncturing pattern πΉπΉ from Singer difference set. Then,
ππmax ππ β€ 2ππ + 1.
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Proof of correlation bound
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First observation:
π π ππ ππ = οΏ½ππππ ππ if tr1ππ πΌπΌππ = 10 if tr1ππ πΌπΌππ = 0
=12
1 β β1 tr1ππ(πΌπΌππ) ππππ[ππ]
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Proof of correlation bound
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Second observation:π π ππ ππ = 1
21 β β1 tr1ππ(πΌπΌππ) ππππ ππ
= 12ππππ ππ β β1 tr1ππ(πΌπΌππ)ππππ ππ = 1
2ππππ ππ β ππππ+1 ππ
Since
β1 tr1ππ(πΌπΌππ)ππππ ππ = β1 tr1ππ πΌπΌππ β1 ππ trππππ πΌπΌππ +tr1ππ (ππ+ππ)πΌπΌππ
= β1 ππ trππππ πΌπΌππ +tr1ππ (ππ+1+ππ)πΌπΌππ
= ππππ+1 ππ
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Proof of correlation bound
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For ππ, ππ β Ξ, the correlation of π π ππ and π π ππ at time shift ππ is given byπππ π ππ,π π ππ ππ = βππ=0πΏπΏβ1 π π ππ[ππ + ππ]π π ππ[ππ]
= 14βππ=0πΏπΏβ1 ππππ ππ + ππ β ππππ+1 ππ + ππ ππππ ππ β ππππ+1 ππ .
= 14ππππππ,ππππ ππ + ππππππ+1,ππππ+1 ππ β ππππππ+1,ππππ ππ β ππππππ,ππππ+1 ππ
(1) when ππ = ππ, we are checking the values πππ π ππ,π π ππ ππ β 0
= 14ππππππ,ππππ ππ β 0 + ππππππ+1,ππππ+1 ππ β 0 β ππππππ+1,ππππ ππ β 0 β ππππππ,ππππ+1 ππ β 0
Therefore, by triangular inequality, we get
πππ π ππ,π π ππ ππ β 0 β€14 ππmax β¬ + ππmax β¬ + ππmax β¬ + ππmax β¬
= ππmax β¬ β€ 2ππ + 1
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crosscorrelationsautocorrelations
Proof of correlation bound
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For ππ, ππ β Ξ, the correlation of π π ππ and π π ππ at time shift ππ is given byπππ π ππ,π π ππ ππ = βππ=0πΏπΏβ1 π π ππ[ππ + ππ]π π ππ[ππ]
= 14βππ=0πΏπΏβ1 ππππ ππ + ππ β ππππ+1 ππ + ππ ππππ ππ β ππππ+1 ππ .
= 14ππππππ,ππππ ππ + ππππππ+1,ππππ+1 ππ β ππππππ+1,ππππ ππ β ππππππ,ππππ+1 ππ
(2) when ππ β ππ, we are checking the values πππ π ππ,π π ππ ππ for all ππ including 0,
= 14ππππππ,ππππ ππ + ππππππ+1,ππππ+1 ππ β ππππππ+1,ππππ ππ β ππππππ,ππππ+1 ππ
Therefore, similarly,
πππ π ππ,π π ππ ππ β€14 ππmax β¬ + ππmax β¬ + ππmax β¬ + ππmax β¬
= ππmax β¬ β€ 2ππ + 1
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crosscorrelationssince ππ + ππ β 1 implies ππ β ππ + 1 and ππ + 1 β ππ
crosscorrelations
Without the condition that ππ + ππ β 1, it may happen that ππ = ππ + 1and ππ + 1 = ππ. Then these become autocorrelations and the values at ππ = 0 matters!
Example: ππ = 4Punctured bent function sequences
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β’ πΌπΌ β π½π½24 be a primitive element, a root of π₯π₯4 + π₯π₯ + 1β’ Let ππ π₯π₯ = π₯π₯3 over π½π½22β’ Walsh-Hadamard Transform of ππ :
ππ(ππ)=βππβπ½π½22 β1 ππ(ππ)+Tr12(ππππ) over π½π½22
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π₯π₯ ππ π₯π₯ Tr12(0 Β· π₯π₯) Tr12(1 Β· π₯π₯) Tr12(πΌπΌ Β· π₯π₯) Tr12(πΌπΌ2 Β· π₯π₯)
0 0 0 0 0 0
1 1 0 0 1 1
πΌπΌ 1 0 1 1 0
πΌπΌ2 1 0 1 0 1
|ππ(ππ)|=2 for all ππ β π½π½22
ππ π₯π₯ = π₯π₯3 is a bent function over π½π½22
ππ 0 = 1 β 1 β 1 β 1 = β2
ππ 1 = 1 β 1 + 1 + 1 = +2
ππ πΌπΌ = 1 + 1 + 1 β 1 = +2
ππ πΌπΌ2 = 1 + 1 β 1 + 1 = +2
Example: ππ = 4Punctured bent function sequences
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β’ ππ = 2 and ππ = 4β’ πΌπΌ β π½π½24 be a primitive element, a root of π₯π₯4 + π₯π₯ + 1β’ ππ π₯π₯ = π₯π₯3 is a bent function over π½π½22β’ Choose a constant ππ = πΌπΌ β π½π½24\π½π½22 . β’ For any ππ β π½π½22 , the sequence
ππππ ππ = β1 ππ tr24 πΌπΌππ +tr14 ππ+πΌπΌ πΌπΌππ
is a bent function sequence of length 24 β 1 = 15There are 4 of them:
β’ ππ=0: + β + β β β + β + β β + + + β
β’ ππ=1: + β + + β β β + + + β β β β +
β’ ππ=πΌπΌ5 : + + β β + β β + β + β + + β β
β’ ππ=πΌπΌ10 : + + β + + β + β β β β β β + +
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Example: ππ = 4, Continued
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β’ ππ0= + β + β β β + β + β β + + + β
|ππππ0,ππ0| = 22 + 1
Example: ππ = 4, Continued
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β’ ππ0= + β + β β β + β + β β + + + βππ1= + β + + β β β + + + β β β β +
|ππππ0,ππ1| = 22 + 1
Example: ππ = 4, Continued
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β’ ππ0 = + β + β β β + β + β β + + + βπππΌπΌ5=+ + β β + β β + β + β + + β β
|ππππ0,πππΌπΌ5| = 22 + 1
Example: ππ = 4, Continued
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β’ ππ0 = + β + β β β + β + β β + + + βπππΌπΌ10= + + β + + β + β β β β β β + +
|ππππ0,πππΌπΌ10| = 22 + 1
Example: ππ = 4, Continued
β’ ππππππππ ππ = 22 + 1
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Example: ππ = 4, Continued
β’ ππ = {0,πΌπΌ5} be a subset of π½π½ππππ such that ππ + ππ β ππ for any ππ,ππ β ππ.
β’ π π is the puncturing pattern given by ππ β π π iff π‘π‘π‘π‘14 πΌπΌππ = 0.
β’ Note that, π‘π‘π‘π‘14 πΌπΌππ = 0 0 0 1 0 0 1 1 0 1 0 1 1 1 1 .
β’ Therefore, π π = {0, 1, 2, 4, 5, 8, 10}
Finally, the set of punctured bent function sequences ππ = π π ππ:ππ β Ξcontains only two sequences, for ππ=0 and ππ=πΌπΌ5. These are
ππ=0: ππ ππ ππ β ππ ππ + β ππ β ππ + + + β
ππ=πΆπΆππ : ππ ππ ππ β ππ ππ β + ππ + ππ + + β β
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Example: ππ = 4, Continued
β’ ππππππππ πΊπΊ = 4β€ 22 + 1 = ππππππππ ππ
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Example: ππ = 8
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β’ ππ =8 and ππ = 4β’ πΌπΌ β π½π½28 be a primitive element, a root of π₯π₯8 + π₯π₯7 + π₯π₯2 + π₯π₯1 + 1β’ ππ π₯π₯ = Tr14(πΌπΌ17π₯π₯3) is a bent function over π½π½24β’ Choose a constant ππ = πΌπΌ β π½π½28\π½π½24 .β’ Represent correlation only the case ππ=0, πΌπΌ17 β π½π½24 .
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β’ ππ = {0,πΌπΌ17} be a subset of π½π½ππ8 such that ππ + ππ β ππ for any ππ,ππ β ππ. β’ π π is the puncturing pattern given by ππ β π π iff π‘π‘π‘π‘18 πΌπΌππ = 0.
For bent function sequencesβ¦
For punctured bent function sequencesβ¦
Example: ππ = 8, Continued
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Autocorrelation of bent function sequences
Example: ππ = 8, Continued
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β’ ππππππππ ππ = 24 + 1
Crosscorrelation of bent function sequences
Example: ππ = 8, Continued
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Autocorrelation of punctured bent function sequences
Example: ππ = 8, Continued
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β’ ππππππππ πΊπΊ = 16 β€ 24 + 1 = ππππππππ ππ
Crosscorrelation of punctured bent function sequences
Properties of Punctured bent function sequences
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The cardinality of ππ is Ξ = 2ππβ1.β’ Because of Ξ in which ππ + ππ β 1
ππ is optimal in terms of πΆπΆ2.β’ Because puncturing pattern of ππ is Singer difference set.
Any sequence in ππ has the energy πΈπΈ = ππ 0 = 2ππβ1, which is about half the energy of the original bent function sequences.
β’ Because π π = 2ππβ1 β 1 is about the half the length
ππ is asymptotically optimal in terms of πΆπΆ1 also.β’ Both ππ and the original bent function sequences have the same upper bound
on the maximum non trivial correlation magnitudeβ’ Since the energy is reduced by half, this upper bound ππππππππ(ππ) asymptotically
achieves TWO times the Welch bound.
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Some open questions
For the puncturing pattern from the Singerβs difference set, try some other spreading sequences
β’ Gold, Kasami, etc
Optimal puncturing patterns must be from either ACDS or CDS.β’ They all are optimal but some implications might be different when it
applies to some other spreading sequences. β’ Does there any pair of puncturing pattern and spreading sequences
that can be provable mathematically, other than those mentioned in this talk
Main theorem implies: we have constructed a set of 2ππβ1 ternary sequences of length 22ππ β 1 such that
β Number of 0βs is 2ππβ1 β 1 in each sequenceβ‘ Number of non-zeros (either +1 or -1) is 2ππβ1 in each sequenceβ’ Max correlation magnitude is upper bounded by 2 times Welch Bound.
True/False: this is a set of BEST ternary sequence family in terms of Welch Bound.
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Any questions?
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