Mark Goresky and Andrew Klapper

Arithmetic Correlations

Mark Goresky∗ and Andrew Klapper†

∗partially supported by DARPA grant HR0011-04-1-0031†partially supported by NSF grant CCF-0514660

Let c = c0, c1, c2, · · · ci ∈ Z/(p) (p prime)eventually periodic sequence of period T.


Let ζ = e2πi/p.

The Imbalance is the sum over one period,

Z(c) =∑

k

ζck =∑

k

e2πick/p


Let ζ = e2πi/p.


Z(c) =∑

k

ζck =∑

k

e2πick/p

Given two sequences of period T

a = a0, a1, · · · , b = b0, b1, · · · ai, bi ∈ Z/(p)


Let ζ = e2πi/p.


Z(c) =∑

k

ζck =∑

k

e2πick/p


a = a0, a1, · · · , b = b0, b1, · · · ai, bi ∈ Z/(p)

Their cross-correlation with shift τ is

Cτ(a, b) =T−1∑

k=0

ζakζ−bk+τ =T−1∑

k=0

ζak−bk+τ

= Z(a − bτ)

where bτ = bτ , bτ+1, · · · is the τ-shift of b.


Let ζ = e2πi/p.


Z(c) =∑

k

ζck =∑

k

e2πick/p


a = a0, a1, · · · , b = b0, b1, · · · ai, bi ∈ Z/(p)

Their cross-correlation with shift τ is

Cτ(a, b) =T−1∑

k=0

ζakζ−bk+τ =T−1∑

k=0

ζak−bk+τ

= Z(a − bτ)

where bτ = bτ , bτ+1, · · · is the τ-shift of b.

There is another way to say this:

a(x) = a0 + a1x + a2x2 + · · · ∈ Z(p)[[x]]

bτ(x) = bτ + bτ+1x + bτ+2x2 + · · · ∈ Z(p)[[x]]

a(x) = a0 + a1x + a2x2 + · · · ∈ Z(p)[[x]]

bτ(x) = bτ + bτ+1x + bτ+2x2 + · · · ∈ Z(p)[[x]]

Then a − bτ is the coefficient sequence of

c(x) = a(x) − bτ(x) ∈ Z/(p)[[x]]

so: Cτ(a, b) = Z(c) =T−1∑

k=0

ζck =N+T−1∑

k=N

ζck.

a(x) = a0 + a1x + a2x2 + · · · ∈ Z(p)[[x]]

bτ(x) = bτ + bτ+1x + bτ+2x2 + · · · ∈ Z(p)[[x]]


c(x) = a(x) − bτ(x) ∈ Z/(p)[[x]]

so: Cτ(a, b) = Z(c) =T−1∑

k=0

ζck =N+T−1∑

k=N

ζck.

Arithmetic analog:

α = a0 + a1p + a2p2 + · · · ∈ Zp.

βτ = bτ + bτ+1p + bτ+2p2 + · · · ∈ Zp.

γ = α − βτ = γ0 + γ1p + γ2p2 + · · · ∈ Zp.

Note: γ = α − βτ is eventually periodic.

a(x) = a0 + a1x + a2x2 + · · · ∈ Z(p)[[x]]

bτ(x) = bτ + bτ+1x + bτ+2x2 + · · · ∈ Z(p)[[x]]


c(x) = a(x) − bτ(x) ∈ Z/(p)[[x]]

so: Cτ(a, b) = Z(c) =T−1∑

k=0

ζck =N+T−1∑

k=N

ζck.

Arithmetic analog:

α = a0 + a1p + a2p2 + · · · ∈ Zp.

βτ = bτ + bτ+1p + bτ+2p2 + · · · ∈ Zp.

γ = α − βτ = γ0 + γ1p + γ2p2 + · · · ∈ Zp.

Note: γ = α − βτ is eventually periodic.

Set Carithτ (a, b) = Z(γ) =

N+T−1∑

k=N

ζγk.

(sum over periodic part)

• Averaged over all pairs of sequences,

E(Cτ(a, b)) =

T (a = b and τ = 0)

0 (a 6= b or τ 6= 0)


E(Cτ(a, b)) =

T (a = b and τ = 0)

0 (a 6= b or τ 6= 0)

• RMS cross-correlation is

√E(Cτ(a, b)2) =

T (a = b and τ = 0)√2T (a = b, τ 6= 0, p = 2)√T (otherwise)


E(Cτ(a, b)) =

T (a = b and τ = 0)

0 (a 6= b or τ 6= 0)


√E(Cτ(a, b)2) =


• Average Arithmetic cross-correlation is:

E(Carithτ (a, b)) =

T (a = b, τ = 0)

T/pT−gcd(τ,T ) (a = b, τ 6= 0)

T/pT (otherwise)


E(Cτ(a, b)) =

T (a = b and τ = 0)

0 (a 6= b or τ 6= 0)


√E(Cτ(a, b)2) =


• Average Arithmetic cross-correlation is:

E(Carithτ (a, b)) =

T (a = b, τ = 0)

T/pT−gcd(τ,T ) (a = b, τ 6= 0)

T/pT (otherwise)

• RMS Arithmetic cross-correlation is:

√E(Carith

τ (a, b)2) =

T (a = b, τ = 0)√2Te1 (a = b, τ 6= 0, p = 2)√Te2 (a = b, τ 6= 0, p > 2)√Te3 (a 6= b)

(e1, e2, e3 ∼ 1)

m-sequences

Let a = a0, a1, a2, · · · be T -periodic, ai ∈ Z/(p).

Set a(x) = a0 + a1x + a2x2 + · · · ∈ Z/(p)[[x]].

m-sequences


Set a(x) = a0 + a1x + a2x2 + · · · ∈ Z/(p)[[x]].

Reduce to lowest terms:

a(x) =a0 + a1x + · · · + aT−1xT−1

1 − xT=

−h(x)

q(x)

m-sequences


Set a(x) = a0 + a1x + a2x2 + · · · ∈ Z/(p)[[x]].


a(x) =a0 + a1x + · · · + aT−1xT−1

1 − xT=

−h(x)

q(x)

• q(x) = −1 + q1x + · · · + qrxr

is the connection polynomial of a LFSRthat generates the sequence a.

ar−1 ar−2 · · · a0

&%'$q1 &%

'$q2 &%

'$qr· · ·

- -

m-sequences


Set a(x) = a0 + a1x + a2x2 + · · · ∈ Z/(p)[[x]].


a(x) =a0 + a1x + · · · + aT−1xT−1

1 − xT=

−h(x)

q(x)

• q(x) = −1 + q1x + · · · + qrxr

is the connection polynomial of a LFSRthat generates the sequence a.

ar−1 ar−2 · · · a0

&%'$q1 &%

'$q2 &%

'$qr· · ·

- -

• m-sequence ⇔ q(x) is a primitive polynomial

`-sequences


Set α = a0 + a1p + a2p2 + · · · ∈ Zp.

`-sequences


Set α = a0 + a1p + a2p2 + · · · ∈ Zp.


α =a0 + a1p + · · · + aT−1pT−1

1 − pT=

−h

q

`-sequences


Set α = a0 + a1p + a2p2 + · · · ∈ Zp.


α =a0 + a1p + · · · + aT−1pT−1

1 − pT=

−h

q

• q = −1 + q1p + · · · + qrpr ∈ Zis the connection integer of an FCSRthat generates the sequence a.

mr ar−1 ar−2 · · · a0

&%'$q1 &%

'$q2 &%

'$qr· · ·

∑-

� - -

��

�

mod pdiv p

`-sequences


Set α = a0 + a1p + a2p2 + · · · ∈ Zp.


α =a0 + a1p + · · · + aT−1pT−1

1 − pT=

−h

q


mr ar−1 ar−2 · · · a0

&%'$q1 &%

'$q2 &%

'$qr· · ·

∑-

� - -

��

�

mod pdiv p

• h < q and h ↔ initial loading

`-sequences


Set α = a0 + a1p + a2p2 + · · · ∈ Zp.


α =a0 + a1p + · · · + aT−1pT−1

1 − pT=

−h

q


mr ar−1 ar−2 · · · a0

&%'$q1 &%

'$q2 &%

'$qr· · ·

∑-

� - -

��

�

mod pdiv p

• h < q and h ↔ initial loading

• `-sequence ⇔ p is a primitive root modulo q.

Shift and add sequences

Let a = a0, a1, . · · · be T -periodic. (ai ∈ Z/(p))

It is a shift and add sequence if for any shift τ,

a + aτ =

aτ ′ some shift τ ′, or

0




a + aτ =


0

Theorem. (Zierler) The sequence a is a shift and

add sequence if and only if it is an m-sequence.




a + aτ =


0



Arithmetic shift and add

(Shift and add with carry)




a + aτ =


0





Let a = a0, a1, · · · be T -periodic (ai ∈ Z/(p))

It is an arithmetic shift and add sequence if, ∀τ ,

periodic part of (α + ατ) = shift of α

where α = a0 + a1p + a2p2 + · · · ∈ Zp.




a + aτ =


0





Let a = a0, a1, · · · be T -periodic (ai ∈ Z/(p))

It is an arithmetic shift and add sequence if, ∀τ ,

periodic part of (α + ατ) = shift of α

where α = a0 + a1p + a2p2 + · · · ∈ Zp.

Theorem. The sequence a is an arithmetic shift

and add sequence if and only if it is an `-sequence.

Summary of propertiesm-sequence `-sequence


Characterization:shift-and-add shift-and-add with carry



Period:

T = pr − 1 T = q − 1



Period:

T = pr − 1 T = q − 1Connection element:

polynomial q(x) integer q



Period:



Primitive element:α ∈ Fpr root p ∈ Z/(q)



Period:




Reduction mapping:

FprTr→ Fp Z/(q)

mod→ Fp



Period:




Reduction mapping:

FprTr→ Fp Z/(q)

mod→ Fp

Output sequence:

an = Tr(αn) an = p−n (mod q) (mod p)



Period:




Reduction mapping:

FprTr→ Fp Z/(q)

mod→ Fp

Output sequence:

an = Tr(αn) an = p−n (mod q) (mod p)Auto-correlation:

|Cτ(a, a)| ≤ 1|Carith

τ (a, a)| ≤ 1Carith

τ (a, a) = 0 for p = 2



Period:




Reduction mapping:

FprTr→ Fp Z/(q)

mod→ Fp

Output sequence:




τ (a, a) = 0 for p = 2Occurrences of a block b,

s = T/(p|b|)

N(b) ∈ {s, s + 1} N(b) ∈ {s, s + 1}



Period:




Reduction mapping:

FprTr→ Fp Z/(q)

mod→ Fp

Output sequence:




τ (a, a) = 0 for p = 2Occurrences of a block b,

s = T/(p|b|)

N(b) ∈ {s, s + 1} N(b) ∈ {s, s + 1}Decimations:

a 6= aτ for (τ, T ) = 1a 6= aτ for (τ, T ) = 1and q > 13 (conj.)

Generalizations

• Replace Z/(p) with Z/(N)

Generalizations


• Replace Z/(p) with Fpr

Generalizations



alphabet m-sequence `-sequence

Z/(N) N-adic numbers

Generalizations




Z/(N)Galois ringsinteresting

N-adic numbers

Generalizations





N-adic numbers

Fpr m-sequences

Generalizations





N-adic numbers

Fpr m-sequencesp-adic fieldsinteresting

Mark Goresky and Andrew Klapper

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