The p-adic Ergodic Theory - UPJVleve/ejc2011/PDF_EJC2011/cours_An… · Vladimir Anashin (Moscow State University) The p-adic Ergodic Theory Lectures at Universit´e de Picardie 8

The p-adic Ergodic Theoryand applications

Vladimir Anashin

Lomonosov Moscow State University

Ecole Jeunes Chercheurs du GDR Informatique MathematiqueUniversite de Picardie Jules Verne. Amiens, 2011

Vladimir Anashin (Moscow State University) The p-adic Ergodic TheoryLectures at Universite de Picardie 1 /

43

Outline

1 Basics of p-adicsp-adic numbersp-adic distancep-adic Calculus

2 The p-adic ergodic theoryBasicsErgodic transformations on the space of p-adic integersErgodic transformations on p-adic balls and p-adic spheres

3 ApplicationsLatin squaresPseudo random generators and stream ciphersTransducers


43

Reading

Vladimir Anashin and Andrei KhrennikovApplied Algebraic Dynamicsde Gruyter Expositions in Mathematics, vol. 49Walter de Gruyter, Berlin–New York, 2009.xxiv +533 pages.ISBN 978-3-11-020300-4


43

http://www.degruyter.de/cont/fb/ma/detailEn.cfm?id=IS-9783110203004-1

Basics of p-adics

Our first theme is:





43

Basics of p-adics p-adic numbers

So we start with:





43


p-adic integers

The ring Zp of p-adic integers may be thought of as a set of all infinitestrings

. . . α3α2α1α0 = ∙ ∙ ∙+ α3 ∙ p3 + α2 ∙ p

2 + α1 ∙ p+ α0

over alphabet Fp = {0, 1, . . . , p − 1} (the number p > 1 will usually be aprime); so all αi ∈ Fp.The rightmost representation is called a canonical p-adic form of a p-adic in-teger, or the base-p expansion; addition and multiplication of p-adic integerscan be defined via school-textbook algorithms.


43


p-adic integers

The ring Zp of p-adic integers may be thought of as a set of all infinitestrings

. . . α3α2α1α0 = ∙ ∙ ∙+ α3 ∙ p3 + α2 ∙ p

2 + α1 ∙ p+ α0

over alphabet Fp = {0, 1, . . . , p− 1}; addition and multiplication of p-adicintegers can be defined via school-textbook algorithms.

The following example shows that . . . 11111 = −1 in Z2:

. . . 1 1 1 1

+

. . . 0 0 0 1

. . . 0 0 0 0


43


p-adic integers

The following example shows that . . . 1010101 = −13 in Z2

. . . 0 1 0 1 0 1

×

. . . 0 0 0 0 1 1

. . . 0 1 0 1 0 1

+

. . . 1 0 1 0 1

. . . 1 1 1 1 1 1


43


p-adic integers

... and even a simple calculator understands this logic!


43


p-adic integers

Note that Zp ∩Q ' Z.


43


p-adic integers


For instance, in Z2:

Sequences that contain only finite number of 1-s correspond tonon-negative rational integers represented by their base-2 expansions:. . . 00011 = 3.

Sequences that contain only finite number of 0-s correspond tonegative rational integers:. . . 111100 = −4.

Sequences that are (eventually) periodic correspond to rationalnumbers that can be represented by irreducible fractions with odddenominators:. . . 1010101 = −13 .

Non-periodic sequences correspond to no rational numbers.


43


p-adic integers








43


p-adic integers








43


p-adic integers








43


The field Qp

Along with p-adic integers we can consider ‘non-integer’ p-adic numbers ofthe form

∞∑

i=−k

αipi,

where k ∈ N = {1, 2, 3, . . .}, αi ∈ Fp = {0, 1, . . . , p − 1}, define addi-tion and multiplication of these numbers in a similar way we did for p-adicintegers and thus obtain a field Qp of p-adic numbers.

The ring of p-adic integers Zp is a ring of integers of the field Qp.

The field Qp contains a field of rational numbers Q as a subfield.


43


The field Qp


∞∑

i=−k

αipi,





43


The field Qp


∞∑

i=−k

αipi,





43


The field Qp


∞∑

i=−k

αipi,




Although further we deal with Zp rather than with Qp, p-adic Calculus,p-adic dynamics, and p-adic ergodic theory can also be developed for Qp.


43

Basics of p-adics p-adic distance

Next topic:





43


Metric on Zp

Definition (Metric)

Let M be a non-empty set, and let d : M ×M → R≥0 be a 2-variatefunction that is defined on M and valuated in non-negative real numbers.The function d is called a metric (and M is called a metric space)whenever d obeys the following laws:

1 For every pair a, b ∈M : d(a, b) = 0 if and only if a = b.2 For every pair a, b ∈M : d(a, b) = d(b, a).3 For every triple a, b, c ∈M : d(a, b) ≤ d(a, c) + d(c, b).

For example, the set R of all real numbers is a metric space with a metricd(a, b) = |a− b|, where | ∙ | is an absolute value.


43


Metric on ZpA distance (=metric) dp(a, b) between a, b ∈ Zp is p−`, where` = (the length of the longest common prefix of a and b)− 1.Absolute value of a p-adic integer a ∈ Zp is a distance from a to 0: |a|p =dp(a, 0); that is, dp(a, b) = |a− b|p.

For instance, in the case p = 2 we have:

. . . 101010101 = −1

3. . . 000000101 = 5

⇒ d2

(

−1

3, 5

)

=1

24=1

16

In other words, −13 ≡ 5 (mod 16);−13 6≡ 5 (mod 32).

Reduction modulo pk is an epimorphism of Zp onto Z/pkZ:

modpk : Zp → Z/pkZ;

(∞∑

i=0

αipi

)

mod pk =k−1∑

i=0

αipi


43


Metric on ZpA distance (=metric) dp(a, b) between a, b ∈ Zp is p−`, where` = (the length of the longest common prefix of a and b)− 1.Absolute value of a p-adic integer a ∈ Zp is a distance from a to 0: |a|p =dp(a, 0); that is, dp(a, b) = |a− b|p.

The metric dp satisfies strong triangle inequality:

|a− b|p 6 max{|a− c|p, |c− b|p} for all a, b, c ∈ Zp,

The latter relation is called a strong triangle inequality, and a metric thatsatisfies this inequality is called a non-Archimedean metric, or an ultramet-ric.


43


Metric on ZpMetric on the n-th Cartesian power Znp of Zp can be defined in a similarway:

|(a1, . . . , an)− (b1, . . . , bn)|p = max{|ai − bi|p : i = 1, 2, . . . , n}

for every (a1, . . . , an), (b1, . . . , bn) ∈ Znp .

This metric is also an ultrametric.


43


Metric on ZpInteresting: In ultrametric spaces, appending a segment to itself may resultin a segment that is shorter than the original one!


43


Metric on ZpInteresting: In ultrametric spaces, appending a segment to itself may resultin a segment that is shorter than the original one!The strong triangle inequality implies:

All triangles are isosceles!

Any point inside a circle is a center of this circle!


43


Metric on ZpInteresting: In ultrametric spaces, appending a segment to itself may resultin a segment that is shorter than the original one!The strong triangle inequality implies:

All triangles are isosceles!

Any point inside a circle is a center of this circle!


43


p-adic limits

Once a metric is defined we can speak of convergence, limits, continuousfunctions, derivatives, etc.


43


p-adic limits

Once a metric is defined we can speak of convergence, limits, continuousfunctions, derivatives, etc.

Definition (Limit)

A p-adic integer z is said to be a limit of the sequence (zi)∞i=0 if and only

if for every real ε > 0 there exists N such that |zi − z|p < ε for all i > N .


43


p-adic limits

Definition (Limit)



However, according to the definition of the p-adic metric, the value of |zi−z|p is equal to p−` for a suitable ` = 0, 1, 2, . . .; so we may consider onlyε = p−r for r = 0, 1, 2, . . .. and thus re-state the definition:


43


p-adic limits

Definition (Limit)



However, according to the definition of the p-adic metric, the value of |zi−z|p is equal to p−` for a suitable ` = 0, 1, 2, . . .; so we may consider onlyε = p−r for r = 0, 1, 2, . . .. and thus re-state the definition:

Definition (Limit, equivalent form)


if for every (sufficiently large) positive rational integer K there exists Nsuch that zi ≡ z (mod pK) for all i > N .

NB: By the definition of the p-adic metric, |zi − z|p ≤ p−K ⇐⇒ zi ≡ z

(mod pK)


43


p-adic limits




Example: 1, 3, 7, 15, 31, . . . , 2i − 1 . . .2−→− 1


43


p-adic limits




Example: 1, 3, 7, 15, 31, . . . , 2i − 1 . . .2−→− 1

This is not too odd, however

. . . 0 0 0 0 1 = 1

. . . 0 0 0 1 1 = 3

. . . 0 0 1 1 1 = 7

. . . 0 1 1 1 1 = 15∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙. . . 1 1 1 1 1 = −1


43


p-adic limits




In p-adics, students’ dream is getting true:

p-adic series∑∞i=0 xi converges if and only if xi tends to 0 as i→∞:

p

limi→∞

xi = 0.


43


p-adic topology

The base of topology on Zp are p-adic balls.A (closed) ball B 1

pk(a) of radius 1

pkcentered at a ∈ Zp is

B 1

pk(a) =

{

z ∈ Zp : |z − a|p ≤1

pk

}

= {z ∈ Zp : z≡a (mod pk)} = a+pkZp.

That is, B 1

pk(a) is a set of all infinite words over alphabet Fp =

{0, 1, . . . , p − 1} that have the same prefix of length k as the worda ∈ Zp.


43


p-adic topology

A (closed) ball B 1

pk(a) of radius 1


B 1

pk(a) =

{

z ∈ Zp : |z − a|p ≤1

pk

}


That is, B 1

pk(a) is a set of all infinite words over alphabet Fp =

{0, 1, . . . , p−1} that have the same prefix of length k as the word a ∈ Zp.

The set of all infinite binary words of the form∙ ∙ ∙ ∗ ∗ ∗ ∗ ∗ 0101 = 5 + 16 ∙ Z2 = −13 + 16 ∙ Z2is a 2-adic ball of radius 116 centered at 5 (or, which is the same, centeredat −13 , etc.)


43


p-adic topology

A (closed) ball B 1

pk(a) of radius 1


B 1

pk(a) =

{

z ∈ Zp : |z − a|p ≤1

pk

}


A closed ball (of radius 1pk) is an open ball (of radius 1

pk−1) and vice versa!


43


p-adic topology

A (closed) ball B 1

pk(a) of radius 1


B 1

pk(a) =

{

z ∈ Zp : |z − a|p ≤1

pk

}




Indeed,

B 1

pk(a) =

{

z ∈ Zp : |z − a|p ≤1

pk

}

=

{

z ∈ Zp : |z − a|p <1

pk−1

}

since |u− v|p takes only discrete values:

|u− v|p ∈

{

1,1

p,1

p2, . . .

}


43


p-adic topology

A (closed) ball B 1

pk(a) of radius 1


B 1

pk(a) =

{

z ∈ Zp : |z − a|p ≤1

pk

}




A p-adic ball of radius 1pkis a disjoint union of p balls of radius 1

pk+1each.


43


p-adic topology

A (closed) ball B 1

pk(a) of radius 1


B 1

pk(a) =

{

z ∈ Zp : |z − a|p ≤1

pk

}





pk+1each.

Indeed,

B 1

pk(a) = a+ pkZp =

a+pk+1Zp∪a+1+pk+1Zp∪a+2+p

k+1Zp∪ . . .∪a+(p−1)+pk+1Zp


43


p-adic topology

A (closed) ball B 1

pk(a) of radius 1


B 1

pk(a) =

{

z ∈ Zp : |z − a|p ≤1

pk

}





pk+1each.

A p-adic sphere S 1

pk(a) of radius 1

pkcentered at a ∈ Zp is a disjoint union

of p− 1 balls of radius 1pk+1

each.

Indeed, S 1

pk(a) = {z ∈ Zp : |z − a|p = 1

pk} = B 1

pk(a) \B 1

pk+1(a)


43


p-adic topology

A (closed) ball B 1

pk(a) of radius 1


B 1

pk(a) =

{

z ∈ Zp : |z − a|p ≤1

pk

}





pk+1each.

A p-adic sphere S 1

pk(a) of radius 1

pkcentered at a ∈ Zp is a disjoint union

of p− 1 balls of radius 1pk+1

each.

BTW, a 2-adic sphere of radius 12kis a ball of a half-small radius 1

2k+1.


43


p-adic topology

Zp is compact (actually, it is a p-adic ball of radius 1).

Zp is totally disconnected (a connected component ofa point is the point).

Rational integers Z = {0,±1,±2, . . .}, as well aspositive rational integers N = {1, 2, 3, . . .} andnegative rational integers −N = {−1,−2,−3, . . .},are everywhere dense in Zp.


43


Continuous p-adically valued functions on Zp

Definition (Continuous function)

A function f : Zp → Zp is said to be continuous at the point z ∈ Zp if andonly if for every (sufficiently large) positive rational integer M there thereexists a positive rational integer L such that f(x) ≡ f(z) (mod pM )whenever x ≡ z (mod pL)

Note: The function f is said to be uniformly continuous (=equicontinuous)on Zp if and only if f is continuous at every point z ∈ Zp, and L dependsonly on M , and not on z.


43





Important example of uniformly continuous transformations onZp are compatible functions (in algebra, compatible transforma-tions=transformations of an algebraic system that agree with all congru-ences the algebraic system).


43





Definition (Compatible function)

A map f : Zp → Zp is called compatible iff a ≡ b (mod pk) impliesf(a) ≡ f(b) (mod pk).


43





As |zi − z|p ≤ p−K ⇐⇒ zi ≡ z (mod pK), by the definition of the p-adicmetric,

compatibility=1-Lipschitz property:

A map f : Zp → Zp is compatible iff it satisfies p-adic Lipschitz conditionwith a constant 1: |f(a)− f(b)|p ≤ |a− b|p for all a, b ∈ Zp.


43





compatibility=1-Lipschitz property:

A map f : Zp → Zp is compatible iff it satisfies p-adic Lipschitz conditionwith a constant 1: |f(a)− f(b)|p ≤ |a− b|p for all a, b ∈ Zp.

compatibility=triangularity:

A map f : Zp → Zp is called triangular iff it is of the form

∙ ∙ ∙+χ2 ∙p2+χ1 ∙p+χ0f7→ ∙ ∙ ∙+ψ2(χ0, χ1, χ2) ∙p2+ψ1(χ0, χ1) ∙p+ψ0(χ0),

where χ0, χ1, . . . ∈ Fp = {0, 1, . . . , p− 1}, ψi : Fi+1p → Fp, i = 0, 1, 2, . . .


43


Automata maps are continuous p-adic functions

A

〈I, S,O, S,O, s0〉

Automaton A = 〈I, S,O, S,O, s0〉: I – input alphabet; O – outputalphabet; S – state set; S : I× S→ S – transition function;O : I× S→ O – output function; s0 ∈ S – initial state

χi ξi

ξi ∈ O — i-th output symbol

Obvious; however, important: ξi depends only on χ0, . . . , χi ∈ I :

ξi = ψi(χ0, . . . , χi) ∈ O

χi ∈ I — i-th input symbol

Note: Both the input alphabet I and the output alphabet O are assumedto be finite; however, the state set S might be infinite.


43



A

〈I, S,O, S,O, s0〉

Automaton A = 〈I, S,O, S,O, s0〉: I – input alphabet; O – outputalphabet; S – state set; S : I× S→ S – transition function;O : I× S→ O – output function; s0 ∈ S – initial state

χi ξi

ξi ∈ O — i-th output symbol

Obvious; however, important: ξi depends only on χ0, . . . , χi ∈ I :

ξi = ψi(χ0, . . . , χi) ∈ O

χi ∈ I — i-th input symbol

Further we mostly deal with transducers; i.e., with automata that havenon-empty input and output. Moreover, we mostly assume that I = O.


43



Let I = O = Fp = {0, 1, . . . , p− 1}; then, as any automaton (=transducer)A transforms (infinite) input sequence . . . , χ1, χ0 into (infinite) output se-quence . . . , ξi = ψi(χ0, χ1, . . . , χi), . . . , ξ1 = ψ1(χ0, χ1), ξ0 = ψ0(χ0),the automaton A determines a (unique) transformation fA of the ring Zp:

∙ ∙ ∙+χ2 ∙p2+χ1 ∙p+χ0

fA7→ ∙ ∙ ∙+ψ2(χ0, χ1, χ2) ∙p2+ψ1(χ0, χ1) ∙p+ψ0(χ0).

The sequence of maps ψi : Fi+1p → Fp, i = 0, 1, 2, . . ., is completelydetermined by the automaton A, and vice versa.


43



Let I = O = Fp = {0, 1, . . . , p− 1}; then, as any automaton (=transducer)A transforms (infinite) input sequence . . . , χ1, χ0 into (infinite) output se-quence . . . , ξi = ψi(χ0, χ1, . . . , χi), . . . , ξ1 = ψ1(χ0, χ1), ξ0 = ψ0(χ0),the automaton A determines a (unique) transformation fA of the ring Zp:

∙ ∙ ∙+χ2 ∙p2+χ1 ∙p+χ0

fA7→ ∙ ∙ ∙+ψ2(χ0, χ1, χ2) ∙p2+ψ1(χ0, χ1) ∙p+ψ0(χ0).

The sequence of maps ψi : Fi+1p → Fp, i = 0, 1, 2, . . ., is completelydetermined by the automaton A, and vice versa.

As ξi = ψi(χ0, . . . , χi), i = 0, 1, 2, . . ., the transformation fA iscompatible; i.e., satisfies a p-adic Lipschitz condition with a constant 1.Conversely, every 1-Lipschitz transformation f of the space Zp isdetermined by some automaton A: f = fA.


43


Computers think 2-adically!

T-functions are special maps that are used in modern cryptography andcomputer science:

Definition (T-function)

A (univariate) T-function is a mapping that maps binary words (finite orinfinite) to binary words w.r.t. the rulef : (χ0, χ1, χ2, . . .) 7→ (ψ0(χ0);ψ1(χ0, χ1);ψ2(χ0, χ1, χ2); . . .),where χj ∈ {0, 1}, and each ψj(χ0, . . . , χj) is a Boolean function inBoolean variables χ0, . . . , χj .


43





Accordingly, an m-variate T-function function is a mapping

(α↓0, α↓1, α

↓2, . . .) 7→ (Φ

↓0(α

↓0),Φ

↓1(α

↓0, α

↓1),Φ

↓2(α

↓0, α

↓1, α

↓2), . . .).

Here α↓i ∈ Bm is a Boolean columnar m-dimensional vector; B =

{0, 1}; Φ↓i : (Bm)(i+1) → Bn maps (i + 1) Boolean columnar m-

dimensional vectors α↓0, . . . , α↓i to a n-dimensional columnar Boolean vector

Φ↓i (α↓0, . . . , α

↓i ).


43





T-functions may be viewed as maps from Z2 to Z2: e.g., a univariate T-function f sends a 2-adic integer

χ0 + χ1 ∙ 2 + χ2 ∙ 22 + ∙ ∙ ∙

to the 2-adic integer

ψ0(χ0) + ψ1(χ0, χ1) ∙ 2 + ψ2(χ0, χ1, χ2) ∙ 22 + ∙ ∙ ∙


43





Every T-function f is compatible with all congruences modulo 2k:

a ≡ b (mod 2k)⇒ f(a) ≡ f(b) (mod 2k).

Vice versa, every map f : Z2 → Z2 that is compatible with all congruencesmodulo 2k for all k = 1, 2, . . ., is a T-function.


43





Thus we conclude:

T-functions are 1-Lipschitz (whence, continuous and thus uniformlycontinuous) 2-adic functions.


43





Examples of T-functions:

arithmetic (numerical) instructions (integer addition, integermultiplication, etc.);

bitwise logical instructions (e.g., and, or, not, xor);

other machine instructions; e.g., shifts towards more significant bits,masking, etc.

arbitrary compositions of the above instructions.


43











43











43











43





Shifts towards less significant bits are not T-functions; however, they arecontinuous 2-adic maps!


43





This is also a T-function:

f(x) = 1 + x+ 4 ∙

(

1− 2 ∙x and x2 + x3 or x4

3− 4 ∙ (5 + 6x5)x6xorx7

)8+not(

8x8

9+10x9

)


43





While processing straight line programs, digital computersjust make evaluations of continuous 2-adic functions w.r.t.2-adic precision determined by the computers’ bitlengths.

Note that loading a long number into a short registry results in automaticallyreduction of the number modulo 2`, where ` is bitlength of the registry.


43





While processing straight line programs, digital computersjust make evaluations of continuous 2-adic functions w.r.t.2-adic precision determined by the computers’ bitlengths.

Branching programs (i.e., the ones containing jumps) can also be consideredas continuous transformations on certain (more sophisticated) ultrametricspaces; see Part II of Applied Algebraic Dynamics for details.


43

Basics of p-adics p-adic Calculus

Next topic:





43


p-adic derivations

Definition (differentiability at a point)

The differentiability of a univariate function f : Zp → Zp at the pointx ∈ Zp means that for arbitrary M ∈ N0 and sufficiently small h ∈ Zp

∣∣∣∣f(x+ h)− f(x)

h− f ′(x)

∣∣∣∣p

≤1

pM

or, in the equivalent form

f(x+ h) ≡ f(x) + f ′(x) ∙ h (mod pordp h+M ),

where ordp h = − logp |h|p is the p-adic valuation of h.

NB: ordp h is the length of zero prefix in a base-p expansion of h.


43


p-adic derivations

Similarly, a multi-variate function F = (f1, . . . , fm) : Znp → Zmp is said tobe differentiable at the point x = (x1, . . . , xn) ∈ Znp if there exists an n×mmatrix F ′(x) over Qp such that for every positive rational integer M ∈ Nand all sufficiently small h = (h1, . . . , hn) ∈ Znp the following congruenceholds:

F (x+ h) ≡ F (x) + h ∙ F ′(x) (mod pordp h+M )

Here ordp h = min{ordp hi : i = 1, 2, . . . , n} by the definition.

Entries of the matrix F ′(x) are partial derivatives of F at the point x.

If F is 1-Lipschitz, then all partial derivatives are p-adic integers.


43


p-adic derivations

Definition (uniform differentiability)

A function f : Zp → Zp is called uniformly differentiable (or,equidifferentiable) on Zp iff for every sufficiently large M ∈ N0 there existsK ∈ N0 such that once |h|p 6 1

pK, the inequality

∣∣∣f(x+h)−f(x)h − f ′(x)

∣∣∣p6 1pM(or, which is equivalent, the congruence

f(x+ h) ≡ f(x) + f ′(x) ∙ h (mod pordp h+M )) holds for all x ∈ Zp. GivenM , the minimum K with this property is denoted via NM (f).

Similarly: a multi-variate function F = (f1, . . . , fm) : Znp → Zmp is uniformlydifferentiable iff the congruence

F (x+ h) ≡ F (x) + h ∙ F ′(x) (mod pordp h+M )

holds for all x ∈ Znp once ordp h ≥ NM (F ).


43


p-adic derivations



pK, the inequality

∣∣∣f(x+h)−f(x)h − f ′(x)



Polynomials over Zp are uniformly differentiable on Zp.


43


p-adic derivations



pK, the inequality

∣∣∣f(x+h)−f(x)h − f ′(x)



If c ∈ Z then bitwise logical instructions f(x) = x and c,f(x) = x or c, f(x) = x xor c are uniformly differentiable on Z2


43


p-adic derivations

Proposition

The function f(x) = x and c is uniformly differentiable on Z2 for anyc ∈ Z, and

f ′(x) =

{0, if c ≥ 0;

1, if c < 0.


43


p-adic derivations

Proposition


f ′(x) =

{0, if c ≥ 0;

1, if c < 0.

Indeed, take n greater than the bitlength of |c|; then for all s ∈ Z2:

f(x+ 2ns) =

{f(x) , if c ≥ 0,

f(x) + 2ns , if c < 0,


43


p-adic derivations

Proposition


f ′(x) =

{0, if c ≥ 0;

1, if c < 0.

Indeed, take n greater than the bitlength of |c|; then for all s ∈ Z2:

f(x+ 2ns) =

{f(x) , if c ≥ 0,

f(x) + 2ns , if c < 0,

since all most significant bits in a base-2 expansion of a positive rationalinteger are 0; whereas they all are 1 for a negative one.


43


p-adic derivations

The rest of table of derivations of bitwise logical instructions

(notx)′ = −1.

If c ∈ Z, then

(x xor c)′ =

{1, if c ≥ 0;

−1, if c < 0.

If c ∈ Z, then

(x or c)′ =

{1, if c ≥ 0;

0, if c < 0.


43


p-adic derivations


(notx)′ = −1.

If c ∈ Z, then

(x xor c)′ =

{1, if c ≥ 0;

−1, if c < 0.

If c ∈ Z, then

(x or c)′ =

{1, if c ≥ 0;

0, if c < 0.


43


p-adic derivations


(notx)′ = −1.

If c ∈ Z, then

(x xor c)′ =

{1, if c ≥ 0;

−1, if c < 0.

If c ∈ Z, then

(x or c)′ =

{1, if c ≥ 0;

0, if c < 0.


43


p-adic derivations


(notx)′ = −1.

If c ∈ Z, then

(x xor c)′ =

{1, if c ≥ 0;

−1, if c < 0.

If c ∈ Z, then

(x or c)′ =

{1, if c ≥ 0;

0, if c < 0.

Rules of derivation are the same both in ‘Archimedean’ and p-adic cases.


43


p-adic derivations


(notx)′ = −1.

If c ∈ Z, then

(x xor c)′ =

{1, if c ≥ 0;

−1, if c < 0.

If c ∈ Z, then

(x or c)′ =

{1, if c ≥ 0;

0, if c < 0.

Example:

The T-function f(x) = x+ x2 or 5 is uniformly differentiable on Z2;f ′(x) = 1 + 2x.


43


Differentiability modulo pM

Whereas the p-adic differentiability is a direct analog of a common(‘Archimedean’) differentiability, the differentiability modulo pM has no suchanalog.


43



Whereas the p-adic differentiability is a direct analog of a common(‘Archimedean’) differentiability, the differentiability modulo pM has no suchanalog.

Definition (differentiability modulo pM )

A function F = (f1, . . . , fm) : Znp → Zmp is said to be differentiable modulopM at the point x = (x1, . . . , xn) ∈ Znp if there exists an n×m matrixF ′M (u) over Qp such that for every sufficiently smallh = (h1, . . . , hn) ∈ Znp the following congruence holds:

F (x+ h) ≡ F (x) + h ∙ F ′M (x) (mod pordp h+M ).


43






Note that in a contrast to ‘ordinary’ p-adic differentiability, now thepositive rational integer M ∈ N is fixed!


43






Note that in a contrast to ‘ordinary’ p-adic differentiability, now thepositive rational integer M ∈ N is fixed!

Entries ∂Mfi(x)∂Mxjof the matrix F ′M (x) are called partial derivatives modulo

pM ); they are determined up to a summand that is 0 modulo pM . That is,if F is 1-Lipschitz, then all its partial derivatives modulo pM are elementsof Z/pMZ, the residue ring modulo pM .


43






Entries ∂Mfi(x)∂Mxjof the matrix F ′M (x) are called partial derivatives modulo

pM ); they are determined up to a summand that is 0 modulo pM . That is,if F is 1-Lipschitz, then all its partial derivatives modulo pM are elementsof Z/pMZ, the residue ring modulo pM .

Rules of derivation modulo pM are similar to those of ‘ordinary’ case;however, they are congruences modulo pM , and not equalities.


43






Differentiability modulo pM implies differentiability modulo pM−1.

‘Ordinary’ p-adic differentiability implies differentiability modulo pM

for all M = 1, 2, 3, . . ..


43






The function F is said to be uniformly differentiability modulo pM iff given

a sufficiently small h, the above congruence holds for all x ∈ Znp si-

multaneously. The smallest ordp h with this property is denoted via NM (F ).

If F is uniformly differentiable modulo pM , its (partial) derivatives modulopM are periodic; the period is pNM (F ). Thus, the derivatives are welldefined on (Z/pNM (F )Z)n (and valuated in Z/pMZ).


43






Example: f(x, y) = x xor y

The T-function f(x, y) = x xor y is uniformly differentiable modulo 2 onZ22; its partial derivatives modulo 2 are ≡ 1 (mod 2).


43






Example: f(x, y) = x xor y

The T-function f(x, y) = x xor y is uniformly differentiable modulo 2 onZ22; its partial derivatives modulo 2 are ≡ 1 (mod 2).However, x xor y is differentiable modulo 4 at no point of Z22; whence,x xor y is nowhere differentiable on Z22.


43

The p-adic ergodic theory

The next theme is:





43

The p-adic ergodic theory Basics

Next topic:





43


Measure preservation and ergodicity

Definition (measure preservation; ergodicity)

A map f : S→ T of a measure space S onto a measure space T endowedwith probability measures μ and ν respectively is called

measure-preserving iff μ(f−1(T )) = ν(T ) for each ν-measurablesubset T ⊂ T, and

ergodic iff S = T, μ = ν, f is measure-preserving and f has no properinvariant μ-measurable subsets:

f−1(S) = S implies either μ(S) = 0 or μ(S) = 1.


43



A trivial (however, important!) example: Let both S and T be finite,

let both μ(S) =#S

#Sand ν(T ) =

#T

#Tbe “common” probability measures,

and let f map S onto T; then:

f is measure-preserving iff f is balanced, i.e., iff the number off -pre-images of a point from T does not depend on the point:

#f−1(t) =#S#Tfor any t ∈ T.

In particular, in the case when S = T and μ = ν, the map f ismeasure-preserving iff f is bijective, i.e., iff f is a permutation on S.

Finally, f is ergodic iff it is transitive; that is, iff f is a single cyclepermutation on S (clearly, the length of the cycle is #S).


43




let both μ(S) =#S

#Sand ν(T ) =

#T








43




let both μ(S) =#S

#Sand ν(T ) =

#T








43



Definition (Bijectivity, transitivity, and balance modulo pk)

A 1-Lipschitz map F : Znp → Znp is called bijective (resp., transitive)modulo pk iff the reduced map F mod pk : x 7→ F (x) (mod pk) is abijective (resp,. transitive) transformation on the n-th Cartesianpower (Z/pkZ)n of the residue ring Z/pkZ.

A 1-Lipschitz map F : Zmp → Znp is called balanced modulo pk iff the

reduced map F mod pk is balanced; that is, F mod pk maps(Z/pkZ)m onto (Z/pkZ)n and all elements from (Z/pkZ)n haveequal numbers of F -preimages in (Z/pkZ)m.

Bijectivity modulo pk = invertibility of F mod pk

Transitivity modulo pk = single cycle property of F mod pk: thesequence F (x) mod pk, F (F (x)) mod pk, F (F (F (x))) mod pk, . . .has the longest possible period, of length pkn.


43










43










43










43


Main theorem

The space Zp is endowed with a natural probability measure μp, thenormalized Haar measure. Elementary measurable sets are p-adic balls:μp(Bp−k(a)) = p

−k.

Reminder: A p-adic ball Bp−k(a) = a + pkZp of radius p−k centered ata ∈ Zp is the set of all p-adic integers that are congruent to a modulo pk;that is, the ball consists of all infinite strings that have common initial prefixof length k.By the definition, the measure of this ball is μp(a+ p

kZp) = p−k.


43


Main theorem


−k.


kZp) = p−k.

For example, ∙ ∙ ∙ ∗ ∗ ∗ ∗ ∗ 0101 = 5+ 16 ∙Z2 = −13 + 16 ∙Z2 is a 2-adic ball

of radius (and of measure) 116 centered at 5 (or, the same, at −13) consists

of all 2-adic integers whose initial prefix (of length 4) is 0101.

. . . 101010101 = −1

3. . . 000000101 = 5


43


Main theorem


−k.


kZp) = p−k.

That is, μp(Bp−k(a)) is just a probability that a randomly chosen

p-adic integer is congruent to a modulo pk;

or in other words, μp(Bp−k(a)) is just a probability that the initialk-letter segment of an infinite random string over the alphabet{0, 1, . . . , p− 1} agrees with the initial k-letter segment of the stringthat represents the p-adic integer a.


43


Main theorem


−k.


kZp) = p−k.

That is, μp(Bp−k(a)) is just a probability that a randomly chosen

p-adic integer is congruent to a modulo pk;

or in other words, μp(Bp−k(a)) is just a probability that the initialk-letter segment of an infinite random string over the alphabet{0, 1, . . . , p− 1} agrees with the initial k-letter segment of the stringthat represents the p-adic integer a.


43


Main theorem


kZp) = p−k.In a similar way we define the measure μp on Znp :By the definition, the measure of an n-dimensional p-adic ball of radius p−k

is p−kn.


43


Main theorem

By the definition, the measure of an n-dimensional p-adic ball of radius p−k

is p−kn.

Definition (measure preservation and ergodicity on Zp)

A map f : Zp → Zp is called ergodic iff

f preserves the measure μp; that is, μp(f−1(S)) = μp(S) for each

μp-measurable subset S ⊂ Zp, and

f has no proper invariant μp-measurable subsets:

f−1(S) = S implies either μp(S) = 0 or μp(S) = 1.


43


Main theorem

By the definition, the measure of an n-dimensional p-adic ball of radius p−k

is p−kn.

Theorem (V. A., 2002)

A 1-Lipschitz map F : Znp → Zmp preserves the measure μp iff every itsreduction modulo pk

F mod pk : (Z/pkZ)n → (Z/pkZ)m

is balanced, for all k = 1, 2, 3, . . .

Once m = n = 1, the 1-Lipschitz map F is ergodic iff its reductionmodulo pk

F mod pk : Z/pkZ→ Z/pkZ

is transitive, for all k = 1, 2, 3, . . ..


43


Main theorem


A 1-Lipschitz map F : Znp → Zmp preserves the measure μp iff every itsreduction modulo pk

F mod pk : (Z/pkZ)n → (Z/pkZ)m

is balanced, for all k = 1, 2, 3, . . .

Once m = n = 1, the 1-Lipschitz map F is ergodic iff its reductionmodulo pk

F mod pk : Z/pkZ→ Z/pkZ

is transitive, for all k = 1, 2, 3, . . ..

In other words: Measure preservation (resp., ergodicity) of a p-adic 1-Lipschitz transformation f : Zp → Zp is equivalent to bijectivity (resp.,transitivity) of all maps f mod pk : Z/pkZ→ Z/pkZ.


43


Main theorem

Measure preservation (resp., ergodicity) of a p-adic 1-Lipschitz transforma-tion f : Zp → Zp is equivalent to bijectivity (resp., transitivity) of all mapsf mod pk : Z/pkZ→ Z/pkZ.

It turns out that in many cases bijectivity (resp., transitivity) of f modulopk for some k implies measure preservation (resp., ergodicity) of f on Zp;whence by main theorem implies bijectivity (resp., transitivity) of fmodulo pk for all k = 1, 2, 3, . . .


43

The p-adic ergodic theory Ergodic transformations on the space of p-adic integers

Next topic:





43


Ergodicity of uniformly differentiable functions

Theorem (on measure preservation; V. A., 2002)

Let a 1-Lipschitz function F : Znp → Zmp be uniformly differentiable modpon Znp .

The function F preserves the measure μp whenever f is balancedmodulo pk for some (equivalently, for any) k ≥ N1(F ) and the rankof the matrix F ′1(u) modulo p is m for allu = (u1, . . . , un) ∈ (Z/pN1(F )Z)n.

When m = n, these conditions are also necessary: the function Fpreserves the measure μp if and only if F is bijective modulo p

k forsome (equivalently, for any) k ≥ N1(F ) and det(F ′1(u)) 6≡ 0 (mod p)for all u = (u1, . . . , un) ∈ (Z/pN1(F )Z)n.

When m = n, the function F preserves the measure μp if and only ifF is bijective modulo pk for some (equivalently, for any)k ≥ N1(F ) + 1.


43










43










43



Theorem (on ergodicity, V. A., 1994)

Let a 1-Lipschitz function f : Zp → Zp be uniformly differentiable modp2

on Zp. The function f is ergodic if and only if f is transitive modulo pk

for some (equivalently, for any) k > N2(f) + 2.


43


Ergodicity in terms of Mahler expansion

Various p-adic techniques may be used to determine measure preservationand/or ergodicity, and not only the one that uses derivations. Other tech-niques are based on convergent p-adic series, e.g. on Mahler series, van derPut series, Taylor series, etc.

Due to time constraints, in the lecture we consider only Mahler series.


43



Every continuous map f : Zp → Zp can be expressed via Mahler interpola-tion series

f(x) =∞∑

i=0

ai

(x

i

)

.

Here ai ∈ Zp, i = 0, 1, 2, . . ., and

(x

i

)

=x(x− 1) ∙ ∙ ∙ (x− i+ 1)

i!

for i = 1, 2, . . .;(x0

)= 1 by the definition.


43



Every continuous map f : Zp → Zp can be expressed via Mahler interpola-tion series

f(x) =∞∑

i=0

ai

(x

i

)

.

Here ai ∈ Zp, i = 0, 1, 2, . . ., and

(x

i

)

=x(x− 1) ∙ ∙ ∙ (x− i+ 1)

i!

for i = 1, 2, . . .;(x0

)= 1 by the definition.

The series converges uniformly on Zp if and only if

p

limi→∞

ai = 0.

Uniformly continuous Mahler series determines a uniformly continuous func-tion.


43



Theorem (V. A.: 1994 for p = 2, 2002 for p > 2)

The function

f(x) =∞∑

i=0

ai

(x

i

)

determines a 1-Lipschitz ergodic transformation on Zp whenever thefollowing conditions hold simultaneously:

a0 6≡ 0 (mod p);

a1 ≡ 1 (mod p), for p odd;

a1 ≡ 1 (mod 4), for p = 2;

ai ≡ 0 (mod pblogp(i+1)c+1), i = 2, 3, . . . .

Moreover, in the case p = 2 these conditions are necessary.


43


Various classes of ergodic transformations on ZpDenote xi = x(x− 1) ∙ ∙ ∙ (x− i+ 1) the i-th descending factorial power ofx ∈ Zp. Then all functions from the following class B are 1-Lipschitz anduniformly differentiable on Zp:

B =

{

f : Zp → Zp : f(x) =∞∑

i=0

eixi,where ei ∈ Zp; i = 0, 1, 2, . . .

}

.

In other words, if f(x) =∑∞i=0 ai

(xi

)is Mahler expansion for f , then

f ∈ B if and only if aii! are p-adic integers for all i = 0, 1, 2, . . ..

Actually the class B is a Stone-Weierstrass completion of polynomial func-tions over Zp: along with polynomials, it contains analytic functions over Zp(e.g., entire functions, exponential functions) and some other functions.


43


Various classes of ergodic transformations on ZpDenote xi = x(x− 1) ∙ ∙ ∙ (x− i+ 1) the i-th descending factorial power ofx ∈ Zp. Then all functions from the following class B are 1-Lipschitz anduniformly differentiable on Zp:

B =

{

f : Zp → Zp : f(x) =∞∑

i=0

eixi,where ei ∈ Zp; i = 0, 1, 2, . . .

}

.

Actually the class B is a Stone-Weierstrass completion of polynomial func-tions over Zp: along with polynomials, it contains analytic functions overZp (e.g., entire functions, exponential functions) and some other functions.


A B-function f is measure preserving iff f is bijective modulo p2.

A B-function f is ergodic iff f is transitive either modulo p2 whenp /∈ {2, 3} or modulo p3 when p ∈ {2, 3}.


43


Various classes of ergodic transformations on ZpDenote xi = x(x− 1) ∙ ∙ ∙ (x− i+ 1) the i-th descending factorial power ofx ∈ Zp.

B =

{

f : Zp → Zp : f(x) =∞∑

i=0

eixi,where ei ∈ Zp; i = 0, 1, 2, . . .

}

.

Corollary (measure preservation/ergodicity criteria for p = 2)

A T-function f ∈ B is measure preserving if and only if

e1 ≡ 1 (mod 2), e2 ≡ 0 (mod 2), e3 ≡ 0 (mod 2).

A T-function f ∈ B is ergodic on Z2 if and only if

e0 ≡ 1 (mod 2), e1 ≡ 1 (mod 4), e2 ≡ 0 (mod 2), e3 ≡ 0 (mod 4).


43


Various classes of ergodic transformations on ZpThe following theorem, which can be derived from the measure-preservation/ergodicity criteria for Mahler series, yields a general and com-plete characterization of measure preserving/ergodic T-functions:

Theorem ( V. A., 2002)

A T-function f : Z2 → Z2 is measure preserving if and only if it is ofthe form

f(x) = c+ x+ 2 ∙ g(x),

where c ∈ Z2 and g : Z2 → Z2 is arbitrary T-function.

A T-function f : Z2 → Z2 is ergodic if and only if it is of the form

f(x) = 1 + x+ 2 ∙ (g(x+ 1)− g(x)),

where g : Z2 → Z2 is arbitrary T-function.


43


Various classes of ergodic transformations on Zp

Examples

The function f(x) = 1 + x+ p2

1+px is ergodic on Zp; e.g.,

1 + x+ 41+2x is an ergodic T-function.

The function f(x) = (1 + p)x+ (1 + p)x is ergodic on Zp; e.g.,3x+ 3x is an ergodic T-function.

The following T-function is ergodic on Z2:

f(x) =2 +x

3+1

3x+ 2 ∙

((x2 + 2x) xor (1/3)

2x+ 3

) (x+1)and(1/5)1−2x

+ 2 ∙ not

((x2 − 1) xor (1/3)

2x+ 1

)xand(1/5)5−2x

.


43

The p-adic ergodic theory Ergodic transformations on p-adic balls and p-adic spheres

Next topic:





43


Ergodic transformations on p-adic balls

Usually the problem to determine ergodicity of a 1-Lipschitz transformationon a ball Bp−k(a) = a + pkZp from Zp can be reduced to the one on thewhole space Zp.

Indeed, if f : Zp → Zp is a 1-Lipschitz transformation on Zp s.t.f(a+ pkZp) ⊂ a+ pkZp, then necessarily f(a) = a+ pky for asuitable y ∈ Zp.

Thus, f(a+ pkz) = f(a) + pk ∙ g(z) for any z ∈ Zp; so we can relateto f the following 1-Lipschitz transformation on Zp

g : z 7→ g(z) =1

pk(f(a+ pkz)− a− pky); z ∈ Zp.

The transformation f is ergodic on the ball Bp−k(a) if and only if thetransformation g is ergodic on Zp.


43






g : z 7→ g(z) =1




43






g : z 7→ g(z) =1




43






g : z 7→ g(z) =1




43


Ergodic transformations of spheres in ZpThe problem to determine whether a 1-Lipschitz transformation on a p-adicsphere is ergodic is more complicated than the one for balls.


43


Ergodic transformations of spheres in ZpThe problem to determine whether a 1-Lipschitz transformation on a p-adicsphere is ergodic is more complicated than the one for balls.A sphere Sp−k(a) of radius p

−k centered at a ∈ Zp is a disjoint union ofp− 1 (thus, not all) balls of radius 1

pk+1each:

Sp−k(a) =

p−1⋃

s=1

(a+ pks+ pk+1Zp).


43


Ergodic transformations of spheres in ZpA sphere Sp−k(a) of radius p

−k centered at a ∈ Zp is a disjoint union of

p− 1 balls of radius 1pk+1

each: Sp−k(a) =⋃p−1s=1(a+ p

ks+ pk+1Zp).

Lemma

A 1-Lipschitz map f : Zp → Zp is ergodic on the sphere Sp−k(a) if andonly if the following two conditions hold simultaneously:

1 the map z 7→ f(z) mod pk+1 is transitive on the set

Sp−k(a) mod pk+1 = {a+ pks : s = 1, 2, . . . , p− 1} ⊂ Z/pk+1Z; and

2 the map fp−1 (i.e., (p− 1) times iterated map f) is ergodic on theball Bp−(k+1)(a+ p

ks) = a+ pks+ pk+1Zp of radius 1pk+1

centered at

a+ pks, for some (equivalently, for all) s ∈ {1, 2, . . . , p− 1}


43






ks+ pk+1Zp).


A B-function f is ergodic on the p-adic sphere Sp−k(a) of a sufficiently

small radius p−k if and only if one of the following alternative is true:1 If p is odd, the next two conditions hold simultaneously:

|f(a)− a|p ≤ p−k−1,f ′(a) is primitive modulo p2.

2 If p = 2, the next two conditions hold simultaneously:

|f(a)− a|2 = 2−k−1,|f ′(a)− 1|2 ≤ 1

4 .

NB: We say that z ∈ Zp is primitive modulo pk iff z mod pk generates thewhole group (Z/pkZ)∗ of invertible elements of the residue ring Z/pkZ.


43






ks+ pk+1Zp).

Corollary

Let a ∈ Zp be a fixed point of a B-function f , and let p be odd. Then, fis ergodic on all spheres around a of sufficiently small radii if and only if fis ergodic on some sphere around a of a sufficiently small radius.

Corollary (solution to Gundlach-Khrennikov-Lindahl problem)

The perturbed monomial mapping f : x 7→ x` + q(x), whereq(x) = pk+1u(x) and u is a B-function, is ergodic on the sphere Sp−k(1),k > 1, if and only if ` is primitive modulo p2.


43

Applications

Our final theme is:





43

Applications Latin squares

Next topic:





43


(Mutually orthogonal) Latin squares

Reminder: Latin square

A Latin square of order P is a P ×P matrix containing P distinct symbols(usually denoted by 0, 1, . . . , P − 1) such that each row and column of thematrix contains each symbol exactly once.


43





In other words, a Latin square is a 2-variate map f : P2 → P (where P ={0, 1, . . . , P − 1}) that is invertible (i.e., bijective) with respect to eithervariable.


43





In other words, a Latin square is a 2-variate map f : P2 → P (where P ={0, 1, . . . , P − 1}) that is invertible (i.e., bijective) with respect to eithervariable.Latin squares are being used widely: For games (recall sudoku), and formore serious applications as, say, private communication networks (forpassword distribution), in coding theory, in some cryptographic algorithms,etc.


43





In other words, a Latin square is a 2-variate map f : P2 → P (where P ={0, 1, . . . , P − 1}) that is invertible (i.e., bijective) with respect to eithervariable.Note that the problem is not only to produce a number of large Latin squares;another part of the problem is that in some constraint environments (e.g.,in smart cards) we can not store the whole matrix in memory: Given twonumbers a, b ∈ {0, 1, . . . , P − 1} we must calculate the (a, b)-th entry ofthe matrix on-the-fly!


43





In other words, a Latin square is a 2-variate map f : P2 → P (where P ={0, 1, . . . , P − 1}) that is invertible (i.e., bijective) with respect to eithervariable.Note that the problem is not only to produce a number of large Latin squares;another part of the problem is that in some constraint environments (e.g.,in smart cards) we can not store the whole matrix in memory: Given twonumbers a, b ∈ {0, 1, . . . , P − 1} we must calculate the (a, b)-th entry ofthe matrix on-the-fly!

The p-adic ergodic theory yields a solution to either part of the problem.


43





In other words, a Latin square is a 2-variate map f : P2 → P (where P ={0, 1, . . . , P − 1}) that is invertible (i.e., bijective) with respect to eithervariable.

Definition (Latin square modulo pk)

We say that a 1-Lipschitz map f : Z2p → Zp is a Latin square modulo pk iffthe reduced map f mod pk : Z/pkZ× Z/pkZ→ Z/pkZ is a Latin squareon Z/pkZ = {0, 1, . . . , pk − 1}.


43



Definition (Latin square modulo pk)

We say that a 1-Lipschitz map f : Z2p → Zp is a Latin square modulo pk iffthe reduced map f mod pk : Z/pkZ× Z/pkZ→ Z/pkZ is a Latin squareon Z/pkZ = {0, 1, . . . , pk − 1}.

Theorem on measure preservation immediately implies the following

Theorem (on Latin squares modulo pk)

A uniformly differentiable modulo p 1-Lipschitz function f : Z2p → Zp is aLatin square modulo pk for all k = 1, 2, . . . iff the two conditions hold:

f is a Latin square modulo pN1(f), and∂1f(u)∂1xi

6≡ 0 (mod p) for all u ∈ (Z/pN1(f)Z)2, i = 1, 2.

Or, equivalently: iff f is bijective modulo pN1(f)+1 w.r.t. either variable.


43








Proof.

By theorem on measure preservation, the function f is bijective modulo pk

with respect to either variable iff f is bijective modulo pN1(f) with respectto either variable, and both partial derivatives modulo p, ∂1f(x,y)∂1x

and∂1f(x,y)∂1y

, are 0 nowhere; these conditions are equivalent to the bijectivity

modulo pN1(f)+1 of the function f with respect to either variable.


43








Example (Latin square on 2k symbols)

Take arbitrary T-function v(x, y) and arbitrary γ ∈ Z. Then the T-functionf(x, y) = x+ y + γ + 2 ∙ v(x, y) is a Latin square mod2k, k = 1, 2, . . ..

Indeed, f(x, y) ≡ x + y + γ (mod 2) is a Latin square modulo 2, and∂1f(x,y)∂1x

≡ ∂1f(x,y)∂1y

≡ 1 (mod 2).


43








Example (Latin square on 2k ∙ 3` ∙ ∙ ∙ pr symbols)

The function f(x, y) = x+ y + 2 ∙ 3 ∙ ∙ ∙ p ∙ v(x, y), where v(x, y) is anarbitrary polynomial with rational integer coefficients, is a Latin square onN = 2k ∙ 3` ∙ ∙ ∙ pr symbols.


43



Mutually orthogonal Latin squares can be constructed in a similar fashion.

Reminder: Mutually orthogonal Latin squares

Two P ×P Latin squares are said to be orthogonal if when the squares aresuperimposed each of the P 2 ordered pairs of symbols appears exactlyonce.


43





For instance, the following two Latin squares are orthogonal

0 1 2 0 1 21 2 0 2 0 12 0 1 1 2 0

as after superimposition we get a square where all pairs are different:

(0, 0) (1, 1) (2, 2)(1, 2) (2, 0) (0, 1)(2, 1) (0, 2) (1, 0)


43





Mutually orthogonal Latin squares are used in experiment design to provideconsistent testing of samples, as well as in cryptography (e.g., as blockmixers for block ciphers, and as cipher combiners), etc.Yet again, there is no problem to construct a pair of small mutually orthog-onal Latin squares; the problem is to create a software that produces manypairs of large mutually orthogonal Latin squares.


43



Again, the method follows from theorem on measure preservation:

Theorem (on mutually orthogonal Latin squares)

Let g, f : Z2p → Zp be 1-Lipschitz functions that are uniformly differentiablemodulo p and that are Latin squares modulo pk for all k = 1, 2, . . .. ThenI Latin squares gmod pk and fmod pk are orthogonal for allk = 1, 2, . . . iff the function F (x, y) = (f(x, y), g(x, y)) : Z2p → Z2ppreserves the measure.


43




Let g, f : Z2p → Zp be 1-Lipschitz functions that are uniformly differentiablemodulo p and that are Latin squares modulo pk for all k = 1, 2, . . .. ThenI Latin squares gmod pk and fmod pk are orthogonal for allk = 1, 2, . . . iff the function F (x, y) = (f(x, y), g(x, y)) : Z2p → Z2ppreserves the measure.I The latter holds iff fmod pk and gmod pk are orthogonal for somek ≥ max{N1(f), N1(g)}, and

det

(∂1f(x,y)∂1x

∂1g(x,y)∂1x

∂1f(x,y)∂1y

∂1g(x,y)∂1y

)

6≡ 0 (mod p)

for all (x, y) ∈ (Z/pN1(F )Z)2.


43




Let g, f : Z2p → Zp be 1-Lipschitz functions that are uniformly differentiablemodulo p and that are Latin squares modulo pk for all k = 1, 2, . . .. ThenI Latin squares gmod pk and fmod pk are orthogonal for allk = 1, 2, . . . iff the function F (x, y) = (f(x, y), g(x, y)) : Z2p → Z2ppreserves the measure.I The latter holds iff fmod pk and gmod pk are orthogonal for somek ≥ max{N1(f), N1(g)}, and

det

(∂1f(x,y)∂1x

∂1g(x,y)∂1x

∂1f(x,y)∂1y

∂1g(x,y)∂1y

)

6≡ 0 (mod p)

for all (x, y) ∈ (Z/pN1(F )Z)2.I Or, equivalently, iff fmod pk and gmod pk are orthogonal for somek ≥ max{N1(f), N1(g)}+ 1.


43



Example (mutually orthogonal Latin squares of composite order)

Let P be a finite set of odd primes; let v(x, y), w(x, y) be polynomials overZ. Denote

Π =∏

p∈P

p

Then the following Latin squares f and g are mutually orthogonal moduloP for every P whose prime divisors are in P:

f(x, y) = x+ y +Π ∙ v(x, y);

g(x, y) = −x+ y +Π ∙ w(x, y).


43

Applications Pseudo random generators and stream ciphers

Next topic:





43


Pseudorandom generators

A pseudorandom number generator (PRNG) is an algorithm that takes ashort random string (a seed) and stretches it to a much longer string thatlooks like random.

“Looks like random” means “passes prescribed statistical tests”Thus, the very concept of “pseudorandomness” depends on what tests theoutput of the PRNG must pass!

PRNG are being used widely: in cryptography, for computer simulations, innumerical analysis (e.g., in quasi Monte Carlo algorithms), etc.; and testsuits the PRNG must pass depend heavily on applications; however, a com-mon demand is that the output of a PRNG must be uniformly distributed:limit frequencies of occurrences of symbols must be equal for all symbols.


43



xi

f

F

xi+1 = f(xi)

F (xi)

A typical PRNG may be schematically represented as above, where

x0 is a seed, the initial state of the registry;

f is a state update (=state transition) function;

F is an output function.Vladimir Anashin (Moscow State University) The p-adic Ergodic Theory

Lectures at Universite de Picardie 33 /43



xi

f

F

xi+1 = f(xi)

F (xi)

PRNG can be viewed as an autonomous dynamical system:sequence of states=orbit; output sequence=observable

states: x0, x1 = f(x0), x2 = f(x1), . . . , xi+1 = f(xi) = fi+1(x0), . . .

output: F (x0), F (x1), F (x2), . . . , F (xi+1), . . .


43



xi

f

F

xi+1 = f(xi)

F (xi)

Let this dynamical system be ergodic; for instance:

let the state transition function be f = f mod 2k, where f : Z2 → Z2is a 1-Lipschitz ergodic map;

let the output function be F = F mod 2`, where `n = k andF : Zn2 → Zm2 is a 1-Lipschitz measure-preserving map.


43



xi

f

F

xi+1 = f(xi)

F (xi)





43



xi

f

F

xi+1 = f(xi)

F (xi)




Then the output sequence (of `m-bit words)F (x0), F (x1), F (x2), . . . , F (xi+1), . . .is strictly uniformly distributed: That is, the output sequence is purelyperiodic (having a period of length 2k) and all `m-bit words occur at theperiod with equal frequencies.


43



As we just have seen, the p-adic ergodic theory offers effective tools toconstruct measure preserving/ergodic maps.

These maps can be used (and already are being used) to constructboth state transition functions and output functions of variousPRNGs.

In particular, all balanced T-functions, invertible T-functions, andtransitive T-functions can be constructed/determined with the use ofp-adic ergodic theory.


43



Moreover, all previously known results on so-called congruential generators(both linear and non-linear; e.g., polynomial, inversive, exponential, etc.)can be re-proved within the p-adic ergodic theory and considerablygeneralized.


43



Moreover, all previously known results on so-called congruential generators(both linear and non-linear; e.g., polynomial, inversive, exponential, etc.)can be re-proved within the p-adic ergodic theory and considerablygeneralized.

We illustrate the general approach by the example of exponential generator:

When a generator defined by the recursion xi+1 = axi mod pk

has the longest possible period? Find the length of the period.

To simplify considerations, we assume that a ≡ 1 (mod p) and p > 3.


43





To simplify considerations, we assume that a ≡ 1 (mod p) and p > 3.I As a = 1 + pz for suitable z ∈ Zp, the 1-Lipschitz function f(x) =(1 + pz)x maps Zp into the ball Bp−1(1) = 1 + pZp; so we can write1 + p ∙ g(x) = (1 + pz)x and then study the function g(x).


43





To simplify considerations, we assume that a ≡ 1 (mod p) and p > 3.I As a = 1 + pz for suitable z ∈ Zp, the 1-Lipschitz function f(x) =(1 + pz)x maps Zp into the ball Bp−1(1) = 1 + pZp; so we can write1 + p ∙ g(x) = (1 + pz)x and then study the function g(x).I By Newton binomial, (1 + pz)x =

∑∞i=0 p

izi(xi

), so Mahler expansion

for g(x) is: g(x) = zx+ pz2(x2

)+ p2z3

(x3

)+ ∙ ∙ ∙ . Thus, g is a B-function.


43






∑∞i=0 p

izi(xi



)+ p2z3

(x3


I In the case when z 6≡ 0 (mod p), all p-adic spheres around 0 are invariantunder action of g, so the period is the longest possible if g is ergodic onspheres Sp−r(0) around 0.


43






∑∞i=0 p

izi(xi



)+ p2z3

(x3


I In the case when z 6≡ 0 (mod p), all p-adic spheres around 0 are invariantunder action of g, so the period is the longest possible if g is ergodic onspheres Sp−r(0) around 0.I Now using theorem on ergodicity on spheres we conclude that g′(0) ≡z− p2z

2 (mod p2) is primitive modulo p2 whenever z is primitive modulo p2,and that the length of the shortest period is (p−1)pk−2, for all k = 2, 3, . . .


43


Stream ciphers

Stream ciphers can be viewed as pseudorandom generators that are cryp-tographically secure. Several stream ciphers use a state update T-functionfrom the following theorem:

Theorem (Klimov and Shamir, 2003)

The T-function f(x) = x+ (x2 or 5) is transitive modulo 2k for allk = 1, 2, 3, . . ..


43


Stream ciphers



It worth mention that although in their paper Klimov and Shamir cited mytheorem on ergodicity, however, they wrote: “...neither the invertibility northe cycle structure of x+(x2 or 5) could be determined by his techniques.”Just on the contrary, their theorem immediately follows from mine’s.


43


Stream ciphers



Proof.

The function f(x) = x+ (x2 or 5) is uniformly differentiable on Z2:f ′(x) = 1 + 2x ∙ (x or 5)′ = 1 + 2x and N2(f) 6 3 as

(x+ h) or 5 = (x or 5) + h

whenever h ≡ 0 (mod 8); the latter is obvious since 5 =...000101.


43


Stream ciphers



Proof.

The function f(x) = x+ (x2 or 5) is uniformly differentiable on Z2:f ′(x) = 1 + 2x ∙ (x or 5)′ = 1 + 2x and N2(f) 6 3 as

(x+ h) or 5 = (x or 5) + h

whenever h ≡ 0 (mod 8); the latter is obvious since 5 =...000101.Now by the ergodicity theorem it suffices to show that f is transitivemodulo 32; direct calculation of f(0), f(f(0)), . . . , f31(0) modulo 32 endsthe proof.


43


Stream ciphers

Many other crucial cryptographic properties of various stream ciphers canbe examined with the use the p-adic dynamics, and new cryptographic prim-itives can be developed on this base. More about this in Applied AlgebraicDynamics.


43

Applications Transducers

Next topic:





43


The 0-1 law for automata

Given a transducer A with a p-letter input/output alphabets, let f =fA : Zp → Zp be the automaton function (i.e., a 1-Lipschitz function).Consider all the following points of the Euclidean unit square E2 =[0, 1]× [0, 1] ⊂ R2:

efk(x) =

(xmod pk

pk,f(x)mod pk

pk

)

,

x ∈ Zp, k = 1, 2, . . ..


43




efk(x) =

(xmod pk

pk,f(x)mod pk

pk

)

,

x ∈ Zp, k = 1, 2, . . ..

Note that f(x)mod pk is merely a k-letter output word that correspondsto the k-letter input word xmod pk.

A

x mod pk = = f(x) mod pkχk−1 ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙χ1χ0 ξk−1 ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ξ1ξ0︸︷︷︸

(0.χk−1 . . . χ1χ0; 0.ξk−1 . . . ξ1ξ0)


43




efk(x) =

(xmod pk

pk,f(x)mod pk

pk

)

,

x ∈ Zp, k = 1, 2, . . ..Denote α(f) the Lebesgue measure of a closure of all efk(x) in E.


43




efk(x) =

(xmod pk

pk,f(x)mod pk

pk

)

,

x ∈ Zp, k = 1, 2, . . ..Denote α(f) the Lebesgue measure of a closure of all efk(x) in E.

Theorem (automata 0-1 law; V. A., 2009)

Given a 1-Lipschitz function f : Zp → Zp, the following alternative holds:Either α(f) = 0, or α(f) = 1 otherwise.


43





We will say for short that a 1-Lipschitz map (resp., a transducer)f : Zp → Zp is of measure 1 iff α(f) = 1, and of measure 0 otherwise.


43






If being used as “sources of pseudorandomness”, transducers of measure 1are more preferable than the ones of measure 0; so the problem is:

How to construct 1-Lipschitz maps/transducers of measure 1?


43






If being used as “sources of pseudorandomness”, transducers of measure 1are more preferable than the ones of measure 0; so the problem is:

How to construct 1-Lipschitz maps/transducers of measure 1?

The problem turns out to be tightly related to transitivity properties offamilies of word transformations related to a given transducer.


43


Word transformations performed by automata

Further we consider only accessible automata; that is, given a state u,there exist a finite input sequence w0, . . . , wn such that once thesequence has been inputted, the automaton reaches the state u.

Moreover, further the word “automaton” stands for accessibletransducer; the automaton A = 〈I, S,O, S,O, s0〉 whose input/outputalphabets consist of p letters: I = O = Fp = {0, 1, . . . , p− 1}.

To every automaton of this kind we associate a family As, s ∈ S, ofautomata As = 〈I, S,O, S,O, s〉, and a corresponding family FA of1-Lipschitz transformations fAs , s ∈ S, on Zp.


43







43







43






Reminder: transitivity of families of mappings

A family F of transformations on the set M is called transitive whenevergiven a pair (a, b) ∈M ×M , there exists f ∈ F such that f(a) = b.Note that a single transformation f : M →M is said to be transitive, iff fis bijective and the family {e, f±1, f±2, . . .} is transitive.


43



Reminder: transitivity of families of mappings

A family F of transformations on the set M is called transitive whenevergiven a pair (a, b) ∈M ×M , there exists f ∈ F such that f(a) = b.Note that a single transformation f : M →M is said to be transitive, iff fis bijective and the family {e, f±1, f±2, . . .} is transitive.

An automaton A = 〈I, S,O, S,O, s0〉 is said to be

n-word transitive, if fA mod pn is a transitive transformation on the

set Wn = Z/pnZ of all words of length n;

word transitive, if it is n-word transitive for all n = 1, 2, 3, . . .;

completely transitive, if the family fAs mod pn, s ∈ S, is transitive on

Wn, n = 1, 2, 3, . . .;

absolutely transitive, if every automaton As, s ∈ S, is completelytransitive.


43








Wn, n = 1, 2, 3, . . .;


Word transitivity Given two finite words V , W , |V | = |W |,one transforms V into W applying A sufficient number of times.

A A︸︷︷︸V

︸︷︷︸W

∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙


43








Wn, n = 1, 2, 3, . . .;


Complete transitivity: Given finite words V , W , |V | = |W |, there exists afinite word Z such that

A∗ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∗︸︷︷︸

Z

︸︷︷︸V

︸︷︷︸W


43








Wn, n = 1, 2, 3, . . .;


Absolute transitivity: Given finite words U , V , W , |V | = |W |, there existsa finite word Z such that

A︸︷︷︸U

∗ ∙ ∙ ∙ ∗︸︷︷︸Z

︸︷︷︸V

︸︷︷︸W


43



From the p-adic ergodic theory (see main theorem) we already know howto determine whether an automaton A is

invertible (=bijective), since the automaton A is invertible if and onlyif the transformation fA on Zp is measure-preserving;

word transitive, since the automaton A is word transitive if and only ifthe transformation fA on Zp is ergodic.


43



From the p-adic ergodic theory (see main theorem) we already know howto determine whether an automaton A is

invertible (=bijective), since the automaton A is invertible if and onlyif the transformation fA on Zp is measure-preserving;

word transitive, since the automaton A is word transitive if and only ifthe transformation fA on Zp is ergodic.

For instance, in the case when the automaton function fA is uniformly dif-ferentiable both theorem on measure preservation and theorem on ergodicityyield:

Let the automaton function fA : Zp → Zp be uniformly differentiable onZp. The automaton A is invertible (accordingly, word transitive) if andonly if it is invertible on the set of all words of length n (accordingly, isn-word transitive) for a sufficiently large n.


43


Transitivity and measure of automata

Now we will learn how to construct automata of measure 1.

Proposition (V. A., 2009–2010)

An automaton A is completely transitive if and only if α(fA) = 1.

If an automaton A is finite (that is, its state set S is finite) thenα(fA) = 0.


43








43







Theorem (V. A., 2009-2010)

Let a 1-Lipschitz function f = fA : Zp → Zp be differentiable in aneighbourhood of some point v ∈ N0, where f ′′(v) 6= 0; let f(N0) ⊂ N0.Then the automaton A is completely transitive, and whence α(f) = 1.

For instance, the following T-functions satisfy the theorem:

f(x) = 3x+ 3x;

f(x) = x+ (x2 or c), where c ∈ {0, 1, 2, . . .}


43






Theorem (V. A., 2009-2010)

Let a 1-Lipschitz function f = fA : Zp → Zp be differentiable in aneighbourhood of some point v ∈ N0, where f ′′(v) 6= 0; let f(N0) ⊂ N0.Then the automaton A is completely transitive, and whence α(f) = 1.

Theorem (V. A.,2009)

If f is a polynomial over Z and deg f ≥ 2, then f is absolutely transitive,and thus α(f) = 1.


43






The following properties of automata are independent:

to be ergodic/non-ergodic;

to be of measure 0/of measure 1.


43









A polynomial of degree ≥ 2 over Z is always of measure 1; however, itmay be ergodic or non-ergodic.

Affine map f(x) = ax+ b, a, b ∈ Z, is always of measure 0; however,it may be ergodic or non-ergodic.


43









Although a finite automaton is always of measure 0, it can be at thesame time ergodic or non-ergodic: Note that the affine mapf(x) = ax+ b, a, b ∈ Z, may be evaluated by a finite automaton.

Although an automaton of measure 1 must be infinite, infiniteautomaton may be of measure 0: consider e.g. the T-functionf(x) = x+ (x2 or (−13)).Vladimir Anashin (Moscow State University) The p-adic Ergodic Theory










Although a finite automaton is always of measure 0, it can be at thesame time ergodic or non-ergodic.

Although an automaton of measure 1 must be infinite, infiniteautomaton may be of measure 0: consider e.g. the T-functionf(x) = x+ (x2 or (−13)).


43

Other applications of non-Archimedean dynamicswhich were not mentioned:

Information processing: data analysis, clustering.

Image processing: image recognition, computer vision.

Physics: spin glasses; p-adic strings; p-adic time.

Genetics: genetic code on the 2-adic plane.

Biology: protein dynamics.

Cognitive sciences: dynamics of thinking.

Artificial intelligence: psycho-robots.

Formal languages: decidability problems on regular languages.

...and more.


43










...and more.


43










...and more.


43










...and more.


43










...and more.


43










...and more.


43










...and more.


43










...and more.


43










...and more.


43

Messages of the lecture

The non-Archimedean dynamics (in particular, the p-adic ergodictheory) serves as a way to solve ‘discrete’ problems by ‘continual’methods: one lifts finite dynamics to a continuum phase space,studies dynamical behaviour there and then projects the result backonto the initial finite domain.

The approach results in an effective toolbox to handle ‘discrete’problems of computer science (e.g., those connected withT-functions, automata, Latin squares, congruential generators, andstream ciphers) by ‘continual’ methods.

The practical effectiveness is based (among other things) on the factthat basic computer instructions are continuous with respect to 2-adicmetric.


43






43






43






43

Thank you!


43

p = 2: f(x) = 1 + x+ 4((7 + 177x) or (3−

13x));

α(f) = 0.Vladimir Anashin (Moscow State University) The p-adic Ergodic Theory


p = 2: f(x) = 1 + x+ 4x2; α(f) = 1.


43

Documents

The p-adic Ergodic Theory - UPJVleve/ejc2011/PDF_EJC2011/cours_An… · Vladimir Anashin (Moscow State University) The p-adic Ergodic Theory Lectures at Universit´e de Picardie 8