Asymptotic enumeration of correlation-immune functions E. Rodney - - PowerPoint PPT Presentation

Asymptotic enumeration of correlation-immune functions

• E. Rodney Canfield

Jason Gao Catherine Greenhill Brendan D. McKay Robert W. Robinson

Correlation-immune functions 1

Correlation-immune functions

Suppose we have a secret Boolean function of ♥ Boolean variables. Suppose a malicious eavesdropper is able to observe function values while monitoring any ❦ of the variables. We would like these observations to give the eavesdropper as little information as possible.

Correlation-immune functions 2

Correlation-immune functions

Suppose we have a secret Boolean function of ♥ Boolean variables. Suppose a malicious eavesdropper is able to observe function values while monitoring any ❦ of the variables. We would like these observations to give the eavesdropper as little information as possible. The function is correlation-immune of order ❦ if the function value is uncorrelated with any ❦ of the arguments. Suppose the fraction ✕ of all 2♥ argument values give a function value 1. Then correlation-immune means that if any arbitrary ❦ of the arguments are fixed to arbitrary values, the same fraction ✕ of the remaining 2♥❦ argument values give a function value 1. The weight of the function is ✕2♥ — the number of argument lists that give function value 1.

Correlation-immune functions 2

Example (Sloan): ♥ = 12, ❦ = 3, ✕ = 24❂212, weight = 24 = 2❦3. The rows of the table give the argument lists for which the function value is 1.

0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 1 1 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 0 1 0 1 0 0 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 1 1 0 1 0 0 0 1 0 0 1 0 1 1 0 1 1 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 0 1 1 1 1 0 1 0 0 0 1 1 1 0 1 0 1 1 0 1 0 0 0 1 1 1 1 1 0 1 1 0 1 0 0 0 1 1 1 1 1 0 1 1 0 1 0 0 0 1 1 1 1 1 0 1 1 0 1 0 0 0 1 0 1 1 1 0 1 1 0 1 0 0 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 0 0 0 1 1 1 0 1 1

This is an orthogonal array of 2 levels, 12 variables, 24 runs and strength 3.

Correlation-immune functions 3

Our task

Since the weight is a multiple of 2❦, let’s define it to be 2❦q, where 0 ✔ q ✔ 2♥❦.

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Our task

Since the weight is a multiple of 2❦, let’s define it to be 2❦q, where 0 ✔ q ✔ 2♥❦. Define ◆(♥❀ ❦❀ q) to be the number of ♥-variable correlation-immune functions

• f order ❦ and weight 2❦q.

Also define ◆(♥❀ ❦) =

q ◆(♥❀ ❦❀ q).

We seek the asymptotic values of ◆(♥❀ ❦❀ q) and ◆(♥❀ ❦) as ♥ ✦ ✶, with ❦ and q being some functions of ♥.

Correlation-immune functions 4

Our task

Since the weight is a multiple of 2❦, let’s define it to be 2❦q, where 0 ✔ q ✔ 2♥❦. Define ◆(♥❀ ❦❀ q) to be the number of ♥-variable correlation-immune functions

• f order ❦ and weight 2❦q.

Also define ◆(♥❀ ❦) =

q ◆(♥❀ ❦❀ q).

We seek the asymptotic values of ◆(♥❀ ❦❀ q) and ◆(♥❀ ❦) as ♥ ✦ ✶, with ❦ and q being some functions of ♥. Define

▼ =

❦

• ✐=0

♥

✐

• and ◗ =

❦

• ✐=1

✐

♥

✐

• ✿

Theorem (Denisov, 1992) If ❦ ✕ 1 is a constant integer, then as ♥ ✦ ✶,

◆(♥❀ ❦) ✘ 22♥+◗❦(2♥1✙)(▼1)❂2✿

Correlation-immune functions 4

Denisov’s method (translated)

For ❙ ✒ ❢1❀ 2❀ ✿ ✿ ✿ ❀ ♥❣, let ☞❙ be the number of rows (☞1❀ ☞2❀ ✿ ✿ ✿ ❀ ☞♥) of the matrix such that ☞✐ = 1 for ✐ ✷ ❙. Also ☞❀ = 2❦q (i.e., all the rows). Then the matrix is that of a correlation-immune function of weight 2❦q iff

☞❙ = 2❦❥❙❥q

for ❥❙❥ ✔ ❦✿

Correlation-immune functions 5

Denisov’s method (translated)

For ❙ ✒ ❢1❀ 2❀ ✿ ✿ ✿ ❀ ♥❣, let ☞❙ be the number of rows (☞1❀ ☞2❀ ✿ ✿ ✿ ❀ ☞♥) of the matrix such that ☞✐ = 1 for ✐ ✷ ❙. Also ☞❀ = 2❦q (i.e., all the rows). Then the matrix is that of a correlation-immune function of weight 2❦q iff

☞❙ = 2❦❥❙❥q

for ❥❙❥ ✔ ❦✿ Consider

• ☞✷❢0❀1❣♥
• 1 +
• ❥❙❥✔❦

①

Q

✐✷❙ ☞✐

❙

• ❀

where ❢①❙ ❥ ❙ ✒ ❢1❀ 2❀ ✿ ✿ ✿ ❀ ♥❣ ❣ are indeterminates. Then ◆(♥❀ ❦❀ q) is the coefficient of the monomial

• ❥❙❥✔❦

① 2❦❥❙❥q

❙

✿

Denisov extracts ◆(♥❀ ❦) by Fourier inversion.

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The inversion integral is concentrated at two equivalent places, where it is approxi- mately gaussian. Expansion near the critical points together with bounds away from the critical points establishes the asymptotics.

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The inversion integral is concentrated at two equivalent places, where it is approxi- mately gaussian. Expansion near the critical points together with bounds away from the critical points establishes the asymptotics.

Denisov’s retraction

In 2000, Denisov published a retraction of his 1992 result. He wrote that he had “made a mistake”, and gave a new asymptotic value of ◆(♥❀ ❦).

Correlation-immune functions 6

The inversion integral is concentrated at two equivalent places, where it is approxi- mately gaussian. Expansion near the critical points together with bounds away from the critical points establishes the asymptotics.

Denisov’s retraction

In 2000, Denisov published a retraction of his 1992 result. He wrote that he had “made a mistake”, and gave a new asymptotic value of ◆(♥❀ ❦). This is unfortunate, since the 1992 result is correct and the 2000 result is incorrect!

Correlation-immune functions 6

Alternative approach

For a boolean function ❣(①1❀ ✿ ✿ ✿ ❀ ①♥), the Walsh transform of ❣ is the real-valued function ˆ

❣ over ❢0❀ 1❣♥ defined by

ˆ

❣(✇1❀ ✿ ✿ ✿ ❀ ✇♥) =

• (①1❀✿✿✿❀①♥)✷❢0❀1❣♥

❣(①1❀ ✿ ✿ ✿ ❀ ①♥)(1)✇1①1+✁✁✁+✇♥①♥✿

It is known that ❣ is correlation-immune of order ❦ iff ˆ

❣(✇1❀ ✿ ✿ ✿ ❀ ✇♥) = 0 whenever

the number of 1s in ✇1❀ ✿ ✿ ✿ ❀ ✇♥ is between 1 and ❦.

Correlation-immune functions 7

Alternative approach

For a boolean function ❣(①1❀ ✿ ✿ ✿ ❀ ①♥), the Walsh transform of ❣ is the real-valued function ˆ

❣ over ❢0❀ 1❣♥ defined by

ˆ

❣(✇1❀ ✿ ✿ ✿ ❀ ✇♥) =

• (①1❀✿✿✿❀①♥)✷❢0❀1❣♥

❣(①1❀ ✿ ✿ ✿ ❀ ①♥)(1)✇1①1+✁✁✁+✇♥①♥✿

It is known that ❣ is correlation-immune of order ❦ iff ˆ

❣(✇1❀ ✿ ✿ ✿ ❀ ✇♥) = 0 whenever

the number of 1s in ✇1❀ ✿ ✿ ✿ ❀ ✇♥ is between 1 and ❦. Put ❘ = ✕❂(1 ✕). Define

❋ (①) =

• ☛✷❢✝1❣♥
• 1 + ❘
• ❥❙❥✔❦

①☛❙

❙

• ❀

where

☛❙ =

• ✐✷❙

☛✐✿

Theorem:

◆(♥❀ ❦❀ q) is the constant term of (❘①❀)2❦q❋ (①).

Correlation-immune functions 7

Apply the Cauchy coefficient formula, using unit circles as contours, and change variables as ①❙ = ❡✐✒❙ for each ❙. Then

◆(♥❀ ❦❀ q) = (1 + ❘)2♥

(2✙)▼❘2❦q ■(♥❀ ❦❀ q)❀ where

■(♥❀ ❦❀ q) =

✙

✙ ✁ ✁ ✁

✙

✙ ●(✒) ❞✒❀

• (✒) = ❡✐2❦q
• ☛✷❢✝1❣♥

1 + ❘❡✐❢☛(✒) 1 + ❘

❀ ❢☛(✒) =

• ❥❙❥✔❦

☛❙✒❙✿

Here ✒ is a vector of the variables ✒❙, ❥❙❥ ✔ ❦, in arbitrary order.

Correlation-immune functions 8

Analysis of the domain of integration

The integrand

• (✒) = ❡✐2❦q
• ☛✷❢✝1❣♥

1 + ❘❡✐❢☛ 1 + ❘ has greatest absolute value 1 when

❢☛ = ❢☛(✒) =

• ❥❙❥✔❦

☛❙✒❙

is a multiple of 2✙ for each ❙. When does that happen?

Correlation-immune functions 9

Analysis of the domain of integration

The integrand

• (✒) = ❡✐2❦q
• ☛✷❢✝1❣♥

1 + ❘❡✐❢☛ 1 + ❘ has greatest absolute value 1 when

❢☛ = ❢☛(✒) =

• ❥❙❥✔❦

☛❙✒❙

is a multiple of 2✙ for each ❙. When does that happen? Define the difference operator

✍❥❢(☛1❀✿✿✿❀☛❥❀✿✿✿❀☛♥) = ❢(☛1❀✿✿✿❀☛❥❀✿✿✿❀☛♥) ❢(☛1❀✿✿✿❀☛❥❀✿✿✿❀☛♥)✿

and in general ✍❙ =

❥✷❙ ✍❥.

If each ❢☛ is a multiple of 2✙, then so are all the differences. Now we compute

✍❙❢☛ = 2❥❙❥

❚ ✓❙

☛❚✒❚✿

and apply this with decreasing ❥❙❥.

Correlation-immune functions 9

Conclusion:

❥●(✒)❥ = 1 iff there are integers ❥❙ such that

• ❚ ✓❙

✒❚ = 2❥❙❥+1❥❙✙

for every ❙ ✒ ❢1❀ 2❀ ✿ ✿ ✿ ❀ ♥❣ with ❥❙❥ ✔ ❦. There are 2◗ such critical points, where ◗ = ❦

✐=1 ✐

• ♥

✐

• .

Correlation-immune functions 10

Conclusion:

❥●(✒)❥ = 1 iff there are integers ❥❙ such that

• ❚ ✓❙

✒❚ = 2❥❙❥+1❥❙✙

for every ❙ ✒ ❢1❀ 2❀ ✿ ✿ ✿ ❀ ♥❣ with ❥❙❥ ✔ ❦. There are 2◗ such critical points, where ◗ = ❦

✐=1 ✐

• ♥

✐

• .

Define the critical region ❘ to be the set of points ✒ such that, for some critical point ˆ

✒ ❥✒❙ ˆ ✒❙❥ ✔ ∆(2♥)❥❙❥

for each ❙, where ∆ = 2♥❂2+❦+3✕1❂2♥❦+1❂2▼1❂2. These 2◗ cuboids are disjoint and equivalent.

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The integrand outside the critical region

If ✒ is not in the critical region,

❥●(✒)❥ ❁ exp

4

5♥▼

✿

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The integrand outside the critical region

If ✒ is not in the critical region,

❥●(✒)❥ ❁ exp

4

5♥▼

✿ Proof: (1) There is some ❙ such that, outside the critical region, ✍❙❢☛ is at least (2 ❡1❂2)∆♥❥❙❥ from any multiple of 2✙ for all ☛. (2) Divide the 2♥ vectors ☛ into 2♥❥❙❥ classes of size 2❥❙❥, where two vectors are in the same class iff they agree outside ❙. (3) For each class,

• ☛
• 1 + ❘❡✐❢☛

1 + ❘

• ✔ exp(stuff )✿

(4) That does it.

Correlation-immune functions 11

The integrand inside the critical region

Since the 2◗ components of the critical region are all equivalent, consider the component containing the origin. If ✒ is in the critical region near the origin,

• (✒) = exp
• 1

2✕(1 ✕)2♥

❥❙❥✔❦

✒2

❙ + ❖(✕2♥∆3)

• ✿

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The integrand inside the critical region

Since the 2◗ components of the critical region are all equivalent, consider the component containing the origin. If ✒ is in the critical region near the origin,

• (✒) = exp
• 1

2✕(1 ✕)2♥

❥❙❥✔❦

✒2

❙ + ❖(✕2♥∆3)

• ✿

Proof: Use Taylor expansion. The linear term vanishes thanks to the choice of ❘.

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Conclusion

Theorem: If ✦

• 25❦♥6❦+3▼3 ✔ q ✔ 2♥❦ ✦
• 25❦♥6❦+3▼3

, then

◆(♥❀ ❦❀ q) ✘

2◗(♥+1)▼❂2

✙▼❂2✕✕(1 ✕)(1✕)2♥+▼❂2✿

This allows some values of q if ❦ ✔ ❝♥❂ log ♥ (compared to constant ❦ for Denisov). In that case:

◆(♥❀ ❦) ✘ 22♥+◗❦(2♥1✙)(▼1)❂2✿

Correlation-immune functions 13

More on the case ❦ = 1

A balanced colouring of a hypercube is a colouring with two colours such that the center of mass is at the center of the hypercube. 0 0 0 0 1 0 0 0 0 1 1 0 1 1 1 0 1 0 0 1 0 1 0 1 0 0 1 1 1 1 1 1 This corresponds to colouring according to the value of ❢ (①1❀ ①2❀ ✿ ✿ ✿ ❀ ①♥) for a correlation- immune function of order 1. Palmer, Read and Robinson did an exact enumeration (1992) but it seems unsuitable for asymptotics.

Correlation-immune functions 14

Naive estimate (2q uses of first colour): Choose, uniformly at random, a set of 2q distinct elements of ❢0❀ 1❣♥. The event of any particular column having exactly q 0s and q 1s has probability 2♥1

q

• 22♥

2q

• ✿

Therefore, if these ♥ events are close to being independent,

◆(♥❀ 1❀ q) ✘

2♥ 2q

• 

    2♥1

q

• 2

2♥ 2q

• 

   

♥ ✿

Correlation-immune functions 15

Naive estimate (2q uses of first colour): Choose, uniformly at random, a set of 2q distinct elements of ❢0❀ 1❣♥. The event of any particular column having exactly q 0s and q 1s has probability 2♥1

q

• 22♥

2q

• ✿

Therefore, if these ♥ events are close to being independent,

◆(♥❀ 1❀ q) ✘

2♥ 2q

• 

    2♥1

q

• 2

2♥ 2q

• 

   

♥ ✿

For small q, actually q = ♦(2♥❂2), we can estimate ◆(♥❀ 1❀ q) probabilistically: make each column randomly with q zeros and q ones. Then use Bonferroni to show that the rows are distinct with probability 1 ♦(1).

Correlation-immune functions 15

Naive estimate (2q uses of first colour): Choose, uniformly at random, a set of 2q distinct elements of ❢0❀ 1❣♥. The event of any particular column having exactly q 0s and q 1s has probability 2♥1

q

• 22♥

2q

• ✿

Therefore, if these ♥ events are close to being independent,

◆(♥❀ 1❀ q) ✘

2♥ 2q

• 

    2♥1

q

• 2

2♥ 2q

• 

   

♥ ✿

For small q, actually q = ♦(2♥❂2), we can estimate ◆(♥❀ 1❀ q) probabilistically: make each column randomly with q zeros and q ones. Then use Bonferroni to show that the rows are distinct with probability 1 ♦(1). Conclusion: The naive estimate is correct for all q.

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Extensions

(1) Correlation-immune functions can be defined over sets other than ❢0❀ 1❣. The asymptotic techniques can be generalized (in the hands of a student).

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Extensions

(1) Correlation-immune functions can be defined over sets other than ❢0❀ 1❣. The asymptotic techniques can be generalized (in the hands of a student). (2) A Hadamard matrix is an ♥ ✂ ♥ matrix ❍ over ❢1❀ +1❣ such that ❍❚❍ = ♥■.

• 1

1 1 1 1 -1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 1 1 1

• 1

1 -1 -1 1 -1 1 1

• 1 -1

1 -1 1 1 -1 1

• 1 -1 -1

1 1 1 1 -1 Hadamard conjecture: A Hadamard matrix exists iff ♥ = 2 or ♥ is a multiple of 4.

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Extensions

(1) Correlation-immune functions can be defined over sets other than ❢0❀ 1❣. The asymptotic techniques can be generalized (in the hands of a student). (2) A Hadamard matrix is an ♥ ✂ ♥ matrix ❍ over ❢1❀ +1❣ such that ❍❚❍ = ♥■.

• 1

1 1 1 1 -1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 1 1 1

• 1

1 -1 -1 1 -1 1 1

• 1 -1

1 -1 1 1 -1 1

• 1 -1 -1

1 1 1 1 -1 Hadamard conjecture: A Hadamard matrix exists iff ♥ = 2 or ♥ is a multiple of 4. Multiply rows by 1 as needed so the first column is 1 then delete the first column. Then change 1 into 0. The result is a correlation-immune function of ♥1 variables,

• rder 2, and weight ♥.

Are there any??

Correlation-immune functions 16