Decomposition of Objects into Structured and Pseudorandom Components, Study notes of Cryptography and System Security

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Structure and

(pseudo-)randomness in

combinatorics

FOCS 2007 tutorial

October 20, 2007

Terence Tao (UCLA)

Large data

In combinatorics, one often deals with high-complexity objects, such as

• Functions f : Fn 2 → R on a Hamming cube;
• Sets A ⊂ Fn 2 in that Hamming cube Fn 2 ; or
• Graphs G = (V, E) on |V | = N vertices.

One should think of |Fn 2 | = 2n^ and N as being very large, thus these objects have a large amount of informational entropy.

All of the above objects can be modeled as elements of a (real) finite-dimensional Hilbert space H:

• The functions f : Fn 2 → R form a Hilbert space H with inner product 〈f, g〉H := Ex∈Fn 2 f (x)g(x).
• A set A ⊂ Fn 2 can be identified with its indicator function 1A : Fn 2 → { 0 , 1 }, which lies in H.
• A graph G = (V, E) can be identified with a symmetric function 1E : V × V → { 0 , 1 } in the Hilbert space of functions f : V × V → R with norm 〈f, g〉H := Ev,w∈V f (v, w)g(v, w).

The dimension of these Hilbert spaces is finite, but extremely large. Thus these objects have many “degrees of freedom”.

In combinatorics one often has to deal with arbitrary objects in such a class - objects with no obvious usable structure.

Examples of structure:

• Functions f : Fn 2 → R which exhibit linear (Fourier) behaviour;
• Functions f : Fn 2 → R which exhibit low-degree polynomial (Reed-Muller) behaviour;
• Sets A ⊂ Fn 2 which only depend on a few of the coordinates of Fn 2 (dictators, juntas);
• Graphs G = (V, E) which are determined by a low-complexity vertex partition (e.g. complete bipartite graphs).

One might also consider computational complexity notions of structure.

Sometimes it is important to distinguish between several “quality levels” of structure:

• A “100%-structured” object might be one in which some statistic measuring structure is exactly equal to its theoretical maximum;
• A “99%-structured” object might be one in which some statistic measuring structure is very close to its theoretical maximum;
• A “1%-structured” object might be one in which some statistic measuring structure is within a multiplicative constant of its theoretical maximum.

Given a concept of structure, one can often define a dual notion of pseudorandom objects - objects which are “almost orthogonal” or have “low correlation” with structured objects.

One can often show by standard probabilistic, counting, or entropy arguments that random objects tend to be almost orthogonal to all structured objects, thus justifying the terminology “pseudorandom”.

Examples of pseudorandomness as duals of structure:

• Functions f : Fn 2 → R which are Fourier-pseudorandom, i.e. have low Fourier coefficients (dual of Fourier structure);
• Functions f : Fn 2 → R which are polynomially-pseudorandom, i.e. have low correlations with low-degree polynomials (dual of Reed-Muller structure);
• Sets A ⊂ Fn 2 in which each coordinate has small low-height Fourier coefficients (dual of dictators and juntas);
• Graphs G = (V, E) which are ε-regular (dual of

In the previous examples, we began by defining structure and then created a dual notion of pseudorandomness. Thus pseudorandomness is defined “extrinsically”, by measuring its correlation with structured objects. In many cases we have an opposite situation: we begin with an “intrinsically defined” notion of pseudorandomness and wish to discover its dual notion of structure - the “obstructions” to that conception of pseudorandomness.

Computing such duals explicitly can sometimes be difficult, but is also very worthwhile; it provides a way to test whether a given object is structured or pseudorandom, or a combination of both.

Examples of “intrinsic” pseudorandomness:

• Functions f : Fn 2 → R whose pair correlations Ex∈Fn 2 f (x)f (x + h) are small for most h ∈ Fn 2 ;
• Functions f : Fn 2 → R whose k-point correlations Ex∈Fn 2 f (x + h 1 )... f (x + hk) are small for most h 1 ,... , hk ∈ Fn 2 ;
• Functions f : Fn 2 → R whose Gowers norms ‖f ‖U d(Fn 2 ) := (EL:Fd 2 →Fn 2 Ex∈Fn 2

ω∈Fd 2 f^ (x^ +^ Lω))^1 /^2

d are small;

• Graphs with a near-minimal (for a given edge density) number of 4-cycles.

General principles

1. Negligibility: pseudorandom objects tend to have negligible impact on statistics, averages, or correlations.
2. Dichotomy: Objects which are not pseudorandom tend to correlate with a structured object, and vice versa.

2. Structure theorem: Arbitrary objects can be decomposed into pseudorandom and structured components, possibly up to a small error.
3. Rigidity: Objects which are “almost”, “statistically”, or “locally” structured tend to be close to objects which actually are structured.
4. Classification: Structured objects can often be classified algebraically by using various bases.

These principles give a strategy to understand arbitrary objects, by splitting them into their pseudorandom and structured components.

Example: orthogonal projection

Theorem 1. Let V be a subspace of H (con- sisting of the “structured” vectors). Then ev- ery f ∈ H can be uniquely decomposed as f = fstr + fpsd + ferr, where

• fstr lies in V ;
• fpsd is orthogonal to V ; and
• ferr = 0.

We recall that there are two standard proofs of this theorem: the first using the Gram-Schmidt orthogonalisation process, and the other by minimising ‖f − fstr‖^2 H over all fstr ∈ V. The latter proof is more relevant here; it relies on the dichotomy that if f − fstr is not orthogonal to V , then one can adjust fstr in V in order to decrease ‖f − fstr‖^2 H.

One can view this variational approach as a prototype of an “energy decrement argument” approach to structure theorems.