Incidences between points and generalized spheres over finite fields and related problems

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Erdos Distinct Distances Problem and Extensions over Finite Spaces

Van Thang Pham

In this thesis we study a number of problems in Discrete Combinatorial Geometry in finite spaces. The contents in this thesis are structured as follows: In Chapter 1 we will state the main results and the notations which will be used throughout the thesis. Chapter 2 is a version of the paper entitled "Sumsets of the distance sets in finite spaces", which has been submitted for publication, (2017). Chapter 3 is a version of the paper entitled "Three-variable expanding polynomials and higher-dimensional distinct distances", which has been submitted for publication, co-authored with L. A. Vinh and de Zeeuw. The author was one of the main investigators of this chapter. Chapter 4 is a postprint version of the paper entitled "Distinct distances on regular varieties over finite fields", Journal of Number Theory, 173(2017), 602-613, co-authored with D. D. Hieu. The author was one of the main investigators of this chapter. Chapter 5 is a postprint version of the paper entitled " Incidences between points and generalized spheres over finite fields and related problems", Forum Mathematicum, Volume 29, Issue 2 (Mar 2017), co-authored with N. D. Phuong and L. A. Vinh. The author was one of the main investigators of this chapter. Chapter 6 is a version of the paper entitled "Distinct spreads in finite spaces", which has been submitted for publication, co-authored with B. Lund and L. A. Vinh. The author was one of the main investigators of this chapter. Chapter 7 is a version of the paper entitled "Paths in pseudo-random graphs", which has been submitted for publication, co-authored with L. A. Vinh. The author was one of the main investigators of this chapter. Chapter 8 is a version of the paper entitled "Conditional expanding bounds for two-variable functions over arbitrary fields", which has been submitted for publication, co-authored with Hossein Nassajian Mojarrad. The author was one of the main investigators of this chapter. Chapter 9 is a postprint version of the paper entitled "A Szemeredi-Trotter type theorem, sum-product estimates in finite quasifields, and related results", Journal of Combinatorial Theory Series A, 147(2017), 55-74, co-authored with Michael Tait, Craig Timmons, Le Anh Vinh. The author was one of the main investigators of this chapter. The content of this chapter also appears in Michael Tait's Phd thesis. In Chapter 10, we will mention some open problems on Erd\H{o}s distinct distances problem and generalizations.

EPFL2017

On some algebraic and extremal problems in discrete geometry

Seyed Hossein Nassajianmojarrad

In the present thesis, we delve into different extremal and algebraic problems arising from combinatorial geometry. Specifically, we consider the following problems. For any integer

n\ge 3

, we define

e(n)

to be the minimum positive integer such that any set of

e(n)

points in general position in the plane contains

n

points in convex position. In 1935, Erd\H{o}s and Szekeres proved that

e(n) \le {2n-4 \choose n-2}+1

and later in 1961, they obtained the lower bound

2^{n-2}+1 \le e(n)

, which they conjectured to be optimal. We prove that

e(n) \le {2n-5 \choose n-2}-{2n-8 \choose n-3}+2

. In a recent breakthrough, Suk proved that

e(n) \le 2^{n+O\left(n^{2/3}\log n\right)}

. We strengthen this result by extending it to pseudo-configurations and also improving the error term. Combining our results with a theorem of Dobbins et al., we significantly improve the best known upper bounds on the following two functions, introduced by Bisztriczky and Fejes T'{o}th and by Pach and T'{o}th, respectively. Let

c(n)

(and

c'(n)

) denote the smallest positive integer

N

such that any family of

N

pairwise disjoint convex bodies in general position (resp.,

N

convex bodies in general position, any pair of which share at most two boundary points) has an

n

members in convex position. We show that

c(n)\le c'(n)\le 2^{n+O\left(\sqrt{n\log n}\right)}

. Given a point set

P

in the plane, an ordinary circle for

P

is defined as a circle containing exactly three points of

P

. We prove that any set of

n

points in the plane, not all on a line or a circle, determines at least

\frac{1}{4}n^2-O(n)

ordinary circles. We determine the exact minimum number of ordinary circles for all sufficiently large

n

, and characterize all point sets that come close to this minimum. We also consider the orchard problem for circles, where we determine the maximum number of circles containing four points of a given set and describe the extremal configurations. A special case of the Schwartz-Zippel lemma states that given an algebraic curve

C\subset \mathbb{C}^2

of degree

d

and two finite sets

A,B\subset \mathbb{C}

, we have

|C\cap (A\times B)|=O_d(|A|+|B|)

. We establish a two-dimensional version of this result, and prove upper bounds on the size of the intersection

|X\cap (P\times Q)|

for a variety

X\subset \mathbb{C}^4

and finite sets

P,Q\subset \mathbb{C}^2

. A key ingredient in our proofs is a two-dimensional version of a special case of Alon's combinatorial Nullstellensatz. As corollaries, we generalize the Szemer'edi-Trotter point-line incidence theorem and several known bounds on repeated and distinct Euclidean distances. We use incidence geometry to prove some sum-product bounds over arbitrary fields. First, we give an explicit exponent and improve a recent result of Bukh and Tsimerman by proving that

\max \{ |A+A|, |f(A, A)|\}\gg |A|^{6/5}

for any small set

A\subset \mathbb{F}_p

and quadratic non-degenerate polynomial

f(x, y)\in \mathbb{F}_p[x, y]

. This generalizes the result of Roche-Newton et al. giving the best known lower bound for the term

\max \{ |A+A|, |A \cdot A |\}

. Secondly, we improve and generalize the sum-product results of Hegyv'{a}ri and Hennecart on

\max\{ |A+B|, |f(B,C)|\}

, for a specific type of function

f

. Finally, we prove that the number of distinct cubic distances generated by any small set

A\times A\subset \mathbb{F}_p^2

\Omega(|A|^{8/7})

, which improves a result of Yazici et al.

EPFL2017

Cryptographic Hash Functions in Groups and Provable Properties

Juraj Sarinay

We consider several "provably secure" hash functions that compute simple sums in a well chosen group (G,*). Security properties of such functions provably translate in a natural way to computational problems in G that are simple to define and possibly also hard to solve. Given k disjoint lists Li of group elements, the k-sum problem asks for gi ∊ Li such that g1 * g2 ... gk = 1G. Hardness of the problem in the respective groups follows from some "standard" assumptions used in public-key cryptology such as hardness of integer factoring, discrete logarithms, lattice reduction and syndrome decoding. We point out evidence that the k-sum problem may even be harder than the above problems. Two hash functions based on the group k-sum problem, SWIFFTX and FSB, were submitted to NIST as candidates for the future SHA-3 standard. Both submissions were supported by some sort of a security proof. We show that the assessment of security levels provided in the proposals is not related to the proofs included. The main claims on security are supported exclusively by considerations about available attacks. By introducing "second-order" bounds on bounds on security, we expose the limits of such an approach to provable security. A problem with the way security is quantified does not necessarily mean a problem with security itself. Although FSB does have a history of failures, recent versions of the two above functions have resisted cryptanalytic efforts well. This evidence, as well as the several connections to more standard problems, suggests that the k-sum problem in some groups may be considered hard on its own, and possibly lead to provable bounds on security. Complexity of the non-trivial tree algorithm is becoming a standard tool for measuring the associated hardness. We propose modifications to the multiplicative Very Smooth Hash and derive security from multiplicative k-sums in contrast to the original reductions that related to factoring or discrete logarithms. Although the original reductions remain valid, we measure security in a new, more aggressive way. This allows us to relax the parameters and hash faster. We obtain a function that is only three times slower compared to SHA-256 and is estimated to offer at least equivalent collision resistance. The speed can be doubled by the use of a special modulus, such a modified function is supported exclusively by the hardness of multiplicative k-sums modulo a power of two. Our efforts culminate in a new multiplicative k-sum function in finite fields that further generalizes the design of Very Smooth Hash. In contrast to the previous variants, the memory requirements of the new function are negligible. The fastest instance of the function expected to offer 128-bit collision resistance runs at 24 cycles per byte on an Intel Core i7 processor and approaches the 17.4 figure of SHA-256. The new functions proposed in this thesis do not provably achieve a usual security property such as preimage or collision resistance from a well-established assumption. They do however enjoy unconditional provable separation of inputs that collide. Changes in input that are small with respect to a well defined measure never lead to identical output in the compression function.

EPFL2011

On some algebraic and extremal problems in discrete geometry

Seyed Hossein Nassajianmojarrad

In the present thesis, we delve into different extremal and algebraic problems arising from combinatorial geometry. Specifically, we consider the following problems. For any integer

n\ge 3

, we define

e(n)

to be the minimum positive integer such that any set of

e(n)

points in general position in the plane contains

n

points in convex position. In 1935, Erd\H{o}s and Szekeres proved that

e(n) \le {2n-4 \choose n-2}+1

and later in 1961, they obtained the lower bound

2^{n-2}+1 \le e(n)

, which they conjectured to be optimal. We prove that

e(n) \le {2n-5 \choose n-2}-{2n-8 \choose n-3}+2

. In a recent breakthrough, Suk proved that

e(n) \le 2^{n+O\left(n^{2/3}\log n\right)}

c(n)

(and

c'(n)

) denote the smallest positive integer

N

such that any family of

N

pairwise disjoint convex bodies in general position (resp.,

N

convex bodies in general position, any pair of which share at most two boundary points) has an

n

members in convex position. We show that

c(n)\le c'(n)\le 2^{n+O\left(\sqrt{n\log n}\right)}

. Given a point set

P

in the plane, an ordinary circle for

P

is defined as a circle containing exactly three points of

P

. We prove that any set of

n

points in the plane, not all on a line or a circle, determines at least

\frac{1}{4}n^2-O(n)

ordinary circles. We determine the exact minimum number of ordinary circles for all sufficiently large

n

C\subset \mathbb{C}^2

of degree

d

and two finite sets

A,B\subset \mathbb{C}

, we have

|C\cap (A\times B)|=O_d(|A|+|B|)

. We establish a two-dimensional version of this result, and prove upper bounds on the size of the intersection

|X\cap (P\times Q)|

for a variety

X\subset \mathbb{C}^4

and finite sets

P,Q\subset \mathbb{C}^2

\max \{ |A+A|, |f(A, A)|\}\gg |A|^{6/5}

for any small set

A\subset \mathbb{F}_p

and quadratic non-degenerate polynomial

f(x, y)\in \mathbb{F}_p[x, y]

. This generalizes the result of Roche-Newton et al. giving the best known lower bound for the term

\max \{ |A+A|, |A \cdot A |\}

. Secondly, we improve and generalize the sum-product results of Hegyv'{a}ri and Hennecart on

\max\{ |A+B|, |f(B,C)|\}

, for a specific type of function

f

. Finally, we prove that the number of distinct cubic distances generated by any small set

A\times A\subset \mathbb{F}_p^2

\Omega(|A|^{8/7})

, which improves a result of Yazici et al.

EPFL2017

Cryptographic Hash Functions in Groups and Provable Properties

Juraj Sarinay

EPFL2011

Erdos Distinct Distances Problem and Extensions over Finite Spaces

Van Thang Pham

EPFL2017