This document proposes a construction for quantum hash functions based on classical universal hashing. It defines the concept of a quantum hash generator as a family of functions that maps elements to quantum states. The construction combines the robustness of classical error-correcting codes with the compact representation of quantum systems. It presents two examples of quantum hash functions: one based on binary error-correcting codes and one based on a field of elements. The construction is proved to be optimal in terms of the number of qubits needed to represent the quantum states.
Na matemática, o teorema de Green-Tao, demonstrado por Ben Green e Terence Tao em 2004, afirma que a sequência de números primos contém progressões aritméticas arbitrariamente longas. Em outras palavras, para cada número natural k, existe um progressão aritmética formada por k números primos. O Teorema de Green-Tao é um caso particular da conjectura de Erdös sobre progressões aritméticas.
Na matemática, o teorema de Green-Tao, demonstrado por Ben Green e Terence Tao em 2004, afirma que a sequência de números primos contém progressões aritméticas arbitrariamente longas. Em outras palavras, para cada número natural k, existe um progressão aritmética formada por k números primos. O Teorema de Green-Tao é um caso particular da conjectura de Erdös sobre progressões aritméticas.
In this talk, I address two new ideas in sampling geometric objects. The first is a new take on adaptive sampling with respect to the local feature size, i.e., the distance to the medial axis. We recently proved that such samples acn be viewed as uniform samples with respect to an alternative metric on the Euclidean space. The second is a generalization of Voronoi refinement sampling. There, one also achieves an adaptive sample while simultaneously "discovering" the underlying sizing function. We show how to construct such samples that are spaced uniformly with respect to the $k$th nearest neighbor distance function.
Here, we look at the problem of going from a source s to a possible multiple destinations. At them, each of the Lemmas, Theorems and Corollaries used to prove the properties of the
1. Bellman-Ford
2. Dijkstra
are examined in detail.
a graph search algorithm that solves the single-source shortest path problem for a graph with non-negative edge path costs, producing a shortest path tree. This algorithm is often used in routing and as a subroutine in other graph algorithms.
EM 알고리즘을 jensen's inequality부터 천천히 잘 설명되어있다
이것을 보면, LDA의 Variational method로 학습하는 방식이 어느정도 이해가 갈 것이다.
옛날 Andrew Ng 선생님의 강의노트에서 발췌한 건데 5년전에 본 것을
아직도 찾아가면서 참고하면서 해야 된다는 게 그 강의가 얼마나 명강의였는지 새삼 느끼게 된다.
In this talk, I address two new ideas in sampling geometric objects. The first is a new take on adaptive sampling with respect to the local feature size, i.e., the distance to the medial axis. We recently proved that such samples acn be viewed as uniform samples with respect to an alternative metric on the Euclidean space. The second is a generalization of Voronoi refinement sampling. There, one also achieves an adaptive sample while simultaneously "discovering" the underlying sizing function. We show how to construct such samples that are spaced uniformly with respect to the $k$th nearest neighbor distance function.
Here, we look at the problem of going from a source s to a possible multiple destinations. At them, each of the Lemmas, Theorems and Corollaries used to prove the properties of the
1. Bellman-Ford
2. Dijkstra
are examined in detail.
a graph search algorithm that solves the single-source shortest path problem for a graph with non-negative edge path costs, producing a shortest path tree. This algorithm is often used in routing and as a subroutine in other graph algorithms.
EM 알고리즘을 jensen's inequality부터 천천히 잘 설명되어있다
이것을 보면, LDA의 Variational method로 학습하는 방식이 어느정도 이해가 갈 것이다.
옛날 Andrew Ng 선생님의 강의노트에서 발췌한 건데 5년전에 본 것을
아직도 찾아가면서 참고하면서 해야 된다는 게 그 강의가 얼마나 명강의였는지 새삼 느끼게 된다.
Brute force searching, the typical set and guessworkLisandro Mierez
Consider the problem of identifying the value of a discrete
random variable by only asking questions of the sort: is its
value X? That this is a time-consuming task is a cornerstone
of computationally secure ciphers
How to make hash functions go fast inside snarks, aka a guided tour through arithmetisation friendly hash functions (useful for all cryptographic protocols where cost is dominated by multiplications -- e.g. anything using R1CS; secret sharing based multiparty computation protocols; etc)
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
We examine the effectiveness of randomized quasi Monte Carlo (RQMC) to improve the convergence rate of the mean integrated square error, compared with crude Monte Carlo (MC), when estimating the density of a random variable X defined as a function over the s-dimensional unit cube (0,1)^s. We consider histograms and kernel density estimators. We show both theoretically and empirically that RQMC estimators can achieve faster convergence rates in
some situations.
This is joint work with Amal Ben Abdellah, Art B. Owen, and Florian Puchhammer.
Slack (or Teams) Automation for Bonterra Impact Management (fka Social Soluti...Jeffrey Haguewood
Sidekick Solutions uses Bonterra Impact Management (fka Social Solutions Apricot) and automation solutions to integrate data for business workflows.
We believe integration and automation are essential to user experience and the promise of efficient work through technology. Automation is the critical ingredient to realizing that full vision. We develop integration products and services for Bonterra Case Management software to support the deployment of automations for a variety of use cases.
This video focuses on the notifications, alerts, and approval requests using Slack for Bonterra Impact Management. The solutions covered in this webinar can also be deployed for Microsoft Teams.
Interested in deploying notification automations for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.
Connector Corner: Automate dynamic content and events by pushing a buttonDianaGray10
Here is something new! In our next Connector Corner webinar, we will demonstrate how you can use a single workflow to:
Create a campaign using Mailchimp with merge tags/fields
Send an interactive Slack channel message (using buttons)
Have the message received by managers and peers along with a test email for review
But there’s more:
In a second workflow supporting the same use case, you’ll see:
Your campaign sent to target colleagues for approval
If the “Approve” button is clicked, a Jira/Zendesk ticket is created for the marketing design team
But—if the “Reject” button is pushed, colleagues will be alerted via Slack message
Join us to learn more about this new, human-in-the-loop capability, brought to you by Integration Service connectors.
And...
Speakers:
Akshay Agnihotri, Product Manager
Charlie Greenberg, Host
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
Let's dive deeper into the world of ODC! Ricardo Alves (OutSystems) will join us to tell all about the new Data Fabric. After that, Sezen de Bruijn (OutSystems) will get into the details on how to best design a sturdy architecture within ODC.
UiPath Test Automation using UiPath Test Suite series, part 3DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 3. In this session, we will cover desktop automation along with UI automation.
Topics covered:
UI automation Introduction,
UI automation Sample
Desktop automation flow
Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Builder.ai Founder Sachin Dev Duggal's Strategic Approach to Create an Innova...Ramesh Iyer
In today's fast-changing business world, Companies that adapt and embrace new ideas often need help to keep up with the competition. However, fostering a culture of innovation takes much work. It takes vision, leadership and willingness to take risks in the right proportion. Sachin Dev Duggal, co-founder of Builder.ai, has perfected the art of this balance, creating a company culture where creativity and growth are nurtured at each stage.
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
Kubernetes & AI - Beauty and the Beast !?! @KCD Istanbul 2024Tobias Schneck
As AI technology is pushing into IT I was wondering myself, as an “infrastructure container kubernetes guy”, how get this fancy AI technology get managed from an infrastructure operational view? Is it possible to apply our lovely cloud native principals as well? What benefit’s both technologies could bring to each other?
Let me take this questions and provide you a short journey through existing deployment models and use cases for AI software. On practical examples, we discuss what cloud/on-premise strategy we may need for applying it to our own infrastructure to get it to work from an enterprise perspective. I want to give an overview about infrastructure requirements and technologies, what could be beneficial or limiting your AI use cases in an enterprise environment. An interactive Demo will give you some insides, what approaches I got already working for real.
PHP Frameworks: I want to break free (IPC Berlin 2024)Ralf Eggert
In this presentation, we examine the challenges and limitations of relying too heavily on PHP frameworks in web development. We discuss the history of PHP and its frameworks to understand how this dependence has evolved. The focus will be on providing concrete tips and strategies to reduce reliance on these frameworks, based on real-world examples and practical considerations. The goal is to equip developers with the skills and knowledge to create more flexible and future-proof web applications. We'll explore the importance of maintaining autonomy in a rapidly changing tech landscape and how to make informed decisions in PHP development.
This talk is aimed at encouraging a more independent approach to using PHP frameworks, moving towards a more flexible and future-proof approach to PHP development.
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
1. arXiv:1404.1503v1[quant-ph]5Apr2014
Quantum Hashing via Classical ǫ-universal Hashing
Constructions
Farid Ablayev∗
Marat Ablayev†
April 8, 2014
Abstract
In the paper, we define the concept of the quantum hash generator and offer de-
sign, which allows to build a large amount of different quantum hash functions. The
construction is based on composition of classical ǫ-universal hash family and a given
family of functions – quantum hash generator.
The proposed construction combines the properties of robust presentation of in-
formation by classical error-correcting codes together with the possibility of highly
compressed presentation of information by quantum systems.
In particularly, we present quantum hash function based on Reed-Solomon code, and
we proved, that this construction is optimal in the sense of number of qubits needed.
Keywords. hashing, fingerprinting, quantum computations, quantum hashing, ǫ-univer-
sal hashing, error-correcting codes.
1 Introduction
Quantum computing is inherently a very mathematical subject, and discussions of how quan-
tum computers can be more efficient than classical computers in breaking encryption algo-
rithms are started since Peter Shor invented his famous quantum algorithm. The answer of
a cryptography community is a “Post-quantum cryptography”, which refers to research on
problems (usually public-key cryptosystems) that are not efficiently breakable using quan-
tum computers more than classical computer architectures. Currently post-quantum cryp-
tography includes different approaches, in particular, hash-based signature schemes such as
Lamport signatures and Merkle signature scheme.
Hashing itself is an important basic concepts for organization transformation and reliable
transmission of information. The concept known as “universal hashing“ was invented by
Carter and Wegman [6] in 1979. In 1994 it was discovered relationship between ǫ-universal
hash families and error-correcting codes [4]. In [16] Avi Wigderson characterizes universal
∗
Kazan Federal University
†
Kazan Federal University
1
2. hashing as being a tool which “should belong to the fundamental bag of tricks of every
computer scientist”.
Gottesman and Chuang proposed a quantum digital system [8] based on quantum mechan-
ics. Their results based on quantum fingerpriting technique and add “quantum direction” for
the post-quantum cryptography. Quantum fingerprints have been introduced by Buhrman,
Cleve, Watrous and de Wolf in [5]. Gavinsky and Ito [7] viewed quantum fingerprints as
cryptographic primitives.
In [2, 3] we viewed at quantum fingerprinting as a construction for binary hash function
and introduced a non binary hash function. The proposed quantum hashing is a suitable
one-way function for quantum digital signature protocol from [8]. For more introductory
information we refer to the paper [2].
In this paper, we define the concept of the quantum hash generator and offer design, which
allows to build different quantum hash functions. The construction is based on composition
of classical ǫ-universal hash family and a given family of functions – quantum hash generator.
The proposed construction combines the properties of robust presentation of information
by classical error-correcting codes together with the possibility of highly compressed presen-
tation of information by quantum systems. In particularly, using the relationship between
ǫ-universal hash families and error-correcting codes, we presented quantum hash function
ψRS based on Reed-Solomon code, and we proved, that this construction is optimal in the
sense of number of qubits needed for the construction.
1.1 Definitions and notations
We begin by recalling definitions of classical hash families from [13].
Given a domain X, |X| = K, and a range Y, |Y| = M, (typically with K ≥ M), a hash
function f is map
f : X → Y,
that hash long inputs to short outputs.
We let q to be a prime power and Fq be a field. Let Σk
be a set of words of length k over
a finite alphabet Σ. In the paper we let X = Σk
, or X = Fq, or X = (Fq)k
, and Y = Fq.
A hash family is a set F = {f1, . . . , fN } of N hash functions fi : X → Y
ǫ universal hash family. Hash family F is called an ǫ-universal hash family if for
any two distinct elements w, w′
∈ X, there exist at most ǫN functions f ∈ F such that
f(w) = f(w′
). We will use the notation ǫ-U (N; K, M) as an abbreviation for ǫ-universal
hash family.
Clearly we have, that if function f is chosen uniformly at random from a given ǫ-U
(N; K, M) hash family F, then the probability that any two distinct words collide under f
is at most ǫ.
The case ǫ = 1/N is known as universal hashing.
Classical-quantum function. The notion of quantum function was considered in [10].
In the paper we use the following variant of quantum function. Let s ≥ 1. Let (H2
)⊗s
be an
2s
-dimensional Hilbert space of s qubits quantum states. Space (H2
)⊗s
made up of s copies
2
3. of a single qubit space H2
(H2
)⊗s
= H2
⊗ . . . ⊗ H2
= H2s
.
For K = |X| and integer s ≥ 1 we define an (K; s) classical-quantum function to be a
map of elements w ∈ X to quantum states |ψ(w) ∈ (H2
)⊗s
, started from some fixed initial
state |ψ(e) ∈ (H2
)⊗s
ψ : X × {|ψ(e) } → (H2
)⊗s
. (1)
Typically we let |ψ(e) = |0 . In this case we present the definition (1) in the form
ψ : X × {|0 } → (H2
)⊗s
or just ψ : X → (H2
)⊗s
. (2)
We will also use notation ψ : w → |ψ(w) for ψ, which is frequently used in different papers.
Effective functions (classical and quantum). For practical uses, hash functions should
be easy to compute. That is, computing the hash value of an element w ∈ X should be feasible
in time polynomial in the size of w. This consideration leads to the following definitions.
• Effective hash family. Hash family F is called effective if there is polynomial time
algorithm such that for each f ∈ F on input w it outputs f(w).
• Effective function ψ. Function ψ is called effective if there is a polynomial-time
algorithm such that on input w outputs |ψ(w) .
2 Quantum hashing
Property 1 Let ψ be an (K; s) classical-quantum function. If ⌈ log K⌉ ≫ s, then given
|ψ(w) , it is impossible to invert w.
Proof. This pre-image resistance property follows from Holevo bound [11]. Since no more
than s classical bits of information can be extracted from s qubits while the original message
contains ⌈ log K⌉ > s bits.
We come to the notion of one-way function if in addition function ψ is easy to compute.
• Function ψ is called a classical-quantum one-way if
– ψ is effective.
– Hard to invert: given |ψ(w) , it is impossible to invert w by virtue of fundamental
quantum information theory.
Example 1 (One-way function) A word w ∈ {0, 1}k
is encoded by a single qubit:
ψ : w → cos
2πw
2k
|0 + sin
2πw
2k
|1 .
Here we treat w = w0 . . . wk−1 also as a number w = k−1
i=0 wi2i
.
3
4. Clearly, we have that ψ has the one-way property of the definition above. What we
need additionally and what is implicitly assumed in various papers (see for example [2] for
more information) is a collision resistance property. However, there is still no such notion as
quantum collision. The reason why we need to define it is the observation that in quantum
hashing there might be no collisions in the classical sense: since quantum hashes are quantum
states they can store arbitrary amount of data and can be different for unequal messages.
But the procedure of comparing those quantum states implies measurement, which can lead
to collision-type errors.
So, a quantum collision is a situation when a procedure that tests an equality of quantum
hashes outputs true, while hashes are different. This procedure can be a well-known SWAP-
test (see for example [2] for more information and citations) or something that is adapted for
specific hashing function. Anyway, it deals with the notion of distinguishability of quantum
states. And since non-orthogonal quantum states cannot be perfectly distinguished, we
require them to be “nearly orthogonal”.
• For δ ∈ (0, 1/2) we call a function
ψ : X → (H2
)⊗s
δ-resistant if for any pair w, w′
of different elements
| ψ(w)|ψ(w′
) | ≤ δ.
Theorem 1 Let ψ : X → (H2
)⊗s
be a δ-resistant function. Then
s ≥ log log |X| − log log 1 + 2/(1 − δ) − 1.
Proof. First we observe, that from the definition |||ψ || = ψ|ψ of the norm it follows
that
|||ψ − |ψ′
||2
= |||ψ ||2
+ |||ψ′
||2
− 2 ψ|ψ′
.
Hence for arbitrary pair w, w′
of different elements from X we have that
|||ψ(w) − |ψ(w′
) || ≥ 2(1 − δ).
We let ∆ = 2(1 − δ). For short we let (H2
)⊗s
= V in this proof. Consider a set Φ =
{|ψ(w) : w ∈ X} of K = |X| points in V . If we draw a sphere of the radius ∆/2 with the
center |ψ ∈ Φ then all such spheres do not intersect pairwise. All these K spheres are in
large sphere of radius 1 + ∆/2. The volume of a sphere of a radius r in V is cr2s+1
for the
complex space V . Constant c depends on the metric of V . From this we have, that the
number K is bonded by the number of “small spheres” in the “large sphere”
K ≤
c(1 + ∆/2)2s+1
c(∆/2)2s+1 .
Hence
s ≥ log log K − log log 1 + 2/(1 − δ) − 1.
4
5. One-way property and the notion of δ-resistance naturally lead to the following notion of
quantum hash function.
Definition 1 (Quantum hash function) Let K, s be a positive integers and K = |X|. We
call a map
ψ : X → (H2
)⊗s
an δ resistant (K; s) quantum hash function if
• ψ is one-way function and
• ψ is δ resistant function.
We will use the notation δ-R (K; s) as an abbreviation for δ resistant (K; s) quantum hash
function.
3 Generator for quantum hash functions
In this section we present two constructions of quantum hash functions and define a notion
of quantum hash function generator, which generalize these constructions.
3.1 Binary quantum hashing.
One of the first explicit quantum hash functions was defined in [5]. Originally authors in-
vented construction called a “quantum fingerprinting” for testing equality of two words for
quantum communication model. The cryptographical aspects of quantum fingerprinting pre-
sented in [7]. Quantum fingerprinting technique is based on binary error correcting codes.
Later this construction was adopted for cryptographic purposes (in the paper fingerprint-
ing that keeps secrets). Here we present quantum fingerprinting construction via quantum
hashing point of view.
An (n, k, d, ) error-correcting code is a map
C : Σk
→ Σn
such that for any two distinct words w, w′
∈ Σk
the Hamming distance between code words
C(w) and C(w′
) is at least d. The code is binary if Σ = {0, 1}.
The construction of quantum hash function based on quantum fingerprinting in the fol-
lowing.
• Let c > 1 and δ < 1. Let k be a positive integer and n > k. Let E : {0, 1}k
→ {0, 1}n
be an (n, k, d) binary error correcting code with the Hamming distance d ≥ (1 − δ)n.
• Define a family of functions FE = {E1, . . . , En}, where Ei : {0, 1}k
→ F2 defined by
the rule: Ei(w) is the i-th bit of the code word E(w).
5
6. • Define classical-quantum function ψFE
: {0, 1}k
→ (H2
)⊗s
, determined for a word w as
ψFE
(w) =
1
√
n
n
i=1
|i |Ei(w) =
1
√
n
n
i=1
|i cos
πEi(w)
2
|0 + sin
πEi(w)
2
|1 ,
For s = log n + 1 function ψFE
is an δ-R (2k
; s) quantum hash function. That is, due to
the Property 1 ψFE
is the one-way. Next, straightforward computations show, that for two
different words w, w′
we have, that
| ψFE
(w)|ψFE
(w′
) | ≤ δn/n = δ.
Observe, that authors in [5] propose for the first choice of such binary codes a Justesen
codes with n = ck, which give δ < 9/10 + 1/(15c) for any chosen c > 2. Next we observe,
that the above construction of quantum hash function needs log n + 1 qubits for the fixed
δ ≈ 9/10 + 1/(15c). This number of qubits is good enough in the sense of the lower bound
of Theorem 1.
3.2 Non binary quantum hashing.
The non binary quantum hash function is presented in [2] and is based on the construction
from [1]. We present non binary quantum hash function from [2] in the following form. For
a field Fq let B = {b1, . . . , bT } ⊆ Fq. For every bj ∈ B, define a function hj : Fq → Fq by the
rule
hj(w) = bjw (mod q).
Let H = {h1, . . . hT } and t = log T. We define classical-quantum function
ψH : Fq → (H2
)⊗(t+1)
defined by the rule
|ψH(w) =
1
√
T
T
j=1
|j cos
2πhj(w)
q
|0 + sin
2πhj(w)
q
|1 . (3)
The following fact proved in [2].
Theorem 2 Let q be a prime power and Fq be a field. Then for arbitrary δ > 0 there exists
a set B = {b1, . . . , bT } ⊆ Fq (and therefore a corresponding family H = {h1, . . . , hT } of
functions) with T = ⌈(2/δ2
) ln(2q)⌉, such that quantum function ψH (3) is an δ-R (q; t + 1)
quantum hash function.
In the rest of the paper we will use the notation Hδ,q to denote this family H of functions
from Theorem 2 and will use the notation ψHδ,q
to denote the corresponding quantum function
(3).
Remark 1 • Observe, that the construction of function ψHδ,q
needs t + 1 ≤ log log 2q +
2 log 1/δ + 3 qubits. This number of qubits is good enough in the sense of the lower
bound of Theorem 1.
• Efficient implementation of ψHδ,q
is based on results from [12].
• numerical results on ψHδ,q
presented in [2].
6
7. 3.3 Quantum hash generator
The above two constructions of quantum hash functions are perform (quantumly in parallel)
certain controlled transformations of one qubit. Each such transformation generated by a
corresponding discrete function from a specific family (FE and Hδ,q respectively) of functions.
These constructions lead to the following definition of “Quantum hash generator”.
First we introduce the following notion.
Consider a function g : X → Fq. Let ℓ ≥ 1 be an integer. Let ψg be a classical-quantum
function ψg : X → (H2
)⊗ℓ
determined by the rule
ψg : w → |ψg(w) =
2ℓ
i=1
αi(g(w))|i , (4)
where amplitudes (complex numbers) αi(g(w)), i ∈ {1, . . . , 2ℓ
}, of the state |ψg(w) are
determined by g(w) and satisfy (at least) the following general condition
2l
i=1
|αi(g(w))|2
= 1.
Definition 2 (Quantum hash generator) Let K = |X| and G = {g1, . . . , gD} be an ef-
fective family of functions gj : X → Fq. Let ℓ ≥ 1 be an integer. For g ∈ G let ψg be an
effective classical-quantum function ψg : X → (H2
)⊗ℓ
. Let d = log D.
We define a classical-quantum function ψG : X → (H2
)⊗(d+ℓ)
by the rule
ψG : w → |ψG(w) =
1
√
D
D
j=1
|j ψgj
(w) . (5)
We say that family G generates δ-R (K; d + ℓ) quantum hash function ψG and we call G a
δ-R (K; d + ℓ) quantum hash generator, if ψG is an δ-R (K; d + ℓ) quantum hash function.
In according to Definition 2 the family FE = {E1, . . . , En} from section 3.1 is δ-R
(2k
; log n + 1) quantum hash generator and the family Hδ,q from section 3.2 is δ-R (q; t + 1)
quantum hash generator.
4 Quantum hashing via classical ǫ-universal hashing
constructions
In this section we present construction of a quantum hash generator based on composition of
ǫ universal hash family with a given quantum hash generator. We begin with the definitions
and notation that we use in the rest of the paper.
Let K = |X|, M = |Y|. Let F = {f1, . . . , fN } be a family of functions, where
fi : X → Y.
7
8. Let q be a prime power and Fq be a field. Let H = {h1, . . . , hT } be a family of functions,
where
hj : Y → Fq.
For f ∈ F and h ∈ HB, define a composition g = f ◦ h,
g : X → Fq,
by the rule
g(w) = (f ◦ h)(w) = h(f(w)).
Define a composition G of two families F and H as follows.
G = F ◦ H = {g = f ◦ h : f ∈ F, h ∈ H}.
Theorem 3 Let F = {f1, . . . , fN } be an effective ǫ-U (N; K, M) hash family. Let ℓ ≥ 1.
Let H = {h1, . . . hT } be an δ-R (M; log T + ℓ) quantum hash generator. Let log K > log N +
log T + ℓ.
Then composition G = F ◦ H is an ∆-R (K; s) quantum hash generator, where
s = log N + log T + ℓ (6)
and
∆ ≤ ǫ + δ. (7)
Proof. Introduce the following notation for the composition G = F ◦ H.
G = {gij = fi ◦ hj : i ∈ I, j ∈ J},
where I = {1, . . . , N}, J = {1, . . . , T}.
δ-R (M; log T +ℓ) quantum hash generator H generates δ-R (M; log T +ℓ) quantum hash
function
ψH : v →
1
√
T j∈J
|j ψhj
(v) . (8)
For s = log N + log T + ℓ, using family G, define a map
ψG : X → (H2
)⊗s
by the rule
|ψG(w) =
1
√
N i∈I
|i ⊗ |ψH(fi(w)) . (9)
8
9. ∆ resistance of ψG. Consider a pair w,w′
of different elements from X . For the pair w,w′
of elements consider inner product ψG(w)|ψG(w′
) . Using the linearity of the inner product
we have that
ψG(w)|ψG(w′
) =
1
N i∈I
ψH(fi(w))|ψH (fi(w′
)) .
For the the pair w,w′
we define two sets of indexes Ibad and Igood:
Ibad = {i ∈ I : fi(w) = fi(w′
)}, Igood = {i ∈ I : fi(w) = fi(w′
)}.
Then we have
| ψG(w)|ψG(w′
) | ≤
1
N i∈Ibad
| ψH(fi(w))|ψH(fi(w′
)) |
+
1
N i∈Igood
| ψH(fi(w))|ψH(fi(w′
)) |. (10)
Hash family F is ǫ-universal, hence
|Ibad| ≤ ǫN.
Quantum function ψH : Y → (H2
)log T+ℓ
is δ-resistant, hence for arbitrary pair v, v′
of
different elements from Y it holds
| ψH(v)|ψH(v′
) | ≤ δ.
Finally from (10) and the above two inequalities we have that
| ψG(w)|ψG(w′
) | ≤ ǫ +
|Igood|
N
δ ≤ ǫ + δ.
The last inequality proves ∆ resistance of ψG(w) (say for ∆ = ǫ + δ(Igood|)/N) and proves
the inequality (7).
One-way property of ψG. Condition log K > log N + log T + ℓ together with Property 1
proves that given ψG(w), it is impossible to invert w. Family F is effective and the family H
is a quantum hash generator.
These means that for composition G = F ◦H function ψG is a classical-quantum one-way
function.
To finish the proof of the theorem remains show that function ψG can be presented in the
form, which displayed in (5). From (8) and (9) we have that
|ψG(w) =
1
√
N i∈I
|i ⊗
1
√
T j∈J
|j ψhj
(fi(w)) .
Using notation from (4) the above expression can be presented in the following form (5).
|ψG(w) =
1
√
NT i∈I,j∈J
|ij ψgij
(w) .
9
10. 5 Quantum hashing. Explicit constructions
The following statement is the corollary of Theorem 3 and is a basis for explicit constructions
of quantum hash functions in this section. Let q be a prime power and Fq be a field. Let
δ ∈ (0, 1). Let Hδ,q be a family of functions from Theorem 2. Let |X| = K.
Theorem 4 Let F = {f1, . . . , fN } be an effective ǫ-U (N; K, q) hash family, where fi : X →
Fq. Then for arbitrary δ > 0, family G = F ◦ Hδ,q is an ∆-R (K; s) quantum hash generator,
where
s ≤ log N + log log q + 2 log 1/δ + 3
and
∆ ≤ ǫ + δ.
Proof. Family Hδ,q = {h1, . . . , hT }, where hi : Fq → Fq, T = ⌈(2/δ2
) ln(2q)⌉, ℓ = 1, and
s = log T + 1 ≤ log n + log log q + 2 log 1/δ + 3 is the δ-R (q; s) quantum hash generator.
In the rest of the section we present examples of quantum hash functions based on uni-
versal linear hash family and error-correcting codes.
5.1 Quantum hashing from universal linear hash family
The next hash family is folklore and was displayed in several papers and books. See the
paper [14] and the book [15] for more information.
• Let k be a positive integer and let q be a prime power. Let X = (Fq)k
{(0, . . . , 0)}. For
every vector a ∈ (Fq)k
define hash function fa : X → Fq by the rule
fa(w) =
k
i=1
aiwi.
Then Flin = {fa : a ∈ (Fq)k
} is an (1/q)-U (qk
; (qk
− 1); q) hash family (universal hash
family).
Theorem 5 Let k be a positive integer, let q be a prime power. Then for arbitrary δ ∈ (0, 1)
composition G = Flin ◦ Hδ,q is a ∆-R (qk
; s) quantum hash generator with ∆ ≤ (1/q) + δ and
s ≤ k log q + log log q + 2 log 1/δ + 3.
Proof. According to Theorem 4 function ψG is ∆-R (qk
; s) quantum hash function with the
parameters stated in the theorem.
Remark 2 Note, that from Theorem 1 we have that
s ≥ log log |X|+log log 1 + 2/(1 − δ) −1 ≥ log k+log log q−log log 1 + 2/(1 − δ) −1.
This lower bound shows that the quantum hash function ψG is not asymptotically optimal in
the sense of number of qubits used for the construction.
10
11. 5.2 Quantum hashing based on Freivalds fingerprinting
For a positive integer k let X = {0, 1}k
. Let c > 1 be a positive integer and let M = ck ln k.
Let Y = {0, 1, . . . , M − 1}.
For an i-th prime pi ∈ Y define function (fingerprint)
fi : X → Y
by the rule
fi(w) = w (mod pi).
Here we treat a word w = w0w1 . . . wk−1 also as an integer w = w0 + w12 + · · · + wk−12k−1
.
Consider a set
FM = {f1, . . . , fπ(M)}
of fingerprints. Here π(M) denote the number of primes less than M. Note that then
π(M) ∼ M/ ln M as M → ∞. Moreover,
M
ln M
≤ π(M) ≤ 1.26
M
ln M
for M ≥ 17.
The following fact is based on the construction (“Freivalds fingerprinting method”) due
to R. Freivalds [9].
Property 2 A set FM of fingerprints is an (1/c)-U (π(M); 2k
, M) hash family.
Proof (sketch). For any pair w, w′
of distinct words from {0, 1}k
the number N(w, w′
) =
|{fi ∈ FM : fi(w) = fi(w′
)}| is bounded above by k. Thus, if we pick a prime pi (uniformly
at random) from Y then
Pr[fi(w) = fi(w′
)] ≤
k
π(M)
≤
k ln M
M
.
Picking M = ck ln k for a constant c gives Pr[fi(w) = fi(w′
)] ≤ 1
c
+ o(1).
Remark 3 Note that the families FM are nonuniform effective families.
Theorem 4 and Property 2 provide the following statement.
Theorem 6 Let c > 1 be a positive integer and let M = ck ln k. Let q ∈ {M, . . . , 2M} be
a prime. Then for arbitrary δ > 0, family G = FM ◦ Hδ,q is an ∆-R (2k
; s) quantum hash
generator, where
s ≤ log ck + log log k + log log q + 2 log 1/δ + 3
and
∆ ≤
1
c
+ δ.
11
12. Proof. From Theorem 4 we have that
s ≤ log π(M) + log log q + 2 log 1/δ + 3.
From the above agreement we have that M = ck ln k. Thus
s ≤ log ck + log log k + log log q + 2 log 1/δ + 3.
Remark 4 Note, that from Theorem 1 we have that
s ≥ log k + log log q − log log 1 + 2/(1 − δ) − 1.
This lower bound shows that the quantum hash function ψFM
is good enough in the sense of
number of qubits used for the construction.
5.3 Quantum hashing and error-correcting codes
Let q be a prime power and let Fq be a field. An (n, k, d, ) error-correcting code is called
linear, if Σ = Fq, and C = {C(w) : w ∈ Fk
q } is a subspace of (Fq)n
. We will denote such
linear code by an [n, k, d, ]q code.
Theorem 7 Let C be an [n, k, d]q code. Then for arbitrary δ ∈ (0, 1) there exists a ∆-R (qk
; s)
quantum hash generator G, where ∆ = (1 − d/n) + δ and s ≤ log n + log log q + 2 log 1/δ + 4.
Proof. The following fact was observed in [4, 13]. Having an [n, k, d]q code C, we can explicitly
construct a (1 − d/n)-U (n; qk
; q) hash family FC.
By Theorem 4 a composition G = FC ◦ Hδ,q is an ∆-R (qk
; s) quantum hash generator,
where ∆ = (1 − d/n) + δ and s ≤ log n + log log q + 2 log 1/δ + 4.
Quantum hash function via Reed-Solomon code As an example we present construc-
tion of quantum hash function, using Reed-Solomon codes.
Let q be a prime power, let k ≤ n ≤ q, let Fq be a finite field. A Reed-Solomon code (for
short RS-code) is a linear code
CRS : (Fq)k
→ (Fq)n
having parameters [n, k, n − (k − 1)]q. RS-code defined as follows. Each word w ∈ (Fq)k
,
w = w0w1 . . . wk−1 associated with the polynomial
Pw(x) =
k−1
i=0
wixi
.
Pick n distinct elements (evaluation points) A = {a1, . . . , an} of Fq. A common special case
is n = q − 1 with the set of evaluating points being A = Fq{0}. To encode word w we
evaluate Pw(x) at all n elements a ∈ A
CRS(w) = (Pw(a1) . . . Pw(an)).
Using Reed-Solon codes, we obtain the following construction of quantum hash generator.
12
13. Theorem 8 Let q be a prime power and let 1 ≤ k ≤ n ≤ q. Then for arbitrary δ ∈ (0, 1)
there is a ∆-R (qk
; s) quantum hash generator GRS, where ∆ ≤ k−1
n
+δ and s ≤ log (q log q)+
2 log 1/δ + 4.
Proof. Reed-Solomon code CRS is [n, k, n − (k − 1)]q code, where k ≤ n ≤ q. Then according
to Theorem 7 there is a family GRS, which is an ∆-R (qk
; s) quantum hash generator with
stated parameters.
In particular, if we select n ∈ [ck, c′
k] for constants c < c′
, then ∆ ≤ 1/c + δ for δ ∈ (0, 1)
and in according to Theorem 1 we get that
log (q log q) − log log 1 + 2/(1 − ∆) − log c′
/2 ≤ s ≤ log (q log q) + 2 log 1/∆ + 4.
Thus, Reed Solomon codes provides good enough parameters for resistance value ∆ and for
a number s of qubits we need to construct quantum hash function ψRS.
Explicit constructions of GRS and ψGRS
. Define (k−1)/q-U (q; Fk
q ; q) hash family FRS =
{fa : a ∈ A} based on CRS as follows. For a ∈ A define fa : (Fq)k
→ Fq by the rule
fa(w0 . . . wk−1) =
k−1
i=0
wiai
.
Let Hδ,q = {h1, . . . , hT }, where hj : Fq → Fq and T = ⌈(2/δ2
) ln 2q⌉. For s = log n+log T +1
composition GRS = FRS ◦ Hδ,q, defines function
ψGRS
: (Fq)k
→ (H2
)⊗s
for a word w ∈ (Fq)k
by the rule.
ψGRS
(w) =
1
√
n
n
i=1
|i ⊗
1
√
T
T
j=1
|j cos
2πhj(fai
(w))
q
|0 + sin
2πhj(fai
(w))
q
|1 .
Acknowledgments
We are grateful to Juraj Hromkovich for the invitation to visit the ETH under the project
“Randomized and Quantum Computational and Communication Models” and the numerous
discussions on the subject of the paper. The research supported in part by the Russian
Foundation for Basic Research under the grant 14-07-00878.
References
[1] F. Ablayev, A. Vasiliev: Algorithms for quantum branching programs based on finger-
printing. Electronic Proceedings in Theoretical Computer Science 9, pp 1-11, (2009)
[2] F. Ablayev, A. Vasiliev : Quantum Hashing, 2013, arXiv:1310.4922 [quant-ph] (2013)
13
14. [3] F. Ablayev, A. Vasiliev : Cryptographic quantum hashing, Laser Physics Letters Volume
11 Number 2, 2014
[4] J Bierbrauer, T Johansson, G Kabatianskii, B. Smeets : On Families of Hash Functions
via Geometric Codes and Concatenation, Advances in Cryptology CRYPTO 93 Lecture
Notes in Computer Science Volume 773, pp 331-342, (1994)
[5] H. Buhrman, R. Cleve, J. Watrous, and R. de Wolf : Quantum fingerprinting. Phys.
Rev. Lett. 87, 167902 (2001)
[6] J.Carter, M. Wegman, Universal Classes of Hash Functions, J.Computer and System
Sciences 18(1979),143-154, 1979
[7] D. Gavinsky, T. Ito: Quantum fingerprints that keep secrets. Quantum Information &
Computation 13(7-8): 583-606 (2013)
[8] D. Gottesman, I. Chuang: Quantum digital signatures, T echnical report, available at
http://arxiv.org/abs/quant-ph/0105032, 2001.
[9] R. Freivalds: Probabilistic Machines Can Use Less Running Time. IFIP Congress 1977:
839-842
[10] A. Montanaro and T. Osborne: Quantum boolean functions. Chicago Journal of Theo-
retical Computer Science, 1, 2010. arXiv:0810.2435.
[11] A. S. Holevo: Bounds for the quantity of information transmitted by a quantum com-
munication channel. Problemy Peredachi Informatsii, 9(3):311, 1973. English translation
Problems of Information Transmission, vol. 9, pp. 177-183, 1973
[12] A. Razborov, E. Szemeredi, and A. Wigderson: Constructing small sets that are uniform
in arithmetic progressions. Combinatorics, Probability & Computing, 2: 513-518, 1993.
[13] D.R. Stinson. On the connections between universal hashing, combinatorial designs and
error-correcting codes. Congressus Numerantium 114, 727, (1996)
[14] D.R. Stinson. Universal hash families and the leftover hash lemma, and applications to
cryptography and computing. Journal of Combinatorial Mathematics and Combinatorial
Computing, Vol. 42 (2002), pp. 3-31
[15] D.R.Stinson. Cryptography: Theory and Practice, Third Edition (Discrete Mathematics
and Its Applications) (2005)
[16] A. Wigderson, Lectures on the Fusion Method and Derandomization. Technical Re-
port SOCS-95. 2, School of Computer Science, McGill University (file /pub/tech-
reports/library/reports/95/TR95.2.ps.gz at the anonymousftpsiteftp.cs.mcgill.ca
14