Foundations of Cryptography - Volume 1 [Goldreich]

The Foundations of Cryptography - Volume 1

Oded Goldreich

Cryptography is concerned with the construction of schemes that should maintain a desired functionality, even under malicious attempts aimed at making them deviate from it. The design of cryptographic systems has to be based on firm foundations; whereas ad-hoc approaches and heuristics are a very dangerous way to go. This work is aimed at presenting firm foundations for cryptography. The foundations of cryptography are the paradigms, approaches and techniques used to conceptualize, define and provide solutions to natural ``security concerns''. The emphasis of the work is on the clarification of fundamental concepts and on demonstrating the feasibility of solving several central cryptographic problems. The current book is the first volume of this work, and it focuses on the main tools of Modern Cryptography: computational difficulty (one-way functions), pseudorandomness and zero-knowledge proofs. The next volume will focus on the main applications of Cryptography: encryption schemes, signature schemes and secure protocols.
ISBN 0-521-79172-3
Published in US in June 2001.
See purchase information.
Publisher: Cambridge University Press
(see the publisher's page for this volume).

This volume is part of the two-volume work Foundations of Cryptography (see Volume 2).
For a brief introduction to the Foundations of Cryptography, see surveys.
For further suggestions regarding teaching, see notes. A preliminary draft is available here.

This page includes the volume's preface, Teaching Tips, Table of Contents, answers to Frequently Asked Questions, and List of Corrections.

Preface

See preface to the entire work Foundations of Cryptography. We highlight the following points:

Cryptography is concerned with the construction of schemes that should maintain a desired functionality, even under malicious attempts aimed at making them deviate from it.
It makes little sense to make assumptions regarding the specific strategy that the adversary may use. The only assumptions that can be justified refer to the computational abilities of the adversary.
The design of cryptographic systems has to be based on firm foundations; whereas ad-hoc approaches and heuristics are a very dangerous way to go.
This work is aimed at presenting firm foundations for cryptography. The foundations of cryptography are the paradigms, approaches and techniques used to conceptualize, define and provide solutions to natural ``security concerns''.
- We will present some of these paradigms, approaches and techniques as well as some of the fundamental results obtained using them.
- Our emphasis is on the clarification of fundamental concepts and on demonstrating the feasibility of solving several central cryptographic problems. This is done in a way independent of the particularities of some popular number theoretic examples.
- The most relevant background for this work is provided by basic knowledge of algorithms (including randomized ones), computability and elementary probability theory. Background on (computational) number theory, which is required for specific implementations of certain constructs, is not really required here.
Most of modern cryptography relies on the ability to generate instances of hard problems. Such ability is captured in the definition of one-way function.

Preface to the current volume

The current (first) volume consists of an introductory chapter (Chapter 1), followed by chapters on computational difficulty (one-way functions), pseudorandomness and zero-knowledge proofs (Chapters 2-4, respectively). Also included are two appendices: one of them providing a brief summary of Volume 2. The high-level structure of the current volume is as follows:

Chapter 1: Introduction
Chapter 2: Computational Difficulty (One-Way Functions)
Chapter 3: Pseudorandom Generators
Chapter 4: Zero-Knowledge Proofs
Appendix A: Brief Background on Computational Number Theory
Appendix B: Brief Outline of Volume 2
Bibliography and Index

Historical notes, suggestions for further reading, some open problems and some exercises are provided at the end of each chapter. The exercises are mostly designed to help and test the basic understanding of the main text, not to test or inspire creativity. The open problems are fairly well-known, still we recommend to check their current status (e.g., in this website).

Using this work

The work is aimed to serve both as a textbook and a reference text. That is, it is aimed at serving both the beginner and the expert. In order to achieve this aim, the presentation of the basic material is very detailed so to allow a typical CS-undergraduate to follow it. An advanced student (and certainly an expert) will find the pace (in these parts) way too slow. However, an attempt was made to allow the latter reader to easily skip details obvious to him/her. In particular, proofs are typically presented in a modular way. We start with a high-level sketch of the main ideas, and only later pass to the technical details. Passage from high-level descriptions to lower level details is typically marked by phrases such as details follow. More advanced material is typically presented at a faster pace and with less details. Thus, we hope that the attempt to satisfy a wide range of readers will not harm any of them.

Teaching

(See also Teaching Notes.)

The material presented in this work is, on one hand, way beyond what one may want to cover in a course, and on the other hand falls very short of what one may want to know about Cryptography in general. To assist these conflicting needs we make a distinction between basic and advanced material, and provide suggestions for further reading (in the last section of each chapter). In particular, sections, subsections, and subsubsections marked by an asterisk (*) are intended for advanced reading.

Volumes 1 and 2 of this work are supposed to provide all material for a course on Foundations of Cryptography. For a one-semester course, the teacher will definitely need to skip all advanced material (marked by an asterisk) and maybe even some basic material: see suggestion below. This should allow, depending on the class, to cover the basic material at a reasonable level (i.e., cover all material marked as ``main'' and some of the ``optional''). Volumes 1 and 2 can also serve as textbook for a two-semester course. Either way, the current volume only covers the first half of the aforementioned material. The second half is covered in Volume 2.

Following is our suggestion for a one-semester course on Foundations of Cryptography. Each lecture consists of one hour. Lectures 1-15 are covered by the current volume. Lectures 16-28 will be covered by the second volume.

Lecture 1: Introduction, Background, etc (depending on class)
Lecture 2-5: Computational Difficulty (One-Way Functions)
Lecture 6-10: Pseudorandom Generators
Lecture 11-15: Zero-Knowledge Proofs
Lecture 16-20: Encryption Schemes
Lecture 21-24: Signature Schemes
Lecture 25-28: General Cryptographic Protocols

Giving a course solely based on the material that appears in the current volume is indeed possible, but such a course cannot be considered a stand-alone course in Cryptography (for the reason that the current volume does not consider at all the basic tasks of encryption and signatures).

Preface

Chapter 1: Introduction

1.1 Cryptography - Main Topics
1.2 Some Background from Probability Theory
1.3 The Computational Model
1.4 Motivation to the Rigorous Treatment
1.5 Miscellaneous

Chapter 2: Computational Difficulty

2.1 One-Way Functions: Motivation
2.2 One-Way Functions: Definitions
2.3 Weak One-Way Functions Imply Strong Ones
2.4 One-Way Functions: Variations
2.5 Hard-Core Predicates
2.6* Efficient Amplification of One-way Functions
2.7 Miscellaneous

Chapter 3: Pseudorandom Generators

3.1 Motivating Discussion
3.2 Computational Indistinguishability
3.3 Definitions of Pseudorandom Generators
3.4 Constructions based on One-Way Permutations
3.5* Constructions based on One-Way Functions
3.6 Pseudorandom Functions
3.7* Pseudorandom Permutations
3.8 Miscellaneous

Chapter 4: Zero-Knowledge Proof Systems

4.1 Zero-Knowledge Proofs: Motivation
4.2 Interactive Proof Systems
4.3 Zero-Knowledge Proofs: Definitions
4.4 Zero-Knowledge Proofs for NP

4.5* Negative Results
4.6* Witness Indistinguishability and Hiding
4.7* Proofs of Knowledge
4.8* Computationally-Sound Proofs (Arguments)
4.9* Constant-Round Zero-Knowledge Proofs
4.10* Non-Interactive Zero-Knowledge Proofs
4.11* Multi-Prover Zero-Knowledge Proofs
4.12 Miscellaneous

Appendix A: Background in Computational Number Theory

A.1 Prime Numbers
A.2 Composite Numbers

Appendix B: Brief Outline of Volume 2

B.1 Encryption - Brief Summary
B.2 Signatures - Brief Summary
B.3 Cryptographic Protocols - Brief Summary

Bibliography (approximately 200 entries)

Index (approximately 500 entries)

Frequently Asked Questions

Is there a solution manual of this book? Unfortunately, the answer is negative, but we believe that the guidelines for the exercises provide sufficient clues.

When will Volume 2 appear? It has appeared, in Spring 2004.

Can Volume 1 (alone) be used for teaching? Yes, see teaching tips above. But this question seems less relevant today, since Volume 2 is available.

How does this volume relate to the book Modern Cryptography, Probabilistic Proofs and Pseudorandomness? The current volume is only remotely related to the former book, published in 1999 as part (i.e., Vol. 17) of the Algorithms and Combinatorics series of Springer: Chapter 1 of the Springer book provides a 30-page overview (or summary) to the entire work, which holds 350 pages in its current (first) volume only. Chapters 2 and 3 of the Springer book provide a wider perspective on Probabilistic Proof systems and Pseudorandomness, respectively. An extensive treatment of the Cryptographically-related aspects of these areas is provided in the current volume, but the interested reader may benefit from the wider perspective offered in the Springer book.

List of Corrections

The following errors were corrected in Volume 2; for further details see Appendix C (of Volume 2) as well as a draft of the said appendix, Feb. 2003.

Unfortunately, we have to withdraw two claims regarding strong witness indistinguishable proofs as defined in Definition 4.6.2. Specifically, in general, strong witness indistinguishability is not closed under parallel composition (and so Lemma 4.6.7 is wrong). Consequently, in contrary to what is stated in Theorem 4.6.8, we do not know whether there exist constant-round public-coin proofs with negligible error that are strong witness indistinguishable for languages out of BPP. We stress that the flaws pointed out here only refer to strong witness indistinguishability and not to (regular) witness indistinguishability. That is, as stated in Lemma 4.6.6, (regular) witness indistinguishability is closed under parallel composition and thus the part of Theorem 4.6.8 that refers to regular witness indistinguishability is valid (i.e., providing constant-round public-coin proofs with negligible error that are witness indistinguishable for NP).
Remark 4.10.6 as well as Theorems 4.10.10, 4.10.14 and 4.10.16, seem to require trapdoor permutations in which the permutation's domain coincides with the set of all strings of certain length. This special case of Definition 2.4.5 can be implemented by modifying the RSA or the Factoring Trapdoors (cf. Canetti et al (in STOC'96) and [5]). (For further detail on Remark 4.10.6, the reader is referred to [23].)
Added comment (Nov. 2008): Unfortunately, the correction that appeared in Appendix C of Volume 2 is also incorrect. For details see note (or better its revision).
For a discussion of enhanced trapdoor permutations see my paper with Ron Rothblum titled Enhancements of Trapdoor Permutations.
Not exactly an error, yet a recent result by Boaz Barak (see FOCS'01) calls for re-evaluation of the significance of all negative results regarding black-box zero-knowledge (cf. Definition 4.5.10). In particular, relying on standard intractability assumptions, Barak presents round-efficient public-coin zero-knowledge arguments for NP (using non-black-box simulators), whereas only BPP can have such black-box zero-knowledge arguments (cf. comment following Theorem 4.5.11). Interestingly, Barak's simulator works in strict (rather than expected) probabilistic polynomial-time, addressing an open problem mentioned in Section 4.12.3.
In Definition 4.10.15 (adaptive NIZK), the adaptive zero-knowledge condition (page 310) should relate only to input-selecting functions $\Xi$ that are implementable by a family of polynomial-size circuits.
Const. 4.10.7 and Prop. 4.10.9 (pp. 303-304): The current description in terms of two mappings $\pi_1,\pi_2$ is confusing and even inaccurate. Instead one should identify the row and columns of $H$ with $[n]$ and use one permuation/isomorphism $\pi$ of $G$ to $H$. Alternatively, one may compose this $\pi$ with the $\psi_i$'s (which results in $\pi_i$'s as in the text). [Luis von Ahn]
Definition 4.9.3 is carelessly written. One should partition the commit phase into two sub-phases, with the second sub-phase being a proof-of-knowledge referring to the view of the first sub-phase, which in turn should constitute a commitment scheme by itself. Alternatively, a Non-Oblivious Commitment may be defined as the sequential composition of an ordinary commitment scheme with a corresponding proof-of-knowledge such that the composed protocol is also a commitment scheme. [Yehuda Lindell]
The proof of Theorem 4.9.4 is wrong, since it relies on the existence of constant-round strong witness indistinguishability proofs of knowledge for NP (i.e., Theorem 4.6.8). Fortunately, an alternative proof exists (see Apdx C.3.3 in Volume 2).
Addition to Section 4.12.2: In continuation to Sections 4.7 and 4.9.2, we mention that the round-efficient argument system of [77] is actually an ``argument of knowledge'' (with negligible error).
Adding self-credits to Section 4.12.1: the notions of strong witness indistinguishability (Section 4.6) and strong proofs of knowledge (Section 4.7.6), and the Hidden Bit Model (Section 4.10.2) are due to (early versions of) this work.

The foregoing errors were corrected in Volume 2. More recent corrections and additions follow

The specific expression given for Chernoff Bound (in Sec 1.2.2, page 11) may be wrong. The exponent should be either $-(\epsilon^2/(4p(1-p)))\cdot n$ (i.e., a factor of two smaller than stated) or just $-2\epsilon^2\cdot n$. (Both alternatives are correct, but one usually sees the latter.) [Kivanc Mihcak]
Typo on page 35, 2nd paragraph: It should be $\nu(n) > 1/p(n)$. [Eldad Zinger]
There is a typo on page 46 (just before equation 2.7): N0 should be N'.
Formally speaking, Prop. 2.4.1 (as stated) is trivial. The statement asserts that there exists a poly-time computable $f$ such that $f$ is a OWF iff OWF exist. But, as pointed out by Claus Diem, if OWF exist then one may use $f$ as any of them, and otherwiose one may use any poly-time computable $f$. Of course, what I meant was that one can specify a poly-time computable $f$ without knowing if OWF exist or not, but this is not captured by the formal statement. Of course, one way out is to describe $f$ first, and then make the claim about this $f$; but, for obvious reasons, I did not want to do that.
In continuation to Sec 2.4.2 and 2.4.4, see clarifications regarding one-way permutations, 2005.
The lower bound (on the size of $S_n$) in Claim 2.5.2.1 can be improved to $\epsilon(n)\cdot 2^n$. See details HERE. [Noam Livne]
Section 2.5.3 does not provide the best analysis of the security of hard-core functions. A better analysis is provided HERE.
At the end of Section 2.7.3, the question should refer to returning a constant fraction of the bits of $x$ (rather than the first half of $x$).
The presentation in Section 2.6.2 contains a (fix-able) error. The graph $G_{f,n}$ defined on page 82 is a directed graph, and the construction of the function $F$ (Construction 2.6.3) refers to directed walks on it. But Lemma 2.6.5 refers to undirected graphs and hence cannot be applied to it. [Qifu Hu]
Nevertheless, the weaker bound provided in the proof of Lemma 2.6.5 holds also in this case. Specifically, consider the stochastic process captured by the matrix $M_f$ such that $M_f =MR$, where $M$ is the normalized adjacency matrix of the graph $G_n$ and $R$ is the permutation matrix that corresponds to the mapping $x\mapsto f(x)$. Now we want to bound $\|(PMR)^t z\|$ and we do it by bounding $\|PMR z\|$, which reduces to bounding $\|PMz'\|$ for $z'=Rz$ (while noting that $Rz$ and $z$ have the same norm). Hence, we obtain a bound of $1-(1-0.5\mu)^{t/2}$, rather than $1-(1-0.5\mu)^{t}$, and this is the bound we should use for $\beta(n)$ (with $t=k(n)/\ell$) when we invoke Lemma 2.6.6. This means that we establish the corresponding bound of with $\beta(n)=1-(1-0.5\alpha(n))^{k(n)/2\ell}$ also in Prop. 2.6.4 (i.e., we lose a factor of two in the exponent).
Note, however, that the above analysis presumes that $\mu \geq 2\rho^2$, where $\rho$ is an upper bound on the eigenvalue ratio. This is fine when applying Lemma 2.6.6 with a constant value of $\alpha$, but for smaller values we have to use an expander of degree $d=\poly(1/\alpha)$ and a polynomially related expansion bound (where such an expander can be obtained by talking a walk of length $\log d$ on a constant-degree expander). This in turn means that $\ell=O(\log(1/\alpha))$ and that we can only use $k(n)=O(n/\ell)$ in Prop 2.6.4 (in order to maintain $n+k(n)\log d = O(n)$). This only means that in the first applications of Prop 2.6.4 (within the proof of Thm 2.6.2), we only increase the hardness parameter by a factor of $n/\log n$ (rather than by $n$). That is, starting with an $n^{-\delta}$-OWP, for $i\in[\delta]$, after $i$ iterations we derive an $(n^{-(\delta-i)}/(\log n)^i)$-OWP, and in the last (i.e., $\delta+1$-st) iteration we obtain a strong one-way function (since $1-(1-(1/\poly\log n))^{n/\log n}$ is negligible).
I'm currently unsure if the original claim (i.e., with Lemma 2.6.5 applied to the directed graph $G_{f,n}$) is true, but it does not matter much.
There is a typo in the guideline given for Exercise 9 of Chapter 2: The suggestion should be $f(y,z)=(g(y),y\xor z)$.
There is a typo in the guideline given for Exercise 17 of Chapter 2: The statistical distance is stated at the end should by $O(2^{-\ell} + 2^{2\ell+2\log n}\cdot\epsilon)$.
Typo in page 99 line 3 (Exer 28 in Chap 2): the reference should be to [204] not [203].
Clarification for Exer 28 of Chap 2:
- The notation $\ell(n)$ is abused; in Item 2 it denotes the codeword length, which is exponential in the length of the description of a position in the codeword (as used in Item 3).
- In Item 3, use a concatanation code in which symbols of the Reed-Solomon code are encoded via the hadamard code.
Prop. 3.3.8 (on page 124) can be extended to all pseudorandom generators $G$ of arbitrary stretch $ell(n)$, yielding a strong OWF if $\ell(n)-n=\omega(\log n))$ and a weak OWF otherwise. This follows by observing that for any subset $S$ of the image of $G$ (or rather the image of $n$-bit long seeds under $G$), it holds that $prob[G(U_n)\in S] \geq \frac{|S|}{2^n}$ whereas $prob[U_{ell(n)}\in S] = \frac{|S|}(2^{ell(n)}} \leq \frac{|S|}(2^{n+1}}$. [Guang Yang]
The sampling procedure referred to at the end of Sec 3.4.2 is not as obvious; the $O(n^3)$ bound relies on the fact that the expected number of candidates to be tried is $O(n)$. [Qifu Hu]
Sec. 3.5.1.2 (on page 138), right after Prop 3.5.3, contains a typo: $k$ should be set to $O(p(n))$, not to $n$. [Qifu Hu]
Typo on page 142 (last sentence of 2nd paragraph): It should be $m$ not $i$. [Qifu Hu]
Typo on page 143 (1st paragraph of the proof of Prop 3.5.9): The 3rd sentence should have been ``Thus, for every $y$ and $\alpha$, it holds that $\pr[|F^{-1}(y,\alpha,H_{n}^{m(n)-l(n)})| >2^{l(n)+1}] < 2^{-l(n)}$.'' and the rest of this paragraph should have referred to $B_y=\{(\alpha,h):|F^{-1}(y,\alpha,h)| \!>\!2^{l(n)+1}\}$. [Qifu Hu]
Sec. 3.5.3 (on page 147) contains an error. The number of copies should be $n^3$, not $n^2$. In all expressions in Sec. 3.5.3, $n^2$ should be replaced by $n^3$, and $n^3$ by $n^4$. [Yu Yu]
Unfortunately, Theorem 3.5.12 (i.e., OWF implies PRGs) still does not have an easy proof (i.e., one suitable for a textbook). But proofs are getting easier and better. Currently, the most reader-friendly proof can be found in Efficiency Improvements in Constructing Pseudorandom Generators from One-way Functions (by Iftach Haitner, Omer Reingold, and Salil Vadhan).
Typo on page 155 at the beginning of the proof of Claim 3.6.6.1. The queries in the simple case should be the reverses of the first $t$ strings in lex-order; that is, for $t=2^{k+1}$, we use strings of the form $x'0^{n-k-1}$, where $x\in\{0,1\}^{k+1}$. [Qifu Hu]
Typo on page 156 line 12: The text should read "In fact, g(p_i)=g(p_j)=v_i=v_j also in case p_i \neq p_j" (i.e., inequality rather than equality). [Qifu Hu]
Typo on Page 163, in the displaced definition of $f$: In the otherwise-case, one should replace $G_{\sigma_{k+1}}$ by $P_{\sigma_{k+1}}$, where $P_0(s)$ (resp., $P_1(s)$) is defined as the first $n$ bits of $s$ (resp., the next $n$ bits of $s$), just as in the proof of Thm 3.6.6. [Qifu Hu]
The definitions in Sec 3.6.1 fail to require that the function $\ell$ is upper bounded by a polynomial. Furthermore, they also fair to require that $\ell$ is efficiently computible (i.e., $\ell(n)$ can be compurted in $\poly(n)$-time when given $n$). [Dima Kogan and Inbal Livni]
Both conditions are not used in the text, but I agree that it is a good idea to impose them both. I tend to insert these conditions into Def 3.6.1 and the preamble of Def 3.6.4. Alternatively, it may be inserted into Def 3.6.3 and the first item of Def 3.6.4.
There is a typo in Exercise 28 of Chapter 3: The threshold value should be $2^ell \cdot ell$ (rather than $\ell^\ell$ as written).
Regarding Footnote 7 in Sec 4.3.1.1 (page 201), see a recent study of Super-Perfect Zero-Knowledge Proofs (2014).
In continuation to the definition of Proofs Of Knowledge (Sec 4.7.1), see discussion of Probabilistic versus Deterministic Provers in the Definition of Proofs Of Knowledge, 2006.
Thm 4.4.11 fails to clarify that this result holds for any NP-witness relation, and not merely for any NP-set [Roei Tell]. Likewise, in Def 4.3.10, it would have been better to define (efficient) zero-knowledge interactive proofs for any relation $R$ rather than refer to the set of strings $P_L(x)$ that satisfy completenss. In such case, the completeness and zero-knowledge conditions should hold for any $(x,y)\in R$ (rather than for every $x\in L$ and $y\in P_L(x)$), for any $R$ such that $L$ contains all $x$'s for which there exists a $y$ such that $(x,y)\in $R$.
Remark 4.10.6 as well as Theorems 4.10.10, 4.10.14 and 4.10.16, seem to require trapdoor permutations in which the permutation's domain coincides with the set of all strings of certain length. This special case of Definition 2.4.5 can be implemented by modifying the RSA or the Factoring Trapdoors (cf. Canetti et al (in STOC'96) and [5]). As stated above, the correction that appeared in Appendix C of Volume 2 is also incorrect. For details see note.
In Construction 4.10.12, it seems better to use the symbol $u'$ (in the definition of $L_2$) instead of $w'$. The point is to streamline better with the proof of Proposition 4.10.13.
A recent work of Nguyen, Ong, and Vadhan (see 47th FOCS, 2006) provides statistical zero-knowledge arguments for NP from any one-way function. This resolves a central open problem in the area (also mentioned in Sec. 4.12.3). See also a follow-up work by Haitner and Reingold (see 39th STOC, 2007) providing statistical hiding commitment schemes.
Exercise 4 of Chapter 4 (p. 324) is wrong. The guideline establishes that this "shared randomness interactive proof" model is at most as powerful as AM, and in fact it coincides with AM, where a set $S$ is in AM if there exists a set $S'$ in NP and a polynomial $p$ such that for every $x\in S$ it holds that $Prob_{r:|r|=p(|x|)}[(x,r)\in S'] \geq 2/3$, and for every $x\in S$ it holds that $Prob_{r:|r|=p(|x|)}[(x,r)\not\in S'] \geq 2/3$. [Ben Diamond]

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