- g the worst-case hardness of ap-proximating the
- g the worst-case hardness of approxi-mating the length of a shortest nonzero vector in an n-dimensional lattice to within a small poly(n) factor
- Prior cryptosystems with worst-case connections were based either on the shortest vector problem for a special class of lattices (Ajtai and Dwork, STOC 1997; Regev, J. ACM 2004), or on the conjectured hardness of lattice problems for quantum algorithms (Regev, STOC 2005)
- Public-Key Cryptosystems from the Worst-Case Shortest Vector Problem Chris Peikert SRI Georgia Tec
- Public-Key Cryptosystems from the Worst-Case Shortest Vector Problem Chris Peikert Georgia Tech Computer Security & Cryptography Workshop 12 April 2010 1/1
- g the \emph{worst-case} hardness of approximating the length of a shortest nonzero vector in an $n$-dimensional lattice to within a small $\poly(n)$ factor
- g the worst-case hardness of approximating the

- g the *worst-case* hardness of approximating the shortest vector problem on lattices. Prior cryptosystems with worst-case connections (e.g., the Ajtai-Dwork system) were based either on a *special case* of the shortest vector problem, or on the conjectured hardness of lattice problems for *quantum* algorithms
- Public Key Encryption from the Worst-Case Shortest Vector Problem Author =1=Chris Peikertbeamerinnerthemerounded,shadow,shadow,shadow,shadowheight-0.5 -widthheightdepth- widthheightdepthheight Subject: Theoretical Computer Science Created Date: 8/19/2008 4:05:12 P
- g the worst-casehardness of approxi-matingthe lengthof a shortestnonzerovectorin an n-dimensionallattice towithin a smallpoly(n)factor
- Prior cryptosystems with worst-case connections were based either on the shortest vector problem for a special class of lattices (Ajtai and Dwork, STOC 1997; Regev, J. ACM 2004), or on the.
- Keywords: Public-Key Cryptosystems, Lattice Reduction Problems in particular the problem of finding closest vectors in a lattice to a given point (CVP). From this trapdoor function, we then derive a public-key encryption and digital signature methods. These.

* Prior cryptosystems with worst-case connections (e*.g., the Ajtai-Dwork system) were based either on a *special case* of the shortest vector problem, or on the conjectured hardness of lattice. Abstract. We construct public-key cryptosystems that are secure assuming the worst-case hardness of approxi-mating the length of a shortest nonzero vector in an n-dimensional lattice to within a small poly (n) factor. Prior cryptosystems with worst-case connections were based either on the shortest vector problem for a special class of lattices.

Public-key cryptosystems from the worst-case shortest vector problem Public-KeyCryptosystems from Worst-CaseShortest Vector Problem Chris Peikert 2009Abstract We construct public-key cryptosystems secureassuming worst-casehardness ap-proximating minimumdistance n-dimensionallattices withinsmall poly(n) factors Abstract. We construct public-key cryptosystems that are secure assuming the worst-case hardness of approximating the minimum distance on n-dimensional lattices to within small poly (n) factors. Prior cryptosystems with worst-case connections were based either on the shortest vector problem for a special class of lattices (Ajtai and Dwork, STOC.

Figure 1: Efficiency measures and underlying problems for lattice-based cryptosystems with worst-case connections. Expansion is the best known amortized ratio of ciphertext length to plaintext length, for many-bit messages (we omit logn factor improvements that are sometimes possible at the expense of slightly looser approximation factors). The final three rows describe known chosen ciphertext. ** We present a probabilistic public key cryptosystem which is secure unless the worst case of the following lattice problem can be solved in polynomial time: Find the shortest nonzero vector in an n dimensional lattice L where the shortest vector v is unique in the sense that any other vector whose length is at most n' [lull is parallel to v**.

Building on these results, Ajtai and Dwork created a **public-key** **cryptosystem** whose security could be proven using only the **worst** **case** hardness of a certain version of SVP, thus making it the first result to have used **worst-case** hardness to create secure systems. See also. Learning with errors; Short integer solution **problem** Archive, 2008: 481, 2008 ** Public-key cryptosystems from the worst-case shortest vector problem: extended abstract**. Chris Peikert.** Public-key cryptosystems from the worst-case shortest vector problem: extended abstract**. In Michael Mitzenmacher, editor, Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009

- We describe a public-key cryptosystem with worst-case/average case equivalence. The cryptosystem has an amortized plaintext to ciphertext expansion of O(n), relies on the hardness of the O˜(n 2 )-unique shortest vector problem for lattices, and requires
- Public-Key Cryptosystems from the Worst-Case Shortest Vector Problem. Public-Key Cryptosystems from the Worst-Case Shortest Vector Problem
- Public-Key Cryptosystems from the Worst-Case Shortest Vector Problem Chris Peikert March 19, 200
- Shortest vector problem (SVP) [edit | edit source] In SVP, a basis of a vector space V and a norm N (often L 2) are given for a lattice L and one must find the shortest non-zero vector in V, as measured by N, in L.In other words, the algorithm should output a non-zero vector v such that. In the -approximation version , one must find a non-zero lattice vector of length at most

- the lattice is reduced by a factor of q. After nsuch iterations we can solve the problem by using, e.g., Babai's nearest plane algorithm. References [Pei09] Chris Peikert. Public-key cryptosystems from the worst-case shortest vector problem: extended abstract. In 41st Annual ACM Symposium on Theory of Computing, STOC 2009, pages 333{342. ACM.
- Learning with errors (LWE) is a problem in machine learning.A generalization of the parity learning problem, it has recently been used to create public-key cryptosystems based on worst-case hardness of some lattice problems.The problem was introduced by Oded Regev in 2005.. Given access to samples where and , with assurance that, for some fixed linear function with high probability and.
- Abstract. We prove the equivalence, up to a small polynomial approximation factor \(\sqrt{n/\log n}\), of the lattice problems uSVP (unique Shortest Vector Problem), BDD (Bounded Distance Decoding) and GapSVP (the decision version of the Shortest Vector Problem). This resolves a long-standing open problem about the relationship between uSVP and the more standard GapSVP, as well the BDD problem.
- Prior cryptosystems with worst-case connections were based either on the shortest vector problem for a \emph{special class} of lattices (Ajtai and Dwork, STOC 1997; Regev, J.~ACM 2004), or on the conjectured hardness of lattice problems for \emph{quantum} algorithms (Regev, STOC 2005)
- g the worst-case hardness of approximating the
- Public-Key Cryptosystems from the Worst-Case Shortest Vector Problem [Slides, Video] Chris Peikert. In STOC 2009. Awarded Best Paper. Generating Shorter Bases for Hard Random Lattices Joel Alwen, Chris Peikert. Theory of Computing Systems, 48(3):535-553, April 2011. By invitation to special issue on STACS 2009

[Pei09] Chris Peikert. Public-key cryptosystems from the worst-case shortest vector problem: extended abstract. In 41st Annual ACM Symposium on Theory of Computing, STOC 2009, pages 333{342. ACM, 2009. [Reg09] Oded Regev. On lattices, learning with errors, random linear codes, and cryptography. JACM, 56(6), 2009. A public-key cryptosystem was later constructed around this principle [2], and while practically ine -cient it served to motivate further research. Perhaps because the relative di culty of the unique shortest vector problem is poorly understood with respect to other lattice problems, the two most well-establishe from some worst-case lattice problems (the shortest independent vectors problem, the short-est vector problem with a gap) to LWE. With a strong security guarantee, LWE makes versatile cryptographic constructions possible including fully homomorphic encryption, multi-linear map, etcetera. For more details, we refer to the recent survey [44]

Large keys aside, error-correcting code cryptosystems have been touted as a potential saviour of public-key cryptography. These systems are based on the syndrome-decoding problem which was shown to be NP-complete in [BMvT78]. Other public-key cryptosystems, such as RSA, are based on problems such as factoring which are only thought to be di cult ** in the worst case, an approximate version of the shortest vector problem**. The latter can be seen as an NP promise problem. If the latter problem were NP-complete, then we would have a reduction relating the average-case hardness of an NP distributional problem to the worst-case hardness of an NP-complete problem

Recently Ajtai [1] proved that certain lattice problems related to the shortest lattice vector problem (SVP) have essentially the same average case and worst case complexity, and both are conjectured to be extremely hard. This development raises the possibility of public-key cryptosystems which will have a new level of security * C*. Peikert, Public-key cryptosystems from the worst-case shortest vector problem, in: Proceedings of the 41st annual ACM symposium on theory of computing (STOC 2009), pp. 333-342. Google Scholar [8

Public-Key Cryptosystems from the Worst-Case Shortest Vector Problem. Public-Key Cryptosystems from the Worst-Case Shortest Vector Problem Chris Peikert March 19, 2009. برچسبها: Public, Key Cryptosystem, Vector Problem, Chris Peikert, 200 Peikert, C. 2009. Public-key cryptosystems from the worst-case shortest vector problem. In Proc. 41st ACM Symp. on Theory of Computing (STOC). ACM, New York, 333--342. Google Scholar Digital Library; Peikert, C., Vaikuntanathan, V., and Waters, B. 2008. A framework for efficient and composable oblivious transfer. In Advances in cryptology.

unique shortest vector problem [LO85,Fri86]. While the Ajtai-Dwork and the subsequent lattice-based cryptosystems [Reg03,Reg05,Pei09] are as hard to break as the average-case subset sum problem, these schemes are based on subset sum in a somewhat indirect way, and this causes their connection to the subset sum problem to not be as tight as. * since Ajtai's seminal result [Ajt96a] on a one-way function based on the worst-case hard-ness of lattice problems, which initiated the cryptographic use of lattice problems*. Ajtai and Dwork ﬁrst succeeded to construct public-key cryptosystems [AD97] based on the unique shortest vector problem (uSVP) The shortest vector pr oblem (SVP) asks for such a vector: itis the most famous lattice problem, and is one of the very few potentially hard problems currently in use in public-key cryptography (see [26 , 24 ] for surv eys on lattice-based cryptosystems, and [15 , 27 ] for recent de velopments). SVP is also well-kno wn for its applications in.

Public-key cryptosystems from the worst-case shortest vector problem PEIKERT C. STOC, 2009, 333-342, 200 and lattice-based public key cryptosystems was that the former could be based on the hard-ness of classical GapSVP, whereas the latter could not. But our work, as well as the recent work of Peikert [Pei09], shows that there are cryptosystems based on the worst-case hardness of the shortest vector problem in its decision version **Public-Key** **Cryptosystems** **from** **the** **Worst-Case** **Shortest** **Vector** **Problem**, by Chris Peikert. A Constructive Proof of the Lovász Local Lemma, by Robin Moser. STOC 2008 Optimal Hierarchical Decompositions for Congestion Minimization in Networks, by Harald Räcke

- primitives and lattice-based public key cryptosystems was that the former could be based on the hardness of classical GapSVP, whereas the latter could not. But our work, as well as the recent work of Peikert [32], shows that there are cryptosystems based on the worst-case hardness of the shortest vector prob-lem in its decision version
- Bibliographic details on Public-Key Cryptosystems from the Worst-Case Shortest Vector Problem. (sorry, in German only) Betreiben Sie datenintensive Forschung in der Informatik? dblp ist Teil eines sich formierenden Konsortiums für eine nationalen Forschungsdateninfrastruktur, und wir interessieren uns für Ihre Erfahrungen
- Public-key cryptosystems from the worst-case shortest vector problem C Peikert Proceedings of the forty-first annual ACM symposium on Theory of computing , 200
- as solving worst-case instances of a lattice problem called the \unique shortest vector problem. The reduction of subset sum to breaking their scheme is then obtained via the classic reduction from random subset sum to the unique shortest vector problem [LO85,Fri86]. While the Ajtai-Dwork and the subsequent lattice-based cryptosystems [Reg03.

Ajtai and Dwork then designed a public-key cryptosystem which is provably secure unless the worst case of a version of the SVP can be solved in probabilistic polynomial time. However, their cryptosystem suffers from massive data expansion because it encrypts data bit-by-bit. Here we present a public-key cryptosystem based on similar ideas, but. Since the introduction of the ring learning with errors (R-LWE) by Lyubashevsky, Peikert and Regev, many efficient and secure applications were founded in cryptography. In this paper, we mainly present an efficient public-key encryption scheme based on the R-LWE assumption. It is very simple to describe and analyze. As well as it can achieve security against certain key-dependent message (KDM.

- UNIQUE SHORTEST VECTOR PROBLEM FOR MAX NORM IS NP-HARD Than Quang Khoat and Nguyen Hong Tan Faculty of Information Technology, Th´ai Nguyˆen University, Th´ai Nguyˆen City, Vietnam E-mails: {tqkhoat, nhtan}@ictu.edu.vn The unique Shortest vector problem (uSVP) in lattice theory plays a crucial role in many public-key cryptosystems
- Unique Shortest Vector Problem for max norm is NP-hard Than Quang Khoat and Nguyen Hong Tan Faculty of Information Technology, Thai Nguyen University, Thai Nguyen city, Vietnam ftqkhoat, nhtang@ictu.edu.vn Abstract. The unique Shortest vector problem (uSVP) in lattice theory plays a crucial role in many public-key cryptosystems
- and C. Dwork presented a new probabilistic public key encryption algorithm([5]) for which they could proof that breaking a random instance of the cryptosystem is as hard as solving the worst-case of the ciphers base problem, the unique shortest vector problem. A lot of other systems followed like the GGH syste
- Cryptosystems based on disguised subset-sum problem are known as subset-sum cryptosystems or knapsack cryptosystems. The general idea is to start with a secret superincreasing sequence, disguise it using secret modular linear operations, and publish the disguised sequence as the public key
- Abstract: The knapsack cryptography is the public-key cryptography whose security depends mainly on the hardness of the subset sum problem. Many of knapsack schemes were able to break by low-density attacks, which are attack methods to use the situation that a shortest vector or a closest vector in a lattice corresponds to a solution of the subset sum problem
- Keywords: Public-Key Cryptosystems, Lattice Reduction Problems 1 Introduction in particular theproblem of ﬁnding closest vectors in a lattice to a given point (CVP). From this trapdoor function, we then derive a public-key encryption and digital signature methods

- Orsini E, Smart NP (2015) Bootstrapping BGV ciphertexts with a wider choice of p and q. In: Public-key cryptography-PKC 2015. Springer, Berlin, pp 673-698. Peikert C (2009) Public-key cryptosystems from the worst-case shortest vector problem: extended abstract. In: STOC, pp 333-342. Peikert C (2014) Lattice cryptography for the internet
- on the worst-case hardness of the unique shortest vector problem. The main result is a new public key cryptosystem whose security guarantee is considerably stronger than previous results (O(n1.5) instead of O(n7)). This provides the ﬁrst alternative to Ajtai and Dwork's original 1996 cryptosystem
- v2L;v6=0 since we need some entropy on the keys IAverage case hard )worst case hard, but not other way around in general Public key cryptosystems) (1)) 1).

cal public key cryptosystems (PKCs) are breakable using Shor's algorithm [31] on a quantum computer. Although the subset-sum problem is NP-hard in the worst case, Euclidean-norm shortest nonzero vector of the lattice in poly-nomial time Public-key cryptosystems provably secure against chosen ciphertext attacks. In Proceedings of the Twenty Second Annual ACM Symposium on ⁿeory of Computing, - May , Baltimore, Maryland, USA, pages - Several proposed public-key cryptosystems, called knapsack public-key crypto- systems, are based on this problem [ 12, 15, 181. Such cryptosystems give a set of weights (ai: 1 5 i 15 n) as public information. A plaintext message consisting of a O-l vector (e . . . , e,) is encrypted using (1 Quantum key distribution (QKD) can provide information theoretically secure key exchange even in the era of quantum computers. However, QKD requires the classical channel to be authenticated, the.

Public-key cryptosystems from the worst-case shortest vector problem: extended abstract w开头的单词 ww ii wrought iron wrought up ww i wu chia pee wrong answer Written Representation wrong side written law written examination writing desk writing tabl 1 uniform by simply adding to the b component an element whose first i 1 coordi from EECS 6,892 at Massachusetts Institute of Technolog We show that approximating the shortest vector problem NTRU-Like Public Key Cryptosystems beyond Dedekind Domain up to Alternative Algebra. Efficient Collision-Resistant Hashing from Worst-Case Assumptions on Cyclic Lattices. Theory of Cryptography, 145-166. 2006 A COMPLETE WORST-CASE ANALYSIS OF KANNAN'S SHORTEST LATTICE VECTOR ALGORITHM GUILLAUME HANROT∗ AND DAMIEN STEHL´E† Abstract. Computing a shortest nonzero vector of a given euclidean lattice and computing a closest lattice vector to a given target are pervasive problems in computer science, computational mathematics and communication theory and the underlying mathematical problems are the shortest vector problem (SVP) and closest vector problem of lat-tices[25].Incomparisontowidely-knownsystemssuchas RSA [2] or ECC [3], the main advantage of NTRU is that its time complexity quadratic order O(N2) in worst case. This system is considered as the fastest cryptosystem, an

C. Peikert, Public-key cryptosystems from the worst-case shortest vector problem, in Proceedings of the forty-first annual ACM symposium on Theory of computing, ACM, 2009. View at: Google Schola Chris Peikert, Public-key cryptosystems from the worst-case shortest vector problem: extended abstract, in Proceedings of the 41st annual ACM symposium on Theory of computing (Bethesda, MD, USA: ACM, 2009), 333-34

Specific worst-case lattice problems considered in this paper are the shortest independent vector problem SIVP and the guaranteed distance decoding problem GDD (a variant of the closest vector problem, CVP) for approximation factors n 1+ǫ almost linear in the dimension of the lattice (2015) Solving the Shortest Vector Problem in 2 n Time Using Discrete Gaussian Sampling. Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing - STOC '15 , 733-742. 2015

Public-Key Cryptosystems from the Worst-Case Shortest Vector Problem Chris Peikert One-Round Authenticated Key Agreement from Weak Secrets Yevgeniy Dodis and Daniel Wichs Twice-Ramanujan Sparsifiers Joshua D. Batson, Daniel A. Spielman, and Nikhil Srivastav Shortest Vector Problems Public key encryption problem L is hard in the worst case hard f N All L's 's hard L easy f N 's. The Subset-sum Problem a 1 a 7 a 6 a 4 a 10 a 5 a 8 a 9 a 3 a 2 b=a 2 +a 4 +a 5 +a 6 +a 8 Subset Sum Problem: Given weights A = (a Keywords: public-key cryptosystems, lattice-based cryptosystems, error- correcting codes, Ajtai-Dwork lattice cryptosystem, McEliece. 1 Introduction In an attempt to design cryptosystems based upon (believed) intractible prob- lems different from factoring and discrete logarithms, several public-key cryp- tosystems have been presented, including the two systems in [M78,AD97] Public-key Cryptosystem, GGH, Dedekind Domain, Polynomial Representation. A B S T R A C T GGH class of public-key cryptosystems relies on computational problems based on the closest vector problem (CVP) in lattices for their security. The subject of lattice based cryptography is very active and there have recently been ne 18. Algorithms for the Closest and Shortest Vector Problem 19. Coppersmith's Method and Other Applications 19a. Cryptosystems Based on Lattices (does not appear in published version of book) Part V: Cryptography Related to Discrete Logarithms 20. The Diffie-Hellman Problem and Cryptographic Applications 21. The Diffie-Hellman Problem 22

worst-case hardness of standard problems (such as the shortest vector problem) on ideal lattices [15].3 Gentry's construction is quite involved { the secret key, even in the private-key version of his scheme, is a short basis of a \random ideal lattice. Generating pairs of public and secret bases with the right distributions appropriate for th Security from worst case assumptions: In security proofs for cryptosystems we typically assume that some problem is hard on average or more precisely hard to solve for random instances drawn from some particular distribution. For example, we may assume that factoring a product of two random primes cannot be done by a poly-time algorithm constructed LWE-based cryptosystems with improved e ciency or additional functionality (e.g., [72, 105,104,61,34,31,62,24,64]). In particular, in work published in 2011, Lindner and Peikert [83] gave a more e cient LWE-based public-key encryption scheme that uses a square public matrix A 2Z n q instead of an oblong rectangular one

I Shortest vector problem (SVP): Given a basis matrix B for a lattice L nd a non-zero vector v 2L such that kvkis minimal. The norm here is usually the standard Euclidean norm in Rn, but it can be any norm such as the ' 1 norm or ' 1norm. I Closest vector problem (CVP): Given a basis matrix B for a full rank lattice L Rn and an element t. • Gap Shortest vector problem underlying lattice problem. • LWE: - Worst-Case to Average-Case Quantum Reduction - RSA and discrete-log based cryptosystems: public key size is linear in the security parameter. • To reduce the public key size, consider lattice One of the earliest public key cryptosystems is the knapsack cryptosystem, first described by Ralph Merkle & Martin Hellman in 1978 and the underlying scheme implements the subset sum problem. As stated before, the subset sum problem can be unsolvable, however, there are still instances of the problem that are solvable constructed LWE-based cryptosystems with improved e ciency or additional functionality (e.g., [96, 95,55,32,29,56,22,58]). In particular, in work published in 2011, Lindner and Peikert [75] gave a more e cient LWE-based public-key encryption scheme that uses a square public matrix A 2Z n q instead of an oblong rectangular one Worst-case to average-case proofs exhibit a polynomial time reduction between the worst-case and average-case problems. While this gives great insight into the asymptotic hardness of the average-case problem, concrete secu-rity guarantees depend on the polynomial itself, as well as on the hardness of the worst-case problem instance

Hard Lattice problems are assumed to be one of the most promising problems for generating cryptosystems that are secure in quantum computing. The shortest vector problem (SVP) is one of the most famous lattice problems. In this paper, we present three improvements on GPU-based parallel algorithms for solving SVP using the classical enumeration and pruned enumeration Keywords.Public Key Cryptosystem, Closest Vector Problem, Lattice Reduction, Trapdoor, McEliece 1 Introduction Since the invention of public key cryptography in 1976 by Di e and Hellman [DH76] security of most cryptosystems is based on the (assumed) hardness of factoring or computing discrete logarithms. Only a few schemes based on othe

M. Ajtai and C. Dwork, A public key cryptosystem with worst-case/average case equivalence, Proc. of the twenty ninth annual ACM symposium on theory of com-puting, pp. 284-293, 1997. 2. O. Goldreich, S. Goldwasser and S. Halevi,Public key cryptosystems from lattice re-duction problems, Advances in Cryptology, Pro. of CRYPTO' 97, Springer-Verlag [Pei09]Chris Peikert. Public-key cryptosystems from the worst-case shortest vector problem. In Proceedings of the forty-ﬁrst annual ACM symposium on Theory of computing, pages 333-342. ACM, 2009. [Pei14]Chris Peikert. Lattice cryptography for the internet. In Post-Quantum Cryptography, pages 197-219. Springer, 2014. [Poi00]David Pointcheval times the length of a shortest vector [12] (for any ǫ > 0). SVP is of prime importance in cryptography since a now quite large family of public-key cryp-tosystems rely more or less on it. The Ajtai-Dwork cryptosystem [4] relies on dc-SVP for somec > 0, where f(d)-SVP is the problem of ﬁnding the shortest non-zero vector inthe lattice L, knowin