Pdf short exponent diffiehellman problems researchgate. Key exchange and public key cryptosystems sivanagaswathi kallam 29 september 2015 1 introduction the subject of key exchange was one of the rst issues addressed by a cryptographic protocol. We say that a problem a reduces in poly nomial time to another problem b, denoted by a b, if and only if there is an algorithm for a which uses a subroutine for b, and each call to the subroutine for b counts as a single step, and the algorithm for a runs in polynomialtime. Efficiency comparison of various important identitybased. In the field of privacy preserving protocols, private set intersection psi plays an important role. What is the relation between discrete log, computational. Groups where the cdh problem is hard but the ddh problem is easy are called gap diffie hellman gdh groups. Add rogaways recent exposition on the necessity of ethicality for cr dec 15, 2015. We show that it is unlikely that an elliptic curve with the desired properties exists. Discovering the shared secret given g, p, ga mod p and gb mod p would take longer than the lifetime of the universe, using the best known algorithm.
This chapter gives a thorough discussion of the computational diffiehellman problem. Brief comparison of rsa and diffiehellman public key algorithm. If eve can solve the dlp, then she can compute alice and bobs secret exponents a and b from the. This problem is closely related to the usual computational di. It enables one to construct e cien t cryptographic systems with strong securit y prop erties. Technische universit at munc hen june 23, 2011 1 introduction the di e hellman key agreement protocol, is a procedure that allows establishing a shared secret over an insecure connection and was developed by whit eld di e and martin hellman in 1976. The computational diffie hellman cdh assumption is a computational hardness assumption about the diffie hellman problem. The discrete logarithm problem dlp, the computational diffie. Distortion maps are an important tool for solving ddh problems using pairings, and it is known that distortion maps exist for all supersingular elliptic curves. It is known that the weil and tate pairings can be used to solve. Xdh assumption crypto wiki fandom powered by wikia. In most of the cases, psi allows two parties to securely determine the intersection of their private input sets, and no other information. Secure and efficient multiparty private set intersection.
Pdf easy decisiondiffiehellman groups semantic scholar. Security analysis of the strong diffiehellman problem iacr. The decisiondiffiehellman problem ddh is a central computational problem in cryptography. On the complexity of the discrete logarithm and diffiehellman. Easy decision diffie hellman groups volume 7 steven d. The external diffie hellman xdh assumption is a mathematic assumption used in elliptic curve cryptography. For example, they enable encrypting a message, but reversing the encryption is difficult. Pdf in this paper, we study short exponent diffiehellman problems, where. The cdh assumption involves the problem of computing the discrete logarithm in cyclic groups. Bilinear diffie hellman problem bdhp let be a finite cyclic group of order with a generator, and let. The decision di ehellman problem stanford university.
Consider the diffiehellman key exchange protocol 12. The protocols are efficient and provably secure against passive adversaries. The diffie hellman algorithm is being used to establish a shared secret that can be used for secret communications while exchanging data over a public network using the elliptic curve to generate points and get the secret key using the parameters. Explanation of the decision diffie hellman ddh problem. How are the three problems discrete logarithm, computational diffie hellman and decisional diffie hellman related. The basic tools for relating the complexities of various problems are polynomial reductions and transformations. It enables one to construct efficient cryptographic systems with strong security properties. Cheng and uchiyama show that if one is given an elliptic curve, depending on a prime, that is defined over a number field and has certain properties, then one can solve the decision diffie hellman problem ddhp in in polynomial time. For if is given to decide whether now due to bilinear pairing it is easy to solve this problem. It is clear that the dhp is no harder than the dlp. The cdh problem illustrates the attack of an eavesdropper in the diffie hellman key exchange protocol to obtain the exchanged secret key. Hellman abstract two kinds of contemporary developments in cryp communications over an insecure channel order to use cryptogtography are examined. The complexity analysis of our algorithm proves that all ddh problems are easy on the supersingular elliptic curves used in practice.
Both make their public keys, p a mod g and p b mod g, freely known to all. Kryptographische protokolle the decision di ehellman problem. Candidate multilinear maps acm books series by sanjam garg. An encryption scheme based on the diffie hellman problem authors. Modification of diffiehellman algorithm to provide more. New directions in cryptography tel aviv university. Specifically, xdh implies the existence of two distinct groups with the following properties. From my understanding, since the discrete log dl problem is considered hard, then so is cdh. The author has also compared two prominent public key cryptography algorithms 1. The diffiehellman problem dhp is a mathematical problem first proposed by whitfield diffie and martin hellman in the context of cryptography. The scheme is as efficient as elgamal encryption, but has stronger security properties. In this pap er w e surv ey the recen applications of ddh as w ell kno wn results regarding its. An overview of key exchange protocols iosr journals. Electronic voting protocol using identitybased cryptography.
Obstacles to the torsionsubgroup attack on the decision. Separating decision diffiehellman from computational diffie. Implementation of diffiehellman algorithm geeksforgeeks. Introduction to postquantum cryptography and learning.
We survey the recent applications of ddh as well as known results regarding its security. Citeseerx public key exchange using matrices over group. Gallant, the static diffiehellman problem, iacr eprint. The diffie hellman problem is central to modern cryptography, and is crucial to internet security. Jul 11, 2003 in this paper we construct concrete examples of groups where the stronger hypothesis, hardness of the decision diffiehellman problem, no longer holds, while the weaker hypothesis, hardness of computational diffiehellman, is equivalent to the hardness of the discrete logarithm problem and still seems to be a reasonable hypothesis. This was before the innovation of public key cryptography. In human advancement, people around the world attempted to hide data. Also, we are considering variations of the decisional di. In this paper, employing a bloom filter, we propose a multiparty private set intersection cardinality mpsica, where the number of participants in psi is. In this paper we have used rsa algorithm along with diffie hellman to solve the problem. More precisely, we are interested in studying relationship among variations of di. This nested structure of the platform makes computation very efficient for legitimate parties.
Summary intro to postquantum cryptography learning with errors problems lwe, ringlwe, modulelwe, learning with rounding, ntru search, decision with uniform secrets, with short secrets public key encryption from lwe regev lindnerpeikert security of lwe lattice problems gapsvp kems and key agreement from lwe other applications of lwe. First alice and bob agree publicly on a prime modulus and a generator, in this case 17 and 3. The decision diffie hellman problem ddh is a central computational problem in cryptography. The diehellman problem dhp is the problem of computing the value of gab mod.
The design and implementation of datagram tls 2004 modadugu, rescorla. A practical public key cryptosystem provably secure against. These constructions open doors to providing solutions to a number of important open problems. It is known that the weil and tate pairings can be used to solve many ddh problems on elliptic curves. Foundations of computer security university of texas at austin. Distortion maps are an important tool for solving ddh problems using pairings and it is known that distortion maps exist for all supersingular elliptic curves. The twin diffiehellman problem and applications victor shoup. As a side result we show that the decision diffie hellman problem in the group of points on this type of supersingular elliptic curves is efficiently computable, which provides an example of a group where the decision diffie hellman problem is simple, while the diffie hellman and discrete logarithm problems are presumably not. The bdhp is to compute the value of the bilinear pairing whenever and are given.
Our construction works in groups equipped with an efficient bilinear map, or, more generally, an algorithm for the decision diffie hellman problem. It is secure against a passive adversary if the diffie hellman problem is intractable. The decision diffie hellman assumption ddh is a gold mine. We assume throughout the paper that dlp and cdhp are intractable, which means that there does not exist a polynomial time algorithm to. New directions in cryptography invited paper whitfield diffie and martin e. A class of problems where the cdh problem is hard but ddh problem is easy. Lately ive been reading about the diffie hellman keyexchange methods, and specifically about the computational diffie hellman assumption vs. This paper is an effort to solve a serious problem in diffie hellman key exchange, that is, maninmiddle attack. Recent advances on data networks, communications, computers issn.
Brief comparison of rsa and diffiehellman public key algorithm ayan roy department of computer science, st. There are a number of ways for circumventing these technical difficulties. The decisional diffiehellman ddh assumption is a computational hardness assumption about a certain problem involving discrete logarithms in cyclic groups. Specifically im referencing dan bonehs paper on ddh problem. This problem arises again later in the chapter in the context of elgamal encryption. Cliques, which offers complete key agreement services. Im extremely new to crypto, and very much inexperienced. Diffiehellman problem wikipedia republished wiki 2. The diffiehellman algorithm riley lochridge april 11, 2003 overview introduction implementation example applications conclusion introduction discovered by whitfield diffie and martin hellman new directions in cryptography diffiehellman key agreement protocol exponential key agreement allows two users to exchange a secret key requires no prior secrets realtime over an untrusted network. Short signature without random oracles and the sdh. Index termscollaborative work, communication system security, cryptography, decision diffie hellman problem.
The motivation for this problem is that many security systems use oneway functions. The moral character of cryptographic work 2015 rogaway. Distortion maps are an important tool for solving ddh problems using pairings and it is known that distortion maps exist for all supersingular. It is already known that the weil and tate pairings can be used to solve many ddh problems on. The diffie hellman problem dhp is a mathematical problem first proposed by whitfield diffie and martin hellman in the context of cryptography. The multilinear analog of the decision diffie hellman problem appears to be hard in our construction, and this allows for their use in cryptography.
In this paper we survey the recent applications of ddh as well as known results regarding its security. A practical public key cryptosystem provably secure. We present a group key exchange protocol which extends in a natural way the diffie hellman protocol. To verify the feasibility of the modified diffie hellman algorithm, we divided the process of converting a plaintext to ciphertext into three parts, key exchange, encryption and decryption and observed the time taken by each of these parts to execute individually. It is used as the basis to prove the security of many cryptographic protocols, most notably the elgamal and cramershoup cryptosystems. We offer a public key exchange protocol in the spirit of diffie hellman, but we use small matrices over a group ring of a small symmetric group as the platform. The same was done for the original diffie hellman algorithm as well. It is used in protocols such as ipsec and ssh to generate a shared key. The xdh assumption holds that there exist certain subgroups of elliptic curves which have useful properties for cryptography. It is already known that the weil and tate pairings can be used to solve many ddh problems on elliptic curves. The decisional diffie hellman ddh assumption is a computational hardness assumption about a certain problem involving discrete logarithms in cyclic groups.
The security of our scheme depends on a new intractability assumption we call strong diffie hellman sdh, by analogy to the strong rsa assumption with which it shares many properties. It is known that the weil and tate pairings can be used to solve many ddh problems on. As a side result we show that the decision diffie hellman problem in the group of points on this type of supersingular elliptic curves is efficiently computable, which provides an example of a. Cliques is based on multiparty extensions of the wellknown diffie hellman key exchange method. The security of alices and bobs shared key rests on the diculty of the following, potentially easier, problem. Widening applications of teleprocess raphy to insure privacy, however, it currently necessary for the. We say that the group g satisfies the computational diffie hellman assumption cdh if no efficient algorithm can compute. The following computational problem is precisely the problem of determining whether the guess for k is correct. Citeseerx document details isaac councill, lee giles, pradeep teregowda.
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