Elgamal digital signature

Elgamal digital signature. She then takes a message m and creates a signature (r,s). The private key is used to sign the ElGamal signature scheme Author: Devon Ritter Created Date: 3/11/2011 5:45:20 PM The approach takes on simple C programming of the ElGamal digital signature to identify some limits that can be encountered and provide considerations for making more complex code. TL;DR: The main reasons to prefer RSA over ElGamal signatures boild down to speed, signature size, standardization, understandability of the method and historical establishment as the result of a lot of lobby work. With ElGamal signing, we create a secret signing exponent (\(s\)) and then a signing exponent of: \(v=g^s \pmod p\) To sign a document (\(D\)), we create an ephereral key (\(e\)). The above program is tested in the 5. Contribute to volkov7/ElGamal-Digital-Signature development by creating an account on GitHub. Next we calculate two signature values: \(S_1 = g^e \pmod p\) \(S_2 = (D-eS_1) e^{-1} \pmod {p-1}\) We then check that these values are the same: In the Elgamal Digital Signature method, an attacker can easily calculate the signer's private key if the same ephemeral key is used to sign two separate messages. We'll begin with a scheme that derives from the Elgamal encrpytion system. The program also generates two intermediary files – cipher1. 9. Link to RSA Algorithm:https://www. 2 Elgamal digital signatures. Digital Signature Standard (DSS) is a Federal Information Processing Standard (FIPS) which defines algorithms that are used to generate digital signatures with the help of Secure Hash Algorithm (SHA) for the authentication of electronic documents. aes digital-signature hill-cipher elgamal vigenere-cipher rsa-encryption vernam-cipher hmac-sha1 diffie-hellman-algorithm man-in-the-middle-attack euclidean-algorithm playfair-cipher chinese-remainder-theorem des-algorithm diffie-hellman-key elliptic-curve-cryptography ceaser-cipher columnar-transposition-cipher railfence-cipher statistical-attack Nov 15, 2014 · 1 Answer. Oct 11, 2020 · RSA digital signatures use mathematical techniques to guarantee data (messages or documents) integrity. Jan 5, 2024 · A digital signature is a mathematical scheme that is used to verify the integrity and authenticity of digital messages and documents. Unlock. Elliptic Curve Digital Signature Algorithm 11 3. The IDEA algorithm is used to strengthen signatures so that others don't modify them. The signer uses his secret key (called the signature generation key) to generate a signature for the given message, and the verifier Jan 1, 2005 · Examples of mechanisms providing digital signatures with appendix are the ElGamal [9], DSA [27] and Schnorr [11] signature schemes. Answer. May 3, 2023 · Following is the implementation of the ElGamal encryption algorithm in C. Step Order: Choose 2 prime numbers for p and q. Calculate n value: p * q = n. In this paper, we use the elliptic curves method to create a new signature scheme. elgamal digital signature scheme Before examining the NIST Digital Signature standard, it will be helpful to under- stand the ElGamal and Schnorr signature schemes. (a) Sign the message digest M = 3 using your private key and random parameter k = 5. The ElGamal signature scheme provides a cryptographic approach to View the full answer. Her public key is then (y,g,p) and where y is: y=g^a (modp) Digital signature depends upon the message and some information unique to the signer to prevent forgery and denial. To more secure, he had claimed that If any k is used twice in the signing, then the system of equations is uniquely determined and x can be recovered. ECC was the most recently-developed encryption method of the three, with Elliptic Curve Digital Signature Algorithm (ECDSA) becoming accredited in 1999, and Key Agreement and Key Transport Using Elliptic Curve Cryptography following in 2001. • The algorithm uses a key pair consisting of a public key and a private key. The algorithm creates two digital signatures, these two signatures, are used in the verification phase. The security disadvantage of the original ElGamal algorithm is that, it has only one random number. Alice creates her digital signature using S=M^d mod n where M is the message. Calculate Y A = a XA mod q. Additionally, the Digital Signature Algorithm (DSA) variant, in view of the ElGamal algorithm (called the ElGamal signature scheme), is used to sign digital documents. This scheme assure that authenticity of message m sent by sender/signer to verifier. This difference in performances is enough to explain not choosing ElGamal. , 2012). DSA produces a 320 bit signature. May 7, 2023 · RSA : It is the most popular asymmetric cryptographic algorithm. It is computationally very complex to compute S 1 and S 2. The research component of this project provides a list of Feb 15, 2000 · Consider the digital signature schemes based on the well-known public key systems, such as Diffie–Hellman system (Diffie and Hellman, 1976), RSA system (Rivest et al. Overview: elgamal is a python module that lets you encrypt and decrypt text using the ElGamal Cryptosystem. Aug 3, 2023 · The Elgamal digital signature scheme is an asymmetric cryptographic algorithm that provides a means to ensure the authenticity and integrity of digital messages. also known as public key cryptography. The approach takes on simple C programming of the ElGamal digital signature to identify some limits that can be encountered and provide considerations for making more complex code. 10. As far as I know, it Oct 6, 2023 · The Elgamal algorithm is used to encrypt and decrypt signatures. Verifying the authenticity of an Elgamal digital signature involves several key steps that ensure the integrity of the message. With ElGamal signing, Alice has a public key (y,g,p) and a private key of a. Attacks on ECC and Pollard’s rho algorithm 12 3. Reload to refresh your session. Suppose Bob chooses p = 29, chooses α = 2, chooses d = 12, and KE = 5Then, what's public key of Bob? Elgamal Digital Signature solved ExamplePrerequisite:Number Theory:Modular ArithmeticModulus of a negative NumberHow to find out inverse of a given number Created by: @sqeel404. The ElGamal cryptosystem includes three major processes: the key generation, the encryption, and the decryption. Hashed value of original data is encrypted using sender’s private key to generate the digital signature. In this answer, we will discuss these steps in detail and explain how the verification process works. The Digital Signature Algorithm (DSA), proposed by NIST (the National Institute of Standards and Technology) in 1991 and published as a DSS (Digital Signature Standard) in 1994, is a modified version of the Schnorr signature and the ElGamal signature which allows shorter signature compared to the ElGamal signature. In the basic formula for the RSA cryptosystem [ 17 ], a digital signature s is computed on a message m according to the equation ( Modular Arithmetic) $$ s \equiv {m}^ {d} \mathbin {\rm mod}\,\,\ n $$. The second class has the feature that the message signed can be Jan 5, 2021 · It provides equivalent levels of cryptographic strength as RSA and DSA, with shorter key lengths. In this work, we present a new alternative of the elliptic curve ElGamal digital signature scheme. ELGAMAL DIGITAL SIGNATURE SCHEME In the signing process , two functions create two signatures ; in the verifying process the output of two functions are compared for verification . You signed out in another tab or window. Digital signatures are used to detect unauthorized modifications to data and to authenticate the identity of the signatory. Sign message m and create signature using El-gamal digital signature scheme: This scheme used the same keys but a different algorithm. Let us understand how RSA can be used for performing digital signatures step-by-step. Dec 8, 2007 · In 1985, Elgamal proposed a digital signature scheme based on discrete logarithms. This Standard specifies a suite of algorithms that can be used to generate a digital signature. Bob computes M1=S^e mod n. To understand how Jan 4, 2017 · As an electronic analogue of a written signature, a digital signature provides assurance that: the claimed signatory signed the information, and the information was not modified after signature generation. National Institute of Standards and Jan 1, 2014 · An Elgamal-style digital signature scheme was put forward in 1985, and a modified version called the Digital Signature Algorithm (DSA), which allows shorter signatures, was proposed in 1991 and officially published as a national Digital Signature Standard (DSS) in 1994; see . It was described by Taher Elgamal in 1985. Process / Steps. It has then been studied in a more general framework, called Meta-ElGamal Signature Schemes. e. Federal Information Processing Standard (FIPS) 186-4, Digital Signature Standard (DSS), specifies three NIST-approved digital signature algorithms: DSA, RSA, and ECDSA. Recall from Chapter 10, that the ElGamal encryption scheme is designed to enable encryption by a user’s public key with decryption by the user’s private key. Host and manage packages Security. Intended Use: This program was created as an exercise in cryptography in one of my classes at the University of Kentucky. I am trying to implement the El Gamal digital signature scheme, using the BigInteger class for generating large prime numbers. With ElGamal, Alice selects a generator (g), prime number of p and a private key of a. ElGamal Public Key Cryptography CS 303 Alg. Additionally, there is a literature review of the ElGamal digital signature and the brute force attack. You are given Bob's private key Kp's (d)- (67) and the corresponding public key Kpub (p, α, β) = (97, 23, 15) a. The key principles are described below. crypto-textbook. com/playlist?list=PLfd3S8S4nJobN_DNECuLB7TTGfELpNbfOالســـــــــــــلام Aug 1, 2023 · ElGamal and Signatures. 🔗. • The private key is used to generate a digital signature for a message, and such a signature can be verified by using the signer's corresponding public key. Elgamal Encryption Calculator, some basic calculation examples on the process to encrypt and then decrypt using the elgamal cryption technique as well as an example of elgamal exponention encryption/decryption. The problem of ElGamal digital signature scheme is that, the message recovery is not provided and its security is constantly being challenged. We implement the classical and modified ElGamal digital signature scheme to compare and to test their functionality, reliability and security. The ElGamal signature scheme is a digital signature scheme which is based on the difficulty of computing discrete logarithms. It is a signature with an appendix that requires a hash function having h: {0, 1}* → Zp *. In cryptography, a proof of knowledge is an interactive proof in which the prover succeeds in ‘convincing’ a verifier that the prover knows something ‘X’. Jan 19, 2024 · A digital signature secures the integrity of a digital document in a similar way as a fingerprint or attachment. Introduction Until the 1970’s, the encryption process was rather complicated and time-consuming You signed in with another tab or window. Aug 3, 2023 · The Elgamal digital signature scheme is a cryptographic algorithm that provides a mechanism for verifying the authenticity and integrity of digital messages. The Schnorr signature scheme is derived from Schnorr’s identification protocol using the Fiat–Shamir heuristic . If M1=M then Bob accepts the data sent by Alice. For more math, subscribe to my channel: https://www. Mar 13, 2023 · RSA Digital Signature Scheme: In RSA, d is private; e and n are public. The digital signatures use asymmetric cryptography i. Digital certificate is a file that ensures holder’s identity and provides security. Mar 28, 2021 · Many variants of the ElGamal digital signature scheme (ElGamal, 1985) have been proposed including the Digital Signature Algorithm (DSA) (FIPS 186-4, 2013), Schnorr signature scheme (Schnorr, 1991), Nyberg-Rueppel signature scheme (Nyberg and Rueppel, 1996), and Edwards-curve Digital Signature Algorithm (EdDSA) (Bernstein et al. Section 5 is devoted to our new digital signature method. May 4, 2011 · Since the traditional ElGamal Digital Signature algorithm used only one random number and therefore had certain security flaws, in order to improve the security of the algorithm, Li et al. The message is part of the input to function 2 when signing; it is part of the input to function 1 when verifying. Taher ElGamal, "A Public-Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms", IEEE Transactions on Information Theory, v. The algorithm uses a key pair consisting of a public key and a private key. Calculate φ (n) value: (p - 1) * (q - 1) = φ (n) Generate e using φ (n) value. It selects a large prime number q and derives its primitive root as p where 1 < p < q − 1. Theory. Number Theory & Cryptography Jeremy Johnson . The signature is verified by recovering the message m with the signer’s RSA Sep 14, 2014 · Presentation Transcript. A known variant [7] of this protocol is described in section 3. As in DSS, the system works in a subgroup of the group \({\mathbb{Z}}_{p}^{{_\ast}}\) for some prime number p. The other revalue b^c will be equal to b^ap. Algoritme Elgamal merupakan salah satu algoritma kriptografi kunci publik yang dibuat oleh Taher ElGamal pada tahun 1984. Generate a random number X A, such that 1 < X A < q-1. Jan 30, 2014 · For slides, a problem set and more on learning cryptography, visit www. In section 2, we review the basic ElGamal digital signature algorithm. Sep 13, 2020 · Follow my blog: https://www. g. It is based on the mathematical problem of computing discrete logarithms in finite fields, which is believed to be computationally hard. txt, which contains the plain text, and generates an output file, results. We start with the Elgamal scheme, which is easier to understand, and Digital signature depends upon the message and some information unique to the signer to prevent forgery and denial. txt, which contains our decrypted text. But unfortunately, there are many misconceptions about these algorithms, and about Asymmetric Cryptography itself. Public keys generated to send out: (n, e) Generate private key d value: de = 1 mod φ (n) : which can also be calculated as: e-1 mod φ (n) = d. Apr 26, 2016 · 7. Algoritme ini pada umumnya digunakan untuk digital signature, tetapi kemudian dimodifikasi sehingga juga bisa digunakan untuk enkripsi dan dekripsi. All three are used to 3. The resulting signatures The world of modern cryptography is built upon the concept of Asymmetric Encryption, and the pillars of Asymmetric Encryption are these three algorithms: RSA, Diffie-Hellman, and DSA (Digital Signature Algorithm). Digital Signature Standard (DSS), Federal Information Processing Standards Publication 186-2. “A public key cryptosystem and a signature scheme based on discrete logarithms. The resulting digital signature scheme is related to the Digital Signature Standard (DSS). Jun 21, 2019 · In 1984, ElGamal [ 7] proposed a new digital signature scheme based on the discrete logarithm problem. Rajinder Singh et al. , the RSA digital signing scheme, the ElGamal digital signing scheme, and many others based on public key cryptosystems. Oct 31, 2017 · El Gamal digital signature construction inexplicably failing. k′ k ′ and m′ m ′ are chosen at random. a) Sign the message digest M = 3 using your private key and random parameter k = 5 (b) If Eve gives you a signed document [M = 9, a = 2, b = 9] and she Question: Bob wants to send a message to Alice, called x. As with the ElGamal digital signature scheme, the Schnorr signature scheme is based on discrete logarithms . Anyone should be able to verify. com/jeffsuzuki1 Dec 25, 2021 · Abroad Education Channel :https://www. (1985). htmlElgamal Scheme | Schnorr Scheme | Which one is better Elgamal or Schnorr Di The digital signature is an electronic signature to sign a document, mail, messages, etc. txt. Like the ElGamal scheme DSA is a digital signature scheme with an appendix meaning that the message cannot be easily recovered from the signature itself. One function is used both for signing and verifying but the function uses different inputs . ElGamal Digital signature scheme realisation. It is widely used to verify a digital message, financial documents, identity cards, etc. The digitally signed algorithm scheme is one of the most well-known digital signature systems, e. 8. Future of ECC 13 Acknowledgments 13 4. Assume that there is a sender (A) and a receiver (B). ” IEEE Transactions on Information Theory, 31, 469–472. In summary Jun 5, 2016 · 1 Answer. May 13, 2023 · The ElGamal Digital Signature Scheme uses a fixed-size hash function to produce a message digest, which is then used to generate and verify digital signatures. Jan 17, 2018 · The Digital Signature Algorithm (DSA) In 1991 the National Institute of Standards and Technology proposed the Digital Signature Algorithm as a standardized general use secure signature scheme. ElGamal encryption can be defined over any cyclic group, like multiplicative group of integers modulo n if and only if n is 1, 2, 4, p k or 2p k, where p is an odd prime and k > 0. The public key remains ( e1,e2, p e 1, e 2, p) and the private key continues Sep 13, 2020 · This lecture is on Elgamal Digital Signature Algorithm Step 1. Find below the steps of Elgamal scheme. The Digital Signature Algorithm (DSA) is a variant of the ElGamal signature scheme, which should not be confused with ElGamal encryption. Note that this scheme is not used in practice because the key sizes are enormous. The paper is organised as follows. Modified 10 months ago. First, you have to generate your key-pair using the following steps: Choose a prime number q and a, such a is a primitive root of q. IT-31, n. Problem: The output always says that Elgamal is a public key scheme similar to the Diffie-Hellman (DH) cryptosystem. It is the same as a handwritten signature, seal, or stamp. Examples of well known digital signature schemes are: DSA, ECDSA, EdDSA, RSA signatures, ElGamal signatures and Schnorr signatures. Aug 3, 2023 · The Elgamal digital signature scheme is a widely used cryptographic algorithm that provides authentication and integrity for digital messages. 3. [38 Mar 21, 2023 · Similarly, digital signature is a way of authenticating a digital data coming from a trusted source. Until now, this scheme is still secure under its cryptographic assumption (discrete logarithm problem). Signature validation key principles. MATH MathSciNet Google Scholar FIPS 186-2 (2000). The main work for signature generation does not depend on the message and can be done during the idle time of the processor. In this context, correctness refers to the ability of the scheme to correctly verify the authenticity and integrity of a message. The ElGamal signature scheme is a digital signature scheme based on the algebraic properties of modular exponentiation, together with the discrete logarithm problem. Viewed 244 times 2 $\begingroup$ Use the ElGamal Digital Nov 14, 2019 · #lakshmichandhanaCryptography and network security Elgamal Digital Signature Scheme. Assuming you don't use counter-measures against this kind of an attack, a chosen-ciphertext attack works as follows: Variables: p p is field prime, α α is the chosen generator, a a is the private key, αa = β α a = β is the public key. This time, it shouldbe signed with the Elgamal digital signature scheme. Designing a digital signature scheme, using Elgamal. The value generates will be Pm=b^p. [1] The ElGamal signature algorithm is rarely used in practice. The above mentioned signature schemes are based on the difficulty of the DLP (discrete logarithm problem) and ECDLP (elliptic-curve discrete logarithm problem) and are quantum-breakable (powerful enough quantum Consider ElGamal digital signature scheme with the following parameters: prime p = 19, generator g = 2, your private key is x = 6, and Alice's public key is (p = 19, g = 2, y = 9). The main technical advantage of RSA is speed. The key generation process is the same as that of EI-gamal algorithms. It is one of the protocols used to implement “Proof Of Knowledge”. Signature creation key principles. This video gives an introduction to the ElGamal digital signature. Note: all the following equations are (mod ( m Aug 3, 2023 · The proof of correctness for the Elgamal digital signature scheme provides assurance of the verification process by demonstrating that the scheme satisfies the desired properties of a secure digital signature scheme. In section 4, we recall the standard definition of the group law defined on elliptic curves. ElGamal digunakan dalam perangkat lunak sekuriti yang Aug 24, 2006 · ElGamal public-key and digital signature scheme were modified from the domain of natural integers, Z, to the domains of Gaussian integers, Z[i], and polynomials over finite fields, F[x]. How this attack works is described in our textbook (Section 10. The signature process begins with generating TN Shankar in 2016 proposed IDS in Manets using ElGamal digital signature in AODV [13], and 2019 came up with research work on wormhole attack [14]. The description of the signature algorithm and an example are given. These fields allow your document to capture a digital signature, usually recording relevant information (like the date, device, location, etc. (b) If Eve gives you a signed document [M = 9, a = 2, b = 9] and she said that Nov 27, 2018 · ElGamal Cryptography in Hindi - Key Generation, Encryption and Decryption Steps with Solved ExampleComputer Network Security(CNS) Lectures – Internet Securit Jul 10, 2023 · The value should be picked in a manner so that Inc passes with a in the particular function will generate the outcome 1. The digital signature scheme allows authenticating documents with non-repudiation and data integrity. 4, 1985, pp469–472 or CRYPTO 84, pp10–18, Springer-Verlag. In addition, the recipient of signed data can use a digital signature as evidence in demonstrating to a third party Again, your work must be shown to earn points. Below is the implementation. Know the user2 will generate some other values that will be used to encrypt the message using the public key. We consider the Elgamal signature scheme. You switched accounts on another tab or window. This signature can then be checked by Bob. Jul 19, 2013 · Abstract. , 1978) and ElGamal system (ElGamal, 1985). We present it here as a starting point for the discussion of the DSA. This method gave rise to several variants as [ 5, 14, 20, 23, 24 ]. A variant developed at the NSA and known as the Digital Signature Algorithm is much more widely used. chiragbhalodia. In fact, using elliptic curves in cryptography was initiated by Koblitz [ 11] and Miller Apr 6, 2023 · ElGamal Digital Signature Scheme question. txt and cipher2. com/watch?v=KNN4KkB3JIc&feature DSA and Schnorr are 6 times faster than ElGamal, and produce signatures which are 6 times smaller. This specific variant of ElGamal has been proposed in 1990 by Agnew, Mullin and Vanstone (the article is called "Improved Digital Signature Scheme based on Discrete Exponentiation"; I could not find a freely downloadable version). Ask Question Asked 10 months ago. com/channel/UC9sgREj-cfZipx65BLiHGmwCompany Specific HR Mock Interview : A seasoned professional with over 18 y Apr 25, 2023 · It is a digital signature scheme known for its simplicity, is efficient and generates short signatures. The program expects an input file, plain. ElGamal/Schnorr/DSA signatures use a per-message secret key and are based on exponentiation. Find and fix vulnerabilities Jan 28, 2022 · مادة التشفير Cryptography بشكل مفصلhttps://www. Sep 20, 2022 · Digital signatures are a segment of digital signature technologies that sign documents using keys and encryption algorithms . ) and locking the document to prevent further changes. I later turned it into a module. com/jeffsuzuki1 Sep 4, 2020 · ElGamal signature scheme The ElGamal digital signature scheme was described by Tahir Elgamal in 1985. [15] in the year 2014 elgamal. ElGamal, T. It is primarily used for encrypting message s but can also be used for performing digital signature over a message. Algoritma Elgamal. 7. I do not recommend you use it to protect any sensitive information. The Schnorr scheme minimizes the message-dependent amount of computation required to generate a signature. ElGamal System on Elliptic Curves 11 3. Please complete the signature and verification process. This article does not intent to describe in deep the cryptography operations, algorithms, schemes used but gives some intuitions . 3 Reuse of the Ephemeral Key). Oct 2, 2023 · ElGamal digital signature is a digital signature scheme which is based on public key cryptography, which uses two keys, one is private, and the other is public. Samantha generates the public key, the private key, chooses a message, signs it and then Victor verifies the signature. It may be considered as a digital version of the handwritten signature or stamped seal. Then they improved the scheme according to the existing problems of ElGamal digital signature scheme, and proposed an implicit ElGamal type digital signature scheme with the function of message recovery. We ameliorated the basic signature equation in order Nov 7, 2021 · Elgamal Scheme (Digital Signature Scheme) This scheme is variant of digital signature algorithm. Alice sends Message M and Signature S to Bob. Share. So for the system to be secure, any value of k should ElGamal-Digital-Signature. Bibliography 13 References 13 1. With RSA you only need to do a small-exponent exponentation to verify a ElGamal Signature Algorithm Parameters, Signing, Veri cation, Correctness ElGamal Signature Scheme Domain Parameters: The prime p and the generator g of Z p Keys: The private key is the integer x 2Z p and the public key y is computed as y = gx (mod p) Signing: The User A forms a message m 2Z p, generates a random number r and computes the An example of the Elgamal Digital Signature algorithm. (1) where ( n, d) is the signer’s RSA private key. Once you have a document with signature fields, you can add your virtual signature to it or send it to other users to their virtual signatures. Step 2. Suppose that the attacker Oscar gotten the following two shifts signed by Bob with the ElGamal public-key and digital signature scheme were modified from the domain of natural integers, Z, to the domains of Gaussian integers, Z[i], and polynomials over finite fields, F [x]. Python3. Question: Consider ElGamal digital signature scheme with the following parameters: prime p = 19, generator g = 2, your private key is x = 6, and Alice's public key is (p = 19, g = 2, y = 9). tant digital signature schemes we mention the elliptic curve digital signature algorithm (ECDSA) [10] and the elliptic curve ElGamal digital signature [17] with its variants as [1]. com Jul 12, 2023 · Since the traditional ElGamal Digital Signature algorithm used only one random number and therefore had certain security flaws, in order to improve the security of the algorithm, Li et al. Indeed, the short size of DSA and Schnorr signatures has long been the selling point for these algorithms when compared to RSA. It validates the authenticity, and integrity of a message or document. proposed an improved ElGamal Digital Signature Scheme that was based on the hard-to-compute property of discrete logarithms over a finite field. com/2021/11/elgamal-schnorr-scheme. youtube. ElGamal Digital Signatures • signature variant of ElGamal, related to D-H • so uses exponentiation in a finite (Galois) • with security based difficulty of computing discrete logarithms, as in D-H • use private key for encryption (signing) • uses public key for decryption (verification) • each user (eg. It is based on the concept of public key cryptography, where a pair of keys, namely the private key and the public key, are used for encryption and decryption operations. This scheme is based on computing assumption of large prime number. Calculate the Elgamal signature (r, s) and the corresponding verification for a message from h the folwine mesage 1 n Bob to Alice with the Chen, Shen & Lv [21] introduced the new modified scheme which is the variant of ElGamal and existing attacks are impossible on it. aq lo gh xs qp hq yt nf vm zx