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# El gamal explained

ElGamal is a public key cryptosystem based on the discrete logarithm problem for a group \( G \), i.e. every person has a key pair \( (sk, pk) \), where \( sk \) is the secret key and \( pk \) is the public key, and given only the public key one has to find the discrete logarithm (solve the discrete logarithm problem) to get the secret key ElGamal encryption is an public-key cryptosystem. It uses asymmetric key encryption for communicating between two parties and encrypting the message. This cryptosystem is based on the difficulty of finding discrete logarithm in a cyclic group that is even if we know g a and g k, it is extremely difficult to compute g ak. Idea of ElGamal cryptosyste

Introduction to ElGamal Encryption. ElGamal cryptosystem can be defined as the cryptography algorithm that uses the public and private key concepts to secure communication between two systems. It can be considered the asymmetric algorithm where the encryption and decryption happen by using public and private keys. In order to encrypt the message, the public key is used by the client, while the message could be decrypted using the private key on the server end. This is considered an efficient. The ElGamal cryptographic algorithm is a public key system like the Diffie-Hellman system. It is mainly used to establish common keys and not to encrypt messages. Introduction. The ElGamal cryptographic algorithm is comparable to the Diffie-Hellman system. Although the inventor, Taher Elgamal, did not apply for a patent on his invention, the owners of the Diffie-Hellman patent (US patent 4,200,770) felt this system was covered by their patent. For no apparent reason everyone calls this the.

### Cryptography Academy - The ElGamal cryptosyste

1. The ElGamal encryption system is a public key encryption algorithm by Taher Elgamal in 1985 that is based on the Diffie-Hellman key exchange. We give an introduction to the ElGamal Encryption System and an example in the video in Figure 16.3.1
2. Cryptography ElGamal The ElGamal algorithm is used as a part of the free GNU Privacy Guard Software, late forms of PGP, and di erent cryptosystems. Additionally, the DigitalSignatureAlgorithm(DSA)arianvt, in view of the ElGamal algorithm (called the ElGamal signature scheme), is used to sign digital documents.Th
3. The security of ElGamal is based on the discrete logarithm problem. To encrypt and respectively decrypt a message, a discrete power is executed. This operation is e cient to compute. An attacker that seeks to decrypt an intercepted message may try to recover the private key. To this end a loga-rithm needs to be computed. No actual method exists for this, given certai
4. Hash Elgamal could also refer to the Fujisaki-Okamoto heuristic applied to Elgamal. This prevents malleability but can also lose the CPA-security of Elgamal. Other Elgamal variants that use a hash function are Cramer-Shoup (mentioned by @jalaj) and DHIES. These are both secure against any malleability (they are CCA-secure) however they aren't typically referred to as hash Elgamal
5. The ElGamal signature scheme is a digital signature scheme which is based on the difficulty of computing discrete logarithms.It was described by Taher Elgamal in 1985.. The ElGamal signature algorithm is rarely used in practice. A variant developed at the NSA and known as the Digital Signature Algorithm is much more widely used. There are several other variants
6. In cryptography, the ElGamal encryption system is an asymmetric key encryption algorithm for public-key cryptography which is based on the Diffie-Hellman key exchange. It was described by Taher Elgamal in 1984. ElGamal encryption is used in the free GNU Privacy Guard software, recent versions of PGP, and other cryptosystems
7. El-Gamal (Simplified) Key generation. Alice has a prime number (p) Special Number (g) and a random number for her private key (a) (p) is the key so needs to be long (1024\2048) (g) must be a primitive element modulo (p) (a) must be bigger than 1 and smaller than p-1; The algorithm is A = g a mod p. Alice's public key is

### ElGamal Encryption Algorithm - GeeksforGeek

ElGamal encryption is one of many encryption schemes which utilizes randomization in the encryption process. Others include McEliece encryption (x8.5), and Goldwas-ser-Micali (x8.7.1), and Blum-Goldwasser (x8.7.2) probabilis-tic encryption. Deterministic encryption schemes such a Visit Our Channel :- https://www.youtube.com/channel/UCxik...Follow Smit Kadvani on :- Facebook :- https://www.facebook.com/smit.kadvaniInstagram :- https://..

### ElGamal Encryption Simple Steps How EIGamal Encryption

- A brief introduction to Elgamel Encryption- Explaining Diffie-Hellman Key Exchange- no details calculation 2 responses to ElGamal Encryption Algorithm in Python Sherry Lee says: March 16, 2021 at 2:45 am. How do I implement and verify the homomorphic property of ElGamal algorithm? I mean how do I allow users to input m1 and m2 from keyboard and verify that E(m1m2)=E(m1)E(m2) by adding some codes to above program? Thank you! Reply. Vybz says: March 31, 2021 at 5:53 am. Could you explain the. The ElGamal signature scheme is known as a signature with appendix: the message is not readily recovered from the signature pair (r, s) and the message m must be included in the verification procedure. This is in contrast to a message recovery scheme wherein the message is easily recoverable from the signature ### The ElGamal public key system (in Technology > Encryption

Is there a simple implementation (using Java BigInteger) of ElGamal elliptic curve cryptosystem with key generation, encryption and decryption functions; one that could be used to explain to university students in a lecture? For example, a Paillier encrypt function can be coded, without loss of generality, as In this paper, a modi ed ELGAMAL key agreement protocol is proposed to enhance the security of the data by adding an increased step of mod inverse of message M with key K. For the illustration let us consider an Elliptic curve E11(1,6) that originates the equation (1) To nd the x and y coordinates of elliptic curve, replace the values of x from 0 to 10 on to equation (1) to extract y2 and. ElGamal ciphertexts are typically at least as many bits as the prime modulus p. If the plaintext size is small comparatively to the size of p(as we can see in many practical scenarios), the relative size overhead becomes worse. For example, assuming pis a 2048-bit prime, in a hybrid encryption scenario where a sym-metric key size is 256-bit, the size overhead of ElGamal is roughly ten times.

The ElGamal signature scheme [] is one of the first digital signature scheme based on an arithmetic modulo a prime (see smash modular arithmetic).It can be viewed as an ancestor of the Digital Signature Standard and Schnorr signature scheme. ElGamal signatures are much longer than DSS and Schnorr signatures. As a result, this signature scheme is not used often and is mostly of interest for. When it comes to RSA, it's super-easy to understand. You take 5-15 minutes with a developer and you can explain him how they work and he can imagine why they are secure. Signature and verification are straight-forward operations and there isn't an awful lot to be done wrong. ElGamal signatures on the other hand are more difficult to understand

The ElGamal signature scheme is a digital signature scheme which is based on the difficulty of computing discrete logarithms.It was described by Taher ElGamal in 1984.. The ElGamal signature algorithm described in this article is rarely used in practice. A variant developed at NSA and known as the Digital Signature Algorithm is much more widely used. There are several other variants The ElGamal Algorithm provides an alternative to the RSA for public key encryption. 1) Security of the RSA depends on the (presumed) difficulty of factoring large integers. 2) Security of the ElGamal algorithm depends on the (presumed) difficulty of computing discrete logs in a large prime modulus. ElGamal has the disadvantage that the ciphertext is twice as long as the plaintext. It has the. El-gamal digital signature scheme: This scheme used the same keys but a different algorithm. The algorithm creates two digital signatures, these two signatures, are used in the verification phase. The key generation process is the same as that of EI-gamal algorithms. The public key remains (\$e_1, e_2, p\$) and the private key continues to be d. Signatur ElGamal Analysis. In ElGamal system, each user has a private key x. and has three components of public key − prime modulus p, generator g, and public Y = g x mod p. The strength of the ElGamal is based on the difficulty of discrete logarithm problem. The secure key size is generally > 1024 bits. Today even 2048 bits long key are used. On the processing speed front, Elgamal is quite slow, it is used mainly for key authentication protocols. Due to higher processing efficiency, Elliptic Curve.

### ElGamal Encryption Algorithm - Tutorialspoint

• El-Gamal Signature Scheme There are some protocol failures that would compromise the El-Gamal signature scheme. The first involves the secret exponent k. Should this become known, then given a signature (m,r,s) the congruence ar = m-ks mod (p-1), has d = gcd(r,p-1) possible solutions for a. The correct one can be found by verifying that β = αa mod p. This gives Alice's secret exponent a and.
• Encryption: The Diffie Hellman key exchange algorithm can be used to encrypt; one of the first schemes to do is ElGamal encryption. One modern example of it is called Integrated Encryption Scheme, which provides security against chosen plain text and chosen clipboard attacks. Password Authenticated Agreement: When two parties share a password.
• ElGamal is now free. I won't try to explain the math or demonstrate why it's a secure set of algorithms. The equations themselves are not too hard to understand. Key Pair Generation. Here's the recipe for generating a key pair: Create a random prime number, p. This number is called the modulus. The size of p is the same as the key size, so a 2048-bit key has a p that is 2048 bits. Choose.
• read. There are some encryption methods which allow us to process encrypted values. In this.
• Elgamal encryption using ECC can be described as analog of the Elgamal cryptosystem and uses Elliptic Curve arithmetic over a finite field. In this project, we visualize some very important aspects of ECC for its use in Cryptography. We explore Elgamal encryption using Elliptic curves and understand its challenges to encrypt data. We also present an approach for fast encryption and compare our.

### ElGamal Encryption System - UNC

• Problems in ElGamal Digital Signature Scheme The problem is that p needs to be very large to guarantee that the discrete log problem is intractable in Zp*. The recommendation is a p of atleast 1024 bits. This could make the signature as large as 2048 bits. To reduce the size of the signatutre , schnorr proposed a new scheme based on ElGamal , but with a reduced signature size. 7. FORGERY IN.
• ElGamal encryption is an example of public-key or asymmetric cryptography. The cryptosystem takes its name from its founder the Egyptian cryptographer Taher Elgamal who introduced the system in his 1985 paper entitled A Public Key Cryptosystem and A Signature Scheme Based on Discrete Logarithms . As this title suggests the security of this cryptosystem is based on the notion of discrete logari
• Mahmoud A. El-Gamal explained that he traded two volumes of lower quality dates for one volume of higher quality. The Messenger of God (pbuh) said: this is precisely the forbidden Rib¯a! Do not do this. Instead, sell the ﬁrst type of dates, and use the proceeds to buy the other. The process of selling one type of dates in the market only to use the proceeds to buy the other type.
• • Examples: RSA, El Gamal, elliptic curve systems, Feistel networks uMany block algorithms are Feistel networks • Examples - DES, Lucifer, FREAL, Khufu, Khafre, LOKI, GOST, CAST, Blowfish, • Feistel network is a standard form for - Iterating a function f on parts of a message - Producing invertible transformation uAES (Rijndael) is related • also a block cipher with.
• I don't explain what is Elgamal or how it works, I don't explain proofs based on games and what semantic security is (and why it is considered insufficient), I don't even explain what I mean by an adversary's advantage. I'm expecting that you will read that on your own and then head here to understand how you can use all of that to prove Elgamal's semantic security
• Andreas V. Meier - The ElGamal Cryptosystem - p.6/23. Public Key Cryptography - Summary Features able to set up a secure channel between two parties based on the Discrete Logarithm Problem Problems vulnerable to the man-in-the-middle attack vulnerable to chosen-plaintext attacks (! signed keys) not useful for one way communication (e.g. email) Andreas V. Meier - The ElGamal Cryptosystem.

So, before I show you the ElGamal system let's do a very brief review of the Diffie-Hellman protocol. So, in my description here, I am going to abstract a little bit from the version that we saw last week. In fact, I just going to use the concept of a finite cyclic group. In fact, we have an arbitrary finite cyclic group, for example, it could be the group (Zp) star, but it could also be the. Explanation: The major disadvantage of the El Gamal cryptosystem is that it doubles the length of any message it encrypts. Therefore, a 2,048-bit plaintext message would yield a 4,096-bit ciphertext message when El Gamal is used for the encryption process

The ElGamal encryption scheme has been proposed several years ago and is one of the few probabilistic encryption schemes. However, its security has never been concretely proven based on clearly understood and accepted primitives. Here we show directly that the decision Diffie-Hellman assumption implies the security of the original ElGamal encryption scheme (with messages from a subgroup. 2 CHAPTER 1. INTRODUCTION The four ground principles of cryptography are Conﬁdentiality Deﬁnes a set of rules that limits access or adds restriction on certain information. Data Integrity Takes care of the consistency and accuracy of data during its entire life-cycle. Authentication Conﬁrms the truth of an attribute of a datum that is claimed to be true by som RSA and ElGamal have the multiplicative homomorphism while ECC and Paillier have the additive homomorphism. Moreover, additive homomorphic property has a wide application, such as pixel average for encrypted image resolution reduction and privacy protection in video surveillance by obtaining the difference image . RSA and ElGamal cryptosystems are the most extensively used encryption methods. Elliptic curve cryptography is used to implement public key cryptography. It was discovered by Victor Miller of IBM and Neil Koblitz of the University of Washington in the year 1985. ECC popularly used an acronym for Elliptic Curve Cryptography. It is based on the latest mathematics and delivers a relatively more secure foundation than the. In the name of Allah, the Most Gracious, the Most Merciful DOWNLOAD CHAPTER-WISE IN PDF FORMAT Surah No. Surah Name Arabic Meaning Revelation Total Verses Order Place 1 Al-Fatihah الفاتحة The Open

ElGamal signature scheme Author: Devon Ritter Created Date: 3/11/2011 5:45:20 PM. ElGamal signatures Choose a large prime p, and a primitive root mod p. Also, take a random integer a and calculate = a mod p The public key is the values of p, , and , while the secret key is the value a Signing uses a random integer k with gcd(k,p 1) = 1, and the signature is the pair (r,s) where (r = k mod p s = k 1(m ar) mod (p 1) (encryption: ( k, m)) Veriﬁcation is done comparing rrs. This simplified curve above is great to look at and explain the general concept of elliptic curves, but it doesn't represent what the curves used for cryptography look like. For this, we have to restrict ourselves to numbers in a fixed range, like in RSA. Rather than allow any value for the points on the curve, we restrict ourselves to whole numbers in a fixed range. When computing the formula.

Asymmetric Encryption Algorithms, Diffie-Hellman, RSA, ECC, ElGamal, DSA. The following are the major asymmetric encryption algorithms used for encrypting or digitally signing data. Diffie-Hellman key agreement: Diffie-Hellman key agreement algorithm was developed by Dr. Whitfield Diffie and Dr. Martin Hellman in 1976 Cybersecurity Essentials 1.1 Chapter 4 Quiz Answers 100% 2018. Learning with Cisco Netacad, there are many exams and lab activities to do. Some instructor require students to complete all Chapter exams, Final Exam and Chapter Quiz. No mater what instructors want you to do, ITexam24.com offers all exams answers with clear explanation

### hash - Malleability of ElGamal and Hashed ElGamal

El-Gamal tries to explain herself. I'm a Muslim and I support the religion from the Qur'an. I'm not over-criticising people, everyone is free to life his or her life. If you are my best. Public-key cryptography (also called asymmetric cryptography) is a cryptographic system that uses a pair of keys - a public key and a private key. The public key may be widely distributed, but the private key is meant to be known only by its owner. Keys are always created in a pair - every public key must have a corresponding private key Explained in Detail. 42. Ash-Shuraa. الشورى . The Consultation. 43. Az-Zukhruf. الزخرف. The Ornaments of Gold. 44. Ad-Dukhan. الدخان. The Smoke. 45. Al-Jathiyah. الجاثية. The Crouching. 46. Al-Ahqaf. الأحقاف. The Wind-Curved Sandhills. 47. Muhammad. محمد. Muhammad. 48. Al-Fath. الفتح. The Victory. 49. Al-Hujurat. الحجرات. The Rooms. 50. Qaf. ق.

El Gamal; DSA; Conclusion. Imagine a game being played by two persons (Symmetrical vs Asymmetrical) in which one tries to catch the other one. Every time the catcher comes close to the runner, the runner increases his/her speed to avoid getting caught. This is exactly what is happening in the world of cyber security right now. In this case, the runner is the developer developing new algorithms. The meet-in-the-middle attack is one of the types of known plaintext attacks. The intruder has to know some parts of plaintext and their ciphertexts. Using meet-in-the-middle attacks it is possible to break ciphers, which have two or more secret keys for multiple encryption using the same algorithm. For example, the 3DES cipher works in this way

### ElGamal signature scheme - Wikipedi

• Explanation of the block diagram. Firstly, each person adopting this scheme has a public-private key pair in cryptography. The key pairs used for encryption or decryption and signing or verifying are different for every signature. Here, the private key used for signing is referred to as the signature key and the public key as the verification key in this algorithm. Then, people take the signer.
• Mr El Gamal is a caring consultant with an attention to detail and who always has time for his patients. I am pleased to be able to recommend him as one of top orthopaedic surgeons in the West Midlands. For specialty: Orthopaedic Surgery. For conditions: Foot Fractures, Ankle Fractures, Ankle Sprain, Arthritis (Ankle), Foot Pain, Foot Sprains.
• How do I send an encrypted email? To properly encrypt emails, businesses should invest in encryption tools designed for email.When choosing the tools, a business can decide on sender encryption or key management. Sender encryption provides tools for users to encrypt their emails, such as flagging as urgent or installing a plug-in with a clickable encryption button
• Cybersecurity Essentials 1.1 Chapter 4 Quiz Answers 100% 2018 What is the name of the method in which letters are rearranged to create the ciphertext? enigma substitution transposition one-time pad Explanation: Ciphertext can be created by using the following: Transposition - letters are rearranged Substitution - letters are replaced One-time pad - plaintext combined with a secret key.
• Der Diffie-Hellman-Schlüsselaustausch oder Diffie-Hellman-Merkle-Schlüsselaustausch bzw.-Schlüsselvereinbarung (auch kurz DHM-Schlüsselaustausch oder DHM-Protokoll) ist ein Protokoll zur Schlüsselvereinbarung.Es ermöglicht, dass zwei Kommunikationspartner über eine öffentliche, abhörbare Leitung einen gemeinsamen geheimen Schlüssel in Form einer Zahl vereinbaren können, den nur.

### ElGamal encryption Crypto Wiki Fando

Elliptic Curve Cryptography Explained Oct 7, 2019 20 minute read Recently, I am learning how Elliptic Curve Cryptography works. I searched around the internet, found so many articles and videos explaining it. Most of them are covering only a portion of it, some of them skip many critical steps how you get from here to there. In the end, I didn't find an article that really explains it from. Download source - 7 Kb; Introduction. Strong Name (further referred to as SN) is a technology introduced with the .NET platform and it brings many possibilities into .NET applications. But many .NET developers still see Strong Names as security enablers (which is very wrong!) and not as a technology uniquely identifying assemblies.There is a lot of misunderstanding about SNs (as we could see.

### El Gamal explained The Biometrics of Failur

In this segment, we're gonna study the security of the ElGamal public key encryption system. So let me remind you that when we first presented the Diffie-Hellman protocol, we said that the security is based on the assumption that says that given G, G to the A, G to the B, it's difficult to compute the Diffie-Hellman secret, G to the AB 20) ElGamal encryption system is an asymmetric key encryption algorithm. Public-key cryptography; Private-key cryptography; Both; None of these; Answer: a) Public-key cryptography. Explanation: The ElGamal encryption scheme in cryptography is an asymmetrical key encryption algorithm based on the Diffie-Hellman key exchange for public-key. ### Video: Elgamal Cryptosystem Asymmetric Key Encryption Algorithm

explained and elaborated by the Sunnah. For example, the Messenger of Allah (s.a.w.) has cursed the one who accepted Riba, the one who paid it, the one who recorded it, and the two witness of it, saying they were all alike.2 It is also reported that the Prophet (s.a.w.) has said to the effect: (Exchange) gold for gold, silver for silver, wheat for wheat, barley for barley, dates for dates. In order to simplify the explanation of how the algorithm works, we will use small positive integers. In reality, the algorithm uses large numbers. In addition, you may find fairly easy explanations on Wikipedia and Khan Academy. Communicating in the clear, Alice and Bob agree on two positive integers, a prime number, and a generator. A generator is a number that, when raised to positive whole.

### Elgamal Encryption (Theory and Concepts) - YouTub

• The OpenSSL EC library provides support for Elliptic Curve Cryptography (ECC).It is the basis for the OpenSSL implementation of the Elliptic Curve Digital Signature Algorithm (ECDSA) and Elliptic Curve Diffie-Hellman (ECDH).. Note: This page provides an overview of what ECC is, as well as a description of the low-level OpenSSL API for working with Elliptic Curves
• What is SSL? SSL, or Secure Sockets Layer, is an encryption-based Internet security protocol.It was first developed by Netscape in 1995 for the purpose of ensuring privacy, authentication, and data integrity in Internet communications
• Here is the weekly answer to a legal question, by the CJSE, which al... lows you to know your rights, thanks to an infographic post! ⚖️ Today it's Hana El Shami, Hania Mahmoud and Malak Yasser who explained the following topic: Women's right in the workplace. ������ Many thanks to them and to the supervisors: Hana El Gamal and Joumana That's all!
• ElGamal 53 As an example, let r be a prime divisor of p − 1, and let m = (p − 1)/r.Suppose that we want to solve for x such that b = ax.The exponent is deﬁned up to multiples of p−1.If we raise both sides to the power m, then for the problem bm = amx a solution x is well-deﬁned up to multiples of r: am(x +r) = amx mr = a mxap−1 = a , since ap−1 = 1. If we now ﬁnd that p−1.
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1. In the ElGamal scheme, explain why the calculation of S1 is done in modulo p, but the calculation of S2 is done in modulo p _ 1. 2. In the Schnorr scheme, explain why the calculation of S1 is done in modulo p, but the calculation of S2 is done in modulo q ELGAMAL Digital Signature Scheme - | Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail | Author : William Stallings Posted On : 20.02.2017 09:53 pm . Chapter: Cryptography and Network Security Principles and Practice - Cryptographic Data Integrity Algorithms - Digital Signatures ELGAMAL Digital Signature Scheme . The ElGamal signature scheme. What does el-gamal-algorithm mean? A popular asymmetric encryption algorithm invented by Taher El Gamal in 1985. Named after its author and based on discre..

Diffie-Hellman (DH) is a key agreement algorithm, ElGamal an asymmetric encryption algorithm. Diffie-Hellman enables two parties to agree a common shared secret that can be used subsequently in a symmetric algorithm like AES. Neither of the partie.. If my understanding is correct, this is all I need to perform the encryption, but I'm struggling to figure out how using Crypto++. For the encryption example, I'm going to cheat a bit and use an RSA encryption example and port it to ElGamal. This is about as difficult as copy and paste because both RSA encryption and ElGamal encryption adhere to the the PK_Encryptor and PK_Decryptor interfaces 4 Elgamal Algorithm Example Correctness Security 5 Diﬃe-Hellman Key Exchange Diﬃe-Hellman Key Exchange Example Correctness Security 6 Summary RSA 10/83 RSA RSA is the best know public-key cryptosystem. Its security is based on the (believed) diﬃculty of factoring large numbers. Plaintexts and ciphertexts are large numbers (1000s of bits) as secure as factoring, is the topic of x8.3. x8.4 considers the ElGamal encryption scheme; related security and implementation issues are also discussed. The McEliece public-key encryption scheme, based on error-correctingcodes, is examined inx8.5. Although known to be insecure, the Merkle-Hellmanknapsackpublic-keyencryptionschemeis presentedin x8.6 for historical reasons - it was the.      Aly El Gamal ECE 301: Signals and Systems Homework Assignment #2 Problem 2 (continued) Using the fact that y 2(1) = 1 and also assuming that T<1, we get for t>T y 2(t) = 1 4 e2(t T) + (1 1 4 e2(1 T))e 2t: Now note that y 2(t) 6= y 1(t T) for t>T. Therefore, the system is not time invariant. (c)In order to show that system is incrementally linear with the auxiliary condition speci ed as y(1. El-Gamal IDEA. Explanation: Elliptic curve cryptography (ECC) uses elliptic curves as part of the algorithm for digital signature generation and key exchange. 10. What is the term used to describe the science of making and breaking secret codes? impersonation spoofing factorization cryptology* jamming. Explanation: Cryptology is the science of making and breaking codes to make sure that cyber. • variant of ElGamal and Schnorr schemes DSA Key GenerationDSA Key Generation • have shared global public key values ( p,q,g ): - choose 160 -bit prime number q - choose a large prime p with 2L-1 <<p<22L • where L=512 1024 bits and is a multiple of 64 • such that qis a 160 bit prime divisor of (p -1) - choose hand find g=h(p -1)/q modmod pp • where 1<h<p-1and h(p -1)/q mod. ElGamal Message Exchange • Bob encrypt a message to send to Alice - Bob represents message Min range 0<= M<= q-1 • longer messages must be sent as blocks - Bob chooses random integer kwith 1<= k<= q-1 - Bob computes one -time key K=yA k mod q - Bob encrypts Mas a pair of integers (C 1,C 2)where •C1=a kmod q and C 2=KM mod q • Alice then recovers message by - recovering key. However they are now widely used, including the El Gamal and Paillier schemes. Both of these schemes use a large prime number for a modulus operation, which is a security parameter. It is important to note however that even though these schemes can provide homomorphic operations, because of the nature of modulus operation, if the input or output values are greater than the modulus, results may.

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