Table of Links Abstract and 1 Introduction Abstract and 1 Introduction Scenario and Requirements


History and Related Work


Concept of Cramer-Shoup with Elliptic Curve and 4.1 Prerequisite
4.2 Public Key Generation by Receiver
4.3 Encryption by Sender
4.4 Decryption by Receiver


Evaluation and 5.1 Proof of Correctness
5.2 Preliminary Performance Comparison


Proof: Secure against adaptive-chosen ciphertext attacks
6.1 DDH Assumption and 6.2 CCA Assumption
6.3 IND-CCA 1 - non-adaptive Security
6.4 IND-CCA 2 - adaptive Security (Validity Checking Failure)


Security discussion: Post-Quantum Cryptography


Summary, References, and Authors Scenario and Requirements Scenario and Requirements Scenario and Requirements History and Related Work History and Related Work History and Related Work Concept of Cramer-Shoup with Elliptic Curve and 4.1 Prerequisite
4.2 Public Key Generation by Receiver
4.3 Encryption by Sender
4.4 Decryption by Receiver Concept of Cramer-Shoup with Elliptic Curve and 4.1 Prerequisite Concept of Cramer-Shoup with Elliptic Curve and 4.1 Prerequisite 4.2 Public Key Generation by Receiver 4.2 Public Key Generation by Receiver 4.3 Encryption by Sender 4.3 Encryption by Sender 4.4 Decryption by Receiver 4.4 Decryption by Receiver Evaluation and 5.1 Proof of Correctness
5.2 Preliminary Performance Comparison Evaluation and 5.1 Proof of Correctness Evaluation and 5.1 Proof of Correctness 5.2 Preliminary Performance Comparison 5.2 Preliminary Performance Comparison Proof: Secure against adaptive-chosen ciphertext attacks
6.1 DDH Assumption and 6.2 CCA Assumption
6.3 IND-CCA 1 - non-adaptive Security
6.4 IND-CCA 2 - adaptive Security (Validity Checking Failure) Proof: Secure against adaptive-chosen ciphertext attacks Proof: Secure against adaptive-chosen ciphertext attacks 6.1 DDH Assumption and 6.2 CCA Assumption 6.1 DDH Assumption and 6.2 CCA Assumption 6.3 IND-CCA 1 - non-adaptive Security 6.3 IND-CCA 1 - non-adaptive Security 6.4 IND-CCA 2 - adaptive Security (Validity Checking Failure) 6.4 IND-CCA 2 - adaptive Security (Validity Checking Failure) Security discussion: Post-Quantum Cryptography Security discussion: Post-Quantum Cryptography Security discussion: Post-Quantum Cryptography Summary, References, and Authors Summary, References, and Authors Summary, References, and Authors 4 Concept of Cramer-Shoup with Elliptic Curve This approach is premised on the general sender-receiver architecture, as shown in Figure 1. Our public-key encryption method is based on the approach of Cramer-Shoup (CS) [42,43]. Here, we adapt the cryptographic strong procedure of CS to the promising base of ECC as an intermediate step to supersingular isogeny EC. The main benefit is that the key length scales linear in relation to the security level [44]. The new security relies on the Elliptic Curve Discrete Logarithm Problem (ECDLP) [45,46]. This mathematical problem is more difficult to solve than the integer factorization problem or the classical DLP. Currently, no algorithm is known that solves the ECDLP in an efficient way. The known fastest approach is the parallelized Pollard-Rho Algorithms with approximately O( √p), where p is the largest prime factor of n [47] Elliptic Curve Discrete Logarithm Problem 4.1 Prerequisite First of all, the following parameters are to be declared. The plain-text message to be secured, is described as the parameter m. Here it is a positive integer value, represented as binary. 4.2 Public Key Generation by Receiver In summary, we obtain the following individual keys: – Public-key - Points: C, D, H – Private-key - Factors: x1, x2, y1, y2, z In addition, we have in common the following parameters, which can also be public: – Function Fp(X) with generator points G1, G2 p – Function Hash Hash For the public format, recommended representation is ANSI X.509, X9.62, and X9.63 syntax following ASN.1 structure. Author:
(1) Peter Hillmann, University of the Bundeswehr Munich, Department of Computer Science, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany. Author: Author: (1) Peter Hillmann, University of the Bundeswehr Munich, Department of Computer Science, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany. This paper is available on arxiv under ATTRIBUTION-NONCOMMERCIAL-SHAREALIKE 4.0 INTERNATIONAL license. This paper is available on arxiv under ATTRIBUTION-NONCOMMERCIAL-SHAREALIKE 4.0 INTERNATIONAL license. available on arxiv available on arxiv

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