TL;DR: The notion of universally composable security was introduced in this paper for defining security of cryptographic protocols, which guarantees security even when a secure protocol is composed of an arbitrary set of protocols, or more generally when the protocol is used as a component of a system.
Abstract: We propose a novel paradigm for defining security of cryptographic protocols, called universally composable security. The salient property of universally composable definitions of security is that they guarantee security even when a secure protocol is composed of an arbitrary set of protocols, or more generally when the protocol is used as a component of an arbitrary system. This is an essential property for maintaining security of cryptographic protocols in complex and unpredictable environments such as the Internet. In particular, universally composable definitions guarantee security even when an unbounded number of protocol instances are executed concurrently in an adversarially controlled manner, they guarantee non-malleability with respect to arbitrary protocols, and more. We show how to formulate universally composable definitions of security for practically any cryptographic task. Furthermore, we demonstrate that practically any such definition can be realized using known techniques, as long as only a minority of the participants are corrupted. We then proceed to formulate universally composable definitions of a wide array of cryptographic tasks, including authenticated and secure communication, key-exchange, public-key encryption, signature, commitment, oblivious transfer, zero knowledge and more. We also make initial steps towards studying the realizability of the proposed definitions in various settings.
TL;DR: In this paper, the authors present a list of figures in the context of digital signatures and message authentication for general cryptographic protocols, including encryption, digital signatures, message authentication, and digital signatures.
Abstract: List of figures Preface Acknowledgements 5. Encryption schemes 6. Digital signatures and message authentication 7. General cryptographic protocols Appendix C: corrections and additions to volume I Bibliography Index.
TL;DR: A new automatic cryptographic protocol verifier based on a simple representation of the protocol by Prolog rules, and on a new efficient algorithm that determines whether a fact can be proved from these rules or not, which proves secrecy properties of the protocols.
Abstract: We present a new automatic cryptographic protocol verifier based on a simple representation of the protocol by Prolog rules, and on a new efficient algorithm that determines whether a fact can be proved from these rules or not. This verifier proves secrecy properties of the protocols. Thanks to its use of unification, it avoids the problem of the state space explosion. Another advantage is that we do not need to limit the number of runs of the protocol to analyze it. We have proved the correctness of our algorithm, and have implemented it. The experimental results show that many examples of protocols of the literature, including Skeme [24], can be analyzed by our tool with very small resources: the analysis takes from less than 0.1 s for simple protocols to 23 s for the main mode of Skeme. It uses less than 2 Mb of memory in our tests.
TL;DR: A mathematical construct which provides a cryptographic protocol to verifiably shuffle a sequence of k modular integers is presented, and its application to secure, universally verifiable, multi-authority election schemes is discussed and shown to be honest-verifier zeroknowledge in a special case and in general.
Abstract: We present a mathematical construct which provides a cryptographic protocol to verifiably shuffle a sequence of k modular integers, and discuss its application to secure, universally verifiable, multi-authority election schemes. The output of the shuffle operation is another sequence of k modular integers, each of which is the same secret power of a corresponding input element, but the order of elements in the output is kept secret. Though it is a trivial matter for the "shuffler" (who chooses the permutation of the elements to be applied) to compute the output from the input, the construction is important because it provides a linear size proof of correctness for the output sequence (i.e. a proof that it is of the form claimed) that can be checked by an arbitrary verifiers. The complexity of the protocol improves on that of Furukawa-Sako[16] both measured by number of exponentiations and by overall size.The protocol is shown to be honest-verifier zeroknowledge in a special case, and is computational zeroknowledge in general. On the way to the final result, we also construct a generalization of the well known Chaum-Pedersen protocol for knowledge of discrete logarithm equality [10], [7]. In fact, the generalization specializes exactly to the Chaum-Pedersen protocol in the case k = 2. This result may be of interest on its own.An application to electronic voting is given that matches the features of the best current protocols with significant efficiency improvements. An alternative application to electronic voting is also given that introduces an entirely new paradigm for achieving Universally Verifiable elections.
TL;DR: In this article, the authors present a rigorous model for secure reactive systems in asynchronous networks with a sound cryptographic semantics, supporting abstract specifications and the composition of secure systems, which is essential in bridging the gap between the rigorous proof techniques of cryptography and tool-supported formal proof techniques.
Abstract: We present a rigorous model for secure reactive systems in asynchronous networks with a sound cryptographic semantics, supporting abstract specifications and the composition of secure systems. This enables modular proofs of security, which is essential in bridging the gap between the rigorous proof techniques of cryptography and tool-supported formal proof techniques. The model follows the general simulatability approach of modern cryptography. A variety of network structures and trust models can be described such as static and adaptive adversaries, some examples of this are given. As an example of our specification methodology we provide an abstract and complete specification for Secure Message Transmission, improving on recent results by Lynch (1999), and verify one concrete implementation. Our proof is based on a general theorem on the security of encryption in a reactive multi-user setting, generalizing a recent result by Bellare et. al (2000).
TL;DR: A key agreement protocol previously proposed by Steer et al. is resurrected and extended to handle dynamic groups and network failures such as network partitions and merges and provides key independence, i.e. a passive adversary who knows any proper subset of group keys cannot discover any other group keys not included in the subset.
Abstract: Traditionally, research in secure group key agreement focuses on minimizing the computational overhead for cryptographic operations, and minimizing the communication overhead and the number of protocol rounds is of secondary concern. The dramatic increased in computation power that we witnessed during the past years exposed network delay in WANs as the primary culprit for a negative performance impact on key agreement protocols. The majority of previously proposed protocols optimize the cryptographic overhead of the protocol. However, high WAN delay negatively impacts their efficiency. The goal of this work is to construct a new protocol that trades off computation with communication efficiency. We resurrect a key agreement protocol previously proposed by Steer et al. We extend it to handle dynamic groups and network failures such as network partitions and merges. The resulting protocol suite is provably secure against passive adversaries and provides key independence, i.e. a passive adversary who knows any proper subset of group keys cannot discover any other group keys not included in the subset. Furthermore, the protocol is simple, fault-tolerant, and well suited for high-delay wide area network.
TL;DR: In this paper, a cryptographic device comprising a processing logic and memory associated with the processing logic is loaded with a first segment of code to control execution of cryptographic functions and hash functions, and a second segment is used to perform cryptographic functions on behalf of a third party having no physical control of hardware employing the cryptographic device.
Abstract: A cryptographic device comprising a processing logic and memory associated with the processing logic. The memory is loaded with a first segment of code to control execution of cryptographic functions and hash functions, and a second segment of code to perform cryptographic functions on behalf of a third party having no physical control of hardware employing the cryptographic device.
TL;DR: The main theorem guarantees that any well-typed protocol is robustly safe, that is, its correspondence assertions are true in the presence of any opponent expressible in spi.
Abstract: We propose a new method to check authenticity properties of cryptographic protocols. First, code up the protocol in the spi-calculus of Abadi and Gordon. Second, specify authenticity properties by annotating the code with correspondence assertions in the style of Woo and Lam. Third, figure out types for the keys, nonces, and messages of the protocol. Fourth, check that the spi-calculus code is well-typed according to a novel type and effect system presented in this paper. Our main theorem guarantees that any well-typed protocol is robustly safe, that is, its correspondence assertions are true in the presence of any opponent expressible in spi.
TL;DR: This work compares two views of symmetric cryptographic primitives in the context of the systems that use them and establishes the soundness of the formal definition of equivalence of systems with respect to eavesdroppers.
Abstract: We compare two views of symmetric cryptographic primitives in the context of the systems that use them. We express those systems in a simple programming language; each of the views yields a semantics for the language. One of the semantics treats cryptographic operations formally (that is, symbolically). The other semantics is more detailed and computational; it treats cryptographic operations as functions on bitstrings. Each semantics leads to a definition of equivalence of systems with respect to eavesdroppers. We establish the soundness of the formal definition with respect to the computational one. This result provides a precise computational justification for formal reasoning about security against eavesdroppers.
TL;DR: This paper shows how some principles that have already been used to make cryptographic protocols more resistant to denial of service can be formalized based on a modification of the Gong-Syverson fail-stop model of cryptographic protocols, and indicates the ways in which existing cryptographic protocol analysis tools could be modified to operate within this formal framework.
Abstract: Denial of service is becoming a growing concern. As computer systems communicate more and more with others that they know less and less, they become increasingly vulnerable to hostile intruders who may take advantage of the very protocols intended for the establishment and authentication of communication to tie up resources and disable servers. This paper shows how some principles that have already been used to make cryptographic protocols more resistant to denial of service by trading off the cost to defender against the cost to the attacker can be formalized based on a modification of the Gong-Syverson fail-stop model of cryptographic protocols, and indicates the ways in which existing cryptographic protocol analysis tools could be modified to operate within this formal framework. We also indicate how this framework could be extended to protocols that do not make use of strong authentication.
TL;DR: A simple technique by which a device that performs private key operations in networked applications and whose local private key is activated with a password or PIN can be immunized to offline dictionary attacks in case the device is captured is presented.
Abstract: We present a simple technique by which a device that performs private key operations (signatures or decryptions) in networked applications, and whose local private key is activated with a password or PIN, can be immunized to offline dictionary attacks in case the device is captured Our techniques do not assume tamper resistance of the device, but rather exploit the networked nature of the device, in that the device's private key operations are performed using a simple interaction with a remote server This server however, is untrusted-its compromise does not reduce the security of the device's private key unless the device is also captured and need not have a prior relationship with the device We further extend this approach with support for key disabling, by which the rightful owner of a stolen device can disable the device's private key even if the attacker already knows the user's password
TL;DR: A cryptographic key split combiner as discussed by the authors includes a number of key split generators and a key split randomizer for randomizing the cryptographic key splits to produce a cryptographic key, and a process for forming cryptographic keys.
Abstract: A cryptographic key split combiner, which includes a number of key split generators for generating cryptographic key splits and a key split randomizer for randomizing the cryptographic key splits to produce a cryptographic key, and a process for forming cryptographic keys. Each of the key split generators generates key splits from seed data. The key split generators may include a random split generator for generating a random key split based on reference data. Other key split generators may include a token split generator for generating a token key split based on label data, a console split generator for generating a console key split based on maintenance data, and a biometric split generator for generating a biometric key split based on biometric data. All splits may further be based on static data, which may be updated, for example by modifying a prime number divisor of the static data. The label data may be read from a storage medium, and may include user authorization data. The resulting cryptographic key may be, for example, a stream of symbols, at least one symbol block, or a key matrix.
TL;DR: In this article, a gateway device for carrying out a data relaying at a transport or upper layer between a first terminal device and a second terminal device which are capable of carrying out communications through networks is described.
Abstract: In a gateway device for carrying out a data relaying at a transport or upper layer between a first terminal device and a second terminal device which are capable of carrying out communications through networks, information regarding a security association set up between the first terminal device and the second terminal device in order to carry out communications with guaranteed data secrecy or in order to carry out communications with guaranteed data authenticity managed, and the decryption/encryption processing or the authentication processing is carried out according to the managed information regarding the security association along with the data relaying at the transport or upper layer.
TL;DR: In this paper, ownership of a secure process is enabled with a cryptographic system and methods for initializing and operating the cryptographic system transfer control from the loading program to the loaded program and, in essence, from cryptographic system vendor to its end-user.
Abstract: Ownership of a secure process is enabled with a cryptographic system. Methods initializing and operating the cryptographic system transfer control from the loading program to the loaded program and, in essence from the cryptographic system vendor to its end-user. As a result, ownership of the secure process can be relinquished to the end-user so that it alone can subsequently use the cryptographic system to control the secure process of loading and running its user-programs. The cryptographic system and methods allow for secure operations and protect against tampering with application software. The application program is retrieved from an encrypted file in external memory and authenticated by the cryptographic system before being executed.
TL;DR: In this paper, the authors propose a method of authenticating a client and a server to each other via a gateway, in which the client uses a first encryption protocol between itself and the gateway, and the server uses a second different encryption protocol in order to confirm to itself that the digital certificate comes from the trusted certification authority.
Abstract: A method of authenticating a client (42) and a server (44) to each other via a gateway (46) in which the client uses a first encryption protocol between itself and the gateway and the server uses a second different encryption protocol between itself and the gateway, the method comprising the steps of:installing in the server that the gateway is a trusted certification authority (48);the gateway issuing a digital certificate authenticating the client; andthe server verifying the digital certificate in order to confirm to itself that the digital certificate comes from the trusted certification authority.
TL;DR: In this article, the authors describe properties of a process calculus that has been developed for the purpose of analyzing security protocols, which is a restricted form of -calculus with bounded replication and probabilistic polynomial time expressions allowed in messages and boolean tests.
TL;DR: An original method is proposed to analyse protocols of the Cliques suites that propose extensions of the Diffie-Hellman key exchange protocol to a group setting and presents a number of unpublished flaws with respect to each of the main security properties claimed in protocol definition.
Abstract: Secure group protocols are not easy to design: this paper will show new attacks found against a protocol suite for sharing key. The method we propose to analyse these protocols is very systematic, and can be applied to numerous protocols of this type. The A-GDH.2 protocols suite analysed throughout this paper is part of the Cliques suites that propose extensions of the Diffie-Hellman key exchange protocol to a group setting. The A-GDH.2 main protocol is intended to allow a group to share an authenticated key while the other protocols of the suite allow to perform dynamic changes in the group constitution (adding and deleting members, fusion of groups, ...). We are proposing an original method to analyse these protocols and are presenting a number of unpublished flaws with respect to each of the main security properties claimed in protocol definition (key authentication, perfect forward secrecy, resistance to known-keys attacks). Most of these flaws arise from the fact that using a group setting does not allow to reason about security properties in the same way as when only two (or three) parties are concerned. Our method has been easily applied on other Cliques protocols and allowed us to pinpoint similar flaws.
TL;DR: The net semantics is formally related to a transition semantics, strand spaces and inductive rules, as well as trace languages and event structures, so unifying a range of approaches, aswell as providing conditions under which particular, more limited, models are adequate for the analysis of protocols.
Abstract: The events of a security protocol and their causal dependency can play an important role in the analysis of security properties. This insight underlies both strand spaces and the inductive method. But neither of these approaches builds up the events of a protocol in a compositional way, so that there is an informal spring from the protocol to its model. By broadening the models to certain kinds of Petri nets, a restricted form of contextual nets, a compositional event-based semantics is given to an economical, but expressive, language for describing security protocols; so the events and dependency of a wide range of protocols are determined once and for all. The net semantics is formally related to a transition semantics, strand spaces and inductive rules, as well as trace languages and event structures, so unifying a range of approaches, as well as providing conditions under which particular, more limited, models are adequate for the analysis of protocols. The net semantics allows the derivation of general properties and proof principles which are demonstrated in establishing an authentication property, following a diagrammatic style of proof.
TL;DR: A typed process calculus for security protocols in which types convey secrecy properties is developed, especially on public-key encryption, which presents special difficulties on asymmetric communication primitives.
Abstract: We develop a typed process calculus for security protocols in which types convey secrecy properties. We focus on asymmetric communication primitives, especially on public-key encryption. These present special difficulties, partly because they rely on related capabilities (e.g., "public" and "private" keys) with different levels of secrecy and scopes.
TL;DR: This work formalizes the notion of a cryptographic counter, which allows a group of participants to increment and decrement a cryptographic representation of a (hidden) numerical value privately and robustly, and shows a general and efficient reduction from any encryption scheme to a general cryptographic counter.
Abstract: We formalize the notion of a cryptographic counter, which allows a group of participants to increment and decrement a cryptographic representation of a (hidden) numerical value privately and robustly. The value of the counter can only be determined by a trusted authority (or group of authorities, which may include participants themselves), and participants cannot determine any information about the increment/decrement operations performed by other parties.
Previous efficient implementations of such counters have relied on fully-homomorphic encryption schemes; this is a relatively strong requirement which not all encryption schemes satisfy. We provide an alternate approach, starting with any encryption scheme homomorphic over the additive group Z2 (i.e., 1-bit xor). As our main result, we show a general and efficient reduction from any such encryption scheme to a general cryptographic counter. Our main reduction does not use additional assumptions, is efficient, and gives a novel implementation of a general counter. The result can also be viewed as an efficient construction of a general n-bit cryptographic counter from any 1-bit counter which has the additional property that counters can be added securely.
As an example of the applicability of our construction, we present a cryptographic counter based on the quadratic residuosity assumption and use it to construct an efficient voting scheme which satisfies universal verifiability, privacy, and robustness.
TL;DR: It is shown that in many cases, the simultaneous use of related keys for two cryptosystems does not compromise their security, and for a variety of public-key encryption schemes that are secure against chosen-ciphertext attacks and against forgery under chosen-message attacks.
Abstract: It is a maxim of sound computer-security practice that a cryptographic key should have only a single use For example, an RSA key pair should be used only for public-key encryption or only for digital signatures, and not for bothIn this paper we show that in many cases, the simultaneous use of related keys for two cryptosystems, eg for a public-key encryption system and for a public-key signature system, does not compromise their security We demonstrate this for a variety of public-key encryption schemes that are secure against chosen-ciphertext attacks, and for a variety of digital signature schemes that are secure against forgery under chosen-message attacks The precise form of the statement of security that we are able to prove depends on the particular cryptographic schemes in question and on the cryptographic assumptions needed for their proofs of security; but in every case, our proof of security does not require any additional cryptographic assumptionsAmong the cryptosystems that we analyze in this manner are the public-key encryption schemes of Cramer and Shoup, Naor and Yung, and Dolev, Dwork, and Naor, which are all defined in them standard model, while in the random-oracle model we analyze plaintext-aware encryption schemes (as defined by Bellare and Rogaway) and in particular the OAEP+ cryptosystem Among public-key signature schemes, we analyze those of Cramer and Shoup and of Gennaro, Halevi, and Rabin in the standard model, while in the random-oracle model we analyze the RSA PSS scheme as well as variants of the El Gamal and Schnorr schemes (See references within)
TL;DR: The net semantics is formally related to a transition semantics, strand spaces and inductive rules, as well as trace languages and event structures, so unifying a range of approaches, aswell as providing conditions under which particular, more limited, models are adequate for the analysis of protocols.
Abstract: The events of a security protocol and their causal dependency can play an important role in the analysis of security properties. This insight underlies both strand spaces and the inductive method. But neither of these approaches builds up the events of a protocol in a compositional way, so that there is an informal spring from the protocol to its model. By broadening the models to certain kinds of Petri nets, a restricted form of contextual nets, a compositional event-based semantics is given to an economical, but expressive, language for describing security protocols; so the events and dependency of a wide range of protocols are determined once and for all. The net semantics is formally related to a transition semantics, strand spaces and inductive rules, as well as trace languages and event structures, so unifying a range of approaches, as well as providing conditions under which particular, more limited, models are adequate for the analysis of protocols. The net semantics allows the derivation of general properties and proof principles which are demonstrated in establishing an authentication property, following a diagrammatic style of proof.
TL;DR: A cryptographic λ-calculus is defined and syntactic logical relations are introduced (in the style of Pitts and Birkedal-Harper) for this calculus that can be used to prove behavioral equivalences between programs that use encryption.
Abstract: The theory of relational parametricity and its logical relations proof technique are powerful tools for reasoning about information hiding in the polymorphic A-calculus. We investigate the application of these tools in the security domain by dejining a cryptographic A-calculus-an extension of the standard simply typed A-calculus with primitives for encryption, decryption, and key generation-and introducing logical relations for this calculus that can be used to prove behavioral equivalences between programs that rely on encryption. We illustrate the framework by encoding some simple security protocols, including the Needham-Schroeder publickey protocol. We give a natural account of the well-known attack on the original protocol and a straighgorward proof that the improved variant of the protocol is secure.
TL;DR: This paper shows how the Dolev-Yao model may be used for protocol analysis, while a further analysis gives a quantitative bound on the extent to which real cryptographic primitives may diverge from the idealized model.
Abstract: Dolev and Yao initiated an approach to studying cryptographic protocols which abstracts from possible problems with the cryptography so as to focus on the structural aspects of the protocol. Recent work in this framework has developed easily applicable methods to determine many security properties of protocols. A separate line of work, initiated by Bellare and Rogaway, analyzes the way specific cryptographic primitives are used in protocols. It gives asymptotic bounds on the risk of failures of secrecy or authentication.In this paper we show how the Dolev-Yao model may be used for protocol analysis, while a further analysis gives a quantitative bound on the extent to which real cryptographic primitives may diverge from the idealized model. We develop this method where the cryptographic primitives are based on Carter-Wegman universal classes of hash functions. This choice allows us to give specific quantitative bounds rather than simply asymptotic bounds.
TL;DR: This paper reviews and discusses the most important issues raised by biometrics and presents a secure authentication protocol skeleton and smart card reveals to be an useful and efficient partner of biometric for such a protocol.
Abstract: Although biometrics can be an useful component for access control, the security they procure is often overestimated, as if they were a magic tool whose simple use will automatically prevent each and every type of attack. Biometrics are not secure unless they are embedded in a strong cryptographic protocol, whose design pays special attention to their specificities. In particular, smart card reveals to be an useful and efficient partner of biometrics for such a protocol. This paper reviews and discusses the most important issues raised by biometrics and presents a secure authentication protocol skeleton.
TL;DR: New classes for the Java 2 Micro-Edition platform are created that offer fundamental cryptographic operations such as message digests and ciphers as well as higher level security protocols like SSL, showing that SSL is a practical solution for ensuring end-to-end security of wireless Internet transactions even within today's technological constraints.
Abstract: Internet-enabled wireless devices continue to proliferate and are expected to surpass traditional Internet clients in the near future. This has opened up exciting new opportunities in the mobile e-commerce market. However, data security and privacy remain major concerns in the current generation of "wireless Web" offerings. All such offerings today use a security architecture that lacks end-to-end security. This unfortunate choice is driven by perceived inadequacies of standard Internet security protocols like SSL on less capable CPUs and low-bandwidth wireless links. This article presents our experiences in implementing and using standard security mechanisms and protocols on small wireless devices. We have created new classes for the Java 2 Micro-Edition platform that offer fundamental cryptographic operations such as message digests and ciphers as well as higher level security protocols like SSL. Our results show that SSL is a practical solution for ensuring end-to-end security of wireless Internet transactions even within today's technological constraints.
TL;DR: Flaws in protocols proposed by Corradi et al, Karjoth et al., and Karnik et al. are identified, including protocols based on secure co-processors.
Abstract: Mobile agent security is still a young discipline and most naturally, the focus up to the time of writing was on inventing new cryptographic protocols for securing various aspects of mobile agents. However, past experience shows that protocols can be flawed, and flaws in protocols can remain unnoticed for a long period of time. The game of breaking and fixing protocols is a necessary evolutionary process that leads to a better understanding of the underlying problems and ultimately to more robust and secure systems. Although, to the best of our knowledge, little work has been published on breaking protocols for mobile agents, it is inconceivable that the multitude of protocols proposed so far are all flawless. As it turns out, the opposite is true. We identify flaws in protocols proposed by Corradi et al., Karjoth et al., and Karnik et al., including protocols based on secure co-processors.
TL;DR: In this article, a method and apparatus provides cryptographic parameters for use in cryptographic applications in response to requests therefor, including pre-computing one or more different types of sets of cryptographic parameters, each the type of set being adapted for use by an associated type of cryptographic application.
Abstract: A method and apparatus provides cryptographic parameters for use in cryptographic applications in response to requests therefor. The method includes the steps of: pre-computing one or more different types of sets of cryptographic parameters, each the type of set being adapted for use by an associated type of cryptographic application; securely storing the pre-computed sets of cryptographic parameters in a memory storage unit; receiving a request for a set of cryptographic parameters having specified characteristics for use in a particular cryptographic application; determining one of the sets of cryptographic parameters stored in the memory storage unit that has specified characteristics; accessing the determined set of cryptographic parameters from the memory storage unit; and providing the determined set of cryptographic parameters with minimal latency.
TL;DR: The paper shows how simulated annealing and genetic algorithms can be used to generate correct and efficient BAN protocols and investigates the use of parsimonious and redundant representations.
Abstract: Protocol security is important. So are efficiency and cost. This paper provides an early framework for handling such aspects in a uniform way based on combinatorial optimisation techniques. The belief logic of Burrows, Abadi and Needham (BAN logic) is viewed as both a specification and proof system and as a ‘protocol programming language’. The paper shows how simulated annealing and genetic algorithms can be used to generate correct and efficient BAN protocols. It also investigates the use of parsimonious and redundant representations.
TL;DR: In this article, the authors propose an apparatus and method for generating pseudo-random cryptographic keys in a cryptographic communications system, whereby, given a common set of initializing configuration data, the pseudoregressive cryptographic keys can be duplicatively generated by various independent pseudorandom key generators of the cryptographic communication system.
Abstract: An apparatus and method for generating pseudo-random cryptographic keys in a cryptographic communications systems, whereby, given a common set of initializing configuration data, the pseudo-random cryptographic keys can be duplicatively generated by various independent pseudo-random key generators of the cryptographic communications system.