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Dynamical Analysis of a Class of Euclidean Algorithms
"... We develop a general framework for the analysis of algorithms of a broad Euclidean type. The averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithm. The methods rely on properti ..."
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Cited by 17 (4 self)
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We develop a general framework for the analysis of algorithms of a broad Euclidean type. The averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithm. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory. As a consequence, we obtain precise averagecase analyses of algorithms for evaluating the Jacobi symbol of computational number theory fame, thereby solving conjectures of Bach and Shallit. These methods also provide a unifying framework for the analysis of an entire class of gcdlike algorithms together with new results regarding the probable behaviour of their cost functions. 1
Computational Alternatives to Random Number Generators
, 1999
"... In this paper, we present a simple method for generating randombased signatures when random number generators are either unavailable or of suspected quality (malicious or accidental). ..."
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Cited by 4 (3 self)
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In this paper, we present a simple method for generating randombased signatures when random number generators are either unavailable or of suspected quality (malicious or accidental).
A cubic publickey transformation
 Circuits, Systems and Signal Processing
"... Abstract. This note proposes the use of the cubic transformation for publickey applications and random event and number generation, in a manner akin to the Rabin cipher. Transformations modulo a prime p or a composite n = pq, where p and q are primes, are so used that each transformed value has onl ..."
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Cited by 3 (1 self)
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Abstract. This note proposes the use of the cubic transformation for publickey applications and random event and number generation, in a manner akin to the Rabin cipher. Transformations modulo a prime p or a composite n = pq, where p and q are primes, are so used that each transformed value has only three roots. Such a transformation, together with additional tag information, makes it possible to uniquely invert each transformed value. The effectiveness of the method as a random number generator (used in a variant with nine roots) comes from the fact that the cryptanalyst must contend with a ninefold branching at each step. Key words: Publickey cryptography, random number generation, cubic transformation. 1.
Averagecase Analyses of three algorithms for computing the Jacobi Symbol.
, 1998
"... We provide here a complete averagecase analysis of the three algorithms for computing the Jacobi symbol, for positive odd integers less than N . We analyse the average number of steps used for each of the algorithms. The average values are shown to be asymptotic to A1 log N or A2 log N for two of ..."
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We provide here a complete averagecase analysis of the three algorithms for computing the Jacobi symbol, for positive odd integers less than N . We analyse the average number of steps used for each of the algorithms. The average values are shown to be asymptotic to A1 log N or A2 log N for two of them, whereas it is asymptotic to A3 log 2 N for the third algorithm. The three constants A i are related to the invariant measure of the PerronFrobenius operator linked to the dynamical system. More precisely, they can be expressed with the entropy of the system. 1 Introduction. The Jacobi symbol, introduced in [24], is a very important tool in algebra, since it is related to quadratic characteristics of modular arithmetics. Interest in its efficient computation is now reawakened with its utilisation in primality tests [40] or more generally in cryptography. The Jacobi symbol intervenes in the definition of the Quadratic Residuality Problem, and many cryptographic primitives are based o...
SET Secure Electronic Transaction Specification
"... Syntax Notation (ASN.1) standard and shall be encoded using the Distinguished Encoding Rules (DER). This permits unambiguous encoding through a wellunderstood and widelyaccepted standard. SET message transport The SET specification does not define how a SET message is transported between entities ..."
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Syntax Notation (ASN.1) standard and shall be encoded using the Distinguished Encoding Rules (DER). This permits unambiguous encoding through a wellunderstood and widelyaccepted standard. SET message transport The SET specification does not define how a SET message is transported between entities. SET messages may be transported using any mechanism that the sender and receiver agree to. It is expected that transport standards will be developed to address the issue of interoperable SET applications. SET environments It is envisioned that SET applications will operate in one of two environments: Interactive  in this environment, the entities communicate in "realtime" with small time delays between the exchange of messages (such as the World Wide Web); and Noninteractive  in this environment, the entities communicate in non "realtime" with large time delays between the exchange of messages (such as EMail). SET Initiation Process In an interactive environment, it is expecte...
Computational Alternatives to Random Number Generators
, 1998
"... In this paper, we present a simple method for generating randombased signatures when random number generators are either unavailable or of suspected quality (malicious or accidental). By opposition to all past statemachine models, we assume that the signer is a memoryless automaton that starts fro ..."
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In this paper, we present a simple method for generating randombased signatures when random number generators are either unavailable or of suspected quality (malicious or accidental). By opposition to all past statemachine models, we assume that the signer is a memoryless automaton that starts from some internal state, receives a message, outputs its signature and returns precisely to the same initial state; therefore, the new technique formally converts randomized signatures into deterministic ones. Finally, we show how to translate the random oracle concept required in security proofs into a realistic set of tamperresistance assumptions.
PC Book 2: Programmer’s Guide SET Secure Electronic Transaction Specification
, 1997
"... This Programmer’s Guide will continue to be reviewed and updated to ensure clarity and consistency with the Formal Protocol Definition in Book 3. MasterCard and Visa are creating a process for applications to be certified as compliant with the SET specifications. ..."
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This Programmer’s Guide will continue to be reviewed and updated to ensure clarity and consistency with the Formal Protocol Definition in Book 3. MasterCard and Visa are creating a process for applications to be certified as compliant with the SET specifications.
A Covert Encryption Method for Applications in Electronic Data Interchange
"... Abstract — A principal weakness of all encryption systems is that the output data can be ‘seen ’ to be encrypted. In other words, encrypted data provides a ‘flag ’ on the potential value of the information that has been encrypted. In this paper, we provide a new approach to ‘hiding ’ encrypted data ..."
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Abstract — A principal weakness of all encryption systems is that the output data can be ‘seen ’ to be encrypted. In other words, encrypted data provides a ‘flag ’ on the potential value of the information that has been encrypted. In this paper, we provide a new approach to ‘hiding ’ encrypted data in a digital image. In conventional (symmetric) encryption, the plaintext is usually represented as a binary stream and encrypted using an XOR type operation with a binary cipher. The algorithm used is ideally designed to: (i) generate a maximum entropy cipher so that there is no bias with regard to any bit; (ii) maximize diffusion in terms of key dependency so that a change in any bit of the key can effect any, and potentially all, bits of the cipher. In the work reported here, we consider an approach in which a binary or lowbit plaintext image is encrypted with a decimal integer or floating point cipher using a convolution operation and the output quantized into a 1bit array generating a binary image ciphertext. This output is then ‘embedded ’ in a host image to hide the encrypted information. Embedding is undertaken either in the lowest 1bit layer or multiple 1bit layers. Decryption is accomplished by: (i) extracting the binary image from the host image; (ii) correlating the result with the original cipher. In principle, any cipher generator can be used for this purpose and the method has been designed to operate with 24bit colour images. The approach has a variety of applications and, in this paper, we focus on the authentication and selfauthentication of edocuments (letters and certificates, for example) that are communicated over the Internet and are thereby vulnerable to attack (e.g. modification, editing, counterfeiting etc.). In addition to document authentication, the approach considered provides a way of propagating disinformation and a solution to scenarios that require ‘plausible deniability’.