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Enforcing and defying associativity, commutativity, totality, and strong noninvertibility for oneway functions in complexity theory
 In ICTCS
, 2005
"... Rabi and Sherman [RS97,RS93] proved that the hardness of factoring is a sufficient condition for there to exist oneway functions (i.e., ptime computable, honest, ptime noninvertible functions) that are total, commutative, and associative but not strongly noninvertible. In this paper we improve th ..."
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Rabi and Sherman [RS97,RS93] proved that the hardness of factoring is a sufficient condition for there to exist oneway functions (i.e., ptime computable, honest, ptime noninvertible functions) that are total, commutative, and associative but not strongly noninvertible. In this paper we improve the sufficient condition to P = NP. More generally, in this paper we completely characterize which types of oneway functions stand or fall together with (plain) oneway functions—equivalently, stand or fall together with P = NP. We look at the four attributes used in Rabi and Sherman’s seminal work on algebraic properties of oneway functions (see [RS97,RS93]) and subsequent papers—strongness (of noninvertibility), totality, commutativity, and associativity—and for each attribute, we allow it to be required to hold, required to fail, or “don’t care. ” In this categorization there are 3 4 = 81 potential types of oneway functions. We prove that each of these 81 featureladen types stand or fall together with the existence of (plain) oneway functions. Key words: computational complexity, complexitytheoretic oneway functions, associativity, 1.1
A Status Report on the P versus NP Question
"... We survey some of the history of the most famous open question in computing: the P versus NP question. We summarize some of the progress that has been made to date, and assess the current situation. ..."
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We survey some of the history of the most famous open question in computing: the P versus NP question. We summarize some of the progress that has been made to date, and assess the current situation.
PostQuantum Signatures
, 2004
"... Digital signatures have become a key technology for making the Internet and other IT infrastructures secure. But in 1994 Peter Shor showed that quantum computers can break all digital signature schemes that are used today and in 2001 Chuang and his coworkers implemented Shor’s algorithm for the firs ..."
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Digital signatures have become a key technology for making the Internet and other IT infrastructures secure. But in 1994 Peter Shor showed that quantum computers can break all digital signature schemes that are used today and in 2001 Chuang and his coworkers implemented Shor’s algorithm for the first time on a 7qubit NMR quantum computer. This paper studies the question: What kind of digital signature algorithms are still secure in the age of quantum computers? 1 1
Every Bit of the Discrete Logarithm is Either Hard Or Easy
"... Let p be a prime such that p = 2 s q +1, where q is odd. Let g be a primitive root of p and let n = dlog(p \Gamma 1)e. If y j g x mod p, then we will show that determining the s least significant bits of x is easy, while determining the any of (n \Gamma s) most significant bits of x is equivalen ..."
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Let p be a prime such that p = 2 s q +1, where q is odd. Let g be a primitive root of p and let n = dlog(p \Gamma 1)e. If y j g x mod p, then we will show that determining the s least significant bits of x is easy, while determining the any of (n \Gamma s) most significant bits of x is equivalent in difficulty to recovering all of x. 1 Introduction In the past decade the applications for intractable problems in the design of secrecy systems, digital signatures, cryptographic protocols and interactive proof techniques have increased dramatically. The impetus for this increase lies in the public key approach to cryptography. In these cryptosystems the encryption and decryption keys are distinct, yet related in an unobvious way, such that the disclosure of one key does not compromise the other. The essence of the idea is that a secret can be hidden in a oneway function, yet the function produces a derivative of the secret that is still useful to the designer. For example, let our se...
Knapsack DieHellman: A New Family of DiffieHellman
, 2005
"... Di#eHellman problems have been widely involved in the design of various cryptographic protocols. Its general family is based on the discrete logarithm over a finite field. Since 2000, its another family which is based on elliptic curve discrete logarithm as well as bilinear pairing, e.g. Weil o ..."
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Di#eHellman problems have been widely involved in the design of various cryptographic protocols. Its general family is based on the discrete logarithm over a finite field. Since 2000, its another family which is based on elliptic curve discrete logarithm as well as bilinear pairing, e.g. Weil or Tate pairing, has been attracted significant studies.
Cryptography from tensor problems (draft)
, 2012
"... We describe a new proposal for a trapdoor oneway function. The new proposal belongs to the “multivariate quadratic ” family but the trapdoor is different from existing methods, and is simpler. Known quantum algorithms do not appear to help an adversary attack this trapdoor. (Beyond the asymptoti ..."
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We describe a new proposal for a trapdoor oneway function. The new proposal belongs to the “multivariate quadratic ” family but the trapdoor is different from existing methods, and is simpler. Known quantum algorithms do not appear to help an adversary attack this trapdoor. (Beyond the asymptotic squarerootspeedup which applies to all oracle search problems.) Keywords: cryptography. Multivariate quadratic cryptosystem, MinRank, tensor rank, postquantum 1
International Journal of Information Security manuscript No. (will be inserted by the editor)
"... Information Abstract In a negative representation a set of elements (the positive representation) is depicted by its complement set. That is, the elements in the positive representation are not explicitly stored, and those in the negative representation are. The concept, feasibility, and properties ..."
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Information Abstract In a negative representation a set of elements (the positive representation) is depicted by its complement set. That is, the elements in the positive representation are not explicitly stored, and those in the negative representation are. The concept, feasibility, and properties of negative representations are explored in the paper; in particular, its potential to address privacy concerns. It is shown that a positive representation consisting of n lbit strings can be represented negatively using only O(ln) strings, through the use of an additional symbol. It is also shown that membership queries for the positive representation can be processed against the negative representation in time no worse than linear in its size, while reconstructing the original positive set from its negative representation is an N Phard problem.
unknown title
"... The literature of cryptography has a curious history. Secrecy, of course, has always played a central role, but until the First World War, important developments appeared in print in a more or less timely fashion and the field moved forward in much the same way as other specialized disciplines. As l ..."
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The literature of cryptography has a curious history. Secrecy, of course, has always played a central role, but until the First World War, important developments appeared in print in a more or less timely fashion and the field moved forward in much the same way as other specialized disciplines. As late as 1918, one of the most influential cryptanalytic papers of the twentieth century, William F. Friedman’s monograph The Index of Coincidence and Its Applications in Cryptography, appeared as a research report of the private Riverbank Laboratories [577]. And this, despite the fact that the work had been done as part of the war effort. In the same year Edward H. Hebern of Oakland, California filed the first patent for a rotor machine [710], the device destined to be a mainstay of military cryptography for nearly 50 years. After the First World War, however, things began to change. U.S. Army and Navy organizations, working entirely in secret, began to make fundamental advances in cryptography. During the thirties and forties a few basic papers did appear in the open literature and several treatises on the subject were published, but the latter were farther and farther behind the state of the art. By the end of the war the transition was complete. With one notable exception, the public literature had died. That exception was Claude Shannon’s paper “The Communication Theory of Secrecy Systems, ” which
Cryptography from tensor problems
, 2012
"... This manuscript describes a proposal for a new trapdoor oneway function of the multivariatequadratic type. It was first posted to the IACR preprint server in May 2012. Subsequently, Enrico Thomae and Christopher Wolf were able to to determine that a smallminors MinRank attack works against this s ..."
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This manuscript describes a proposal for a new trapdoor oneway function of the multivariatequadratic type. It was first posted to the IACR preprint server in May 2012. Subsequently, Enrico Thomae and Christopher Wolf were able to to determine that a smallminors MinRank attack works against this scheme. I would like to thank them for their close study of the proposal. The manuscript follows as originally posted, with the addition of a few references and a brief description of the successful attack (end of Section 4.1). Keywords: cryptography. Multivariate quadratic cryptosystem, MinRank, tensor rank, postquantum 1