
	Where results make sense

Topic: Modular exponentiation

Related Topics

Exponentiation

Odd number

Fermat primality test

Carmichael number

Algorithm

PGP

Demographics of Ireland

List of towns in Denmark

Exponent

	Shor's Algorithm for Quantum Factorization
		of the modular exponentiation can be determined in polynomial time.
		The most difficult part in implementing Shor's algorithm is the construction of an efficient quantum function for modular exponentiation.
		Modular multiplication is merely a composition of conditional additions for each qubit of
	tph.tuwien.ac.at /~oemer/doc/quprog/node18.html (1444 words)

	Modular exponentiation - Wikipedia, the free encyclopedia
		Modular exponentiation is a type of exponentiation performed over a modulus.
		Modular exponentiation problems similar to the one described above are considered easy to do, even if the numbers involved are enormous.
		It is a combination of the previous method and a more general principle called binary exponentiation (also known as exponentiation by squaring).
	en.wikipedia.org /wiki/Modular_exponentiation (1459 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		Modular exponentiation and related mathematical operations are commonly used in a number of applications such as cryptography.
		Modular exponentiation, in turn, is a process of repeated modular multiplication of the form A x B mod M utilizing similarly-sized integers.
		Modular exponentiator 300 of the illustrated embodiment includes a first modular exponentiator 302 and a second modular exponentiator 304 selectively coupled together via a coupling device 306 according to the present invention.
	www.wipo.int /cgi-pct/guest/getbykey5?KEY=03/30015.030410&ELEMENT_SET=DECL (5856 words)

	Encyclopedia: Modular exponentiation
		In mathematics, exponentiation is a process generalized from repeated (or iterated) multiplication, in much the same way that multiplication is a process generalized from repeated addition.
		Modular arithmetic is a system of arithmetic for integers, where numbers wrap around after they reach a certain value â€” the modulus.
		Exponentiating by squaring is an algorithm used for the fast computation of large integer powers of a number x.
	www.nationmaster.com /encyclopedia/Modular-exponentiation (471 words)

	Modular exponentiation calculation apparatus and method - Patent 5870478
		As a result, calculation quantity of the modular exponentiation increases heavily and a method to fastly calculate the modular exponentiation is necessary.
		In case of calculating the modular exponentiation, the modular exponentiation calculation is expanded as a series of modular multiplications.
		It is an object of the present invention to provide a modular exponentiation calculation apparatus and method to quickly calculate the modular exponentiation for modular algorithm in base m (m>2).
	www.freepatentsonline.com /5870478.html (3462 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		In all conventional modular multipliers however, the total number of processing elements required is related to the size of the modular multiplication operands and the number of bits processed per element.
		In one embodiment, the first portion of the described modular multiplication operation is performed during a first time period and the second portion of the described modular multiplication operation is performed during a second time period where the first time period and the second timer period overlap.
		Thereafter, a first portion of a modular multiplication operation is performed using the first computation channel (block 606) and a second portion of the modular multiplication operation is performed using the second computation channel (block 608) prior to the illustrated process terminating (block 610).
	www.wipo.int /cgi-pct/guest/getbykey5?KEY=03/29958.030410&ELEMENT_SET=DECL (6270 words)

	Microsoft Security Bulletin (MS00-024): Frequently Asked Questions
		Modular exponentiation is a mathematical operation that is at the heart of public-key cryptography.
		Modular exponentiation is very computationally-intensive operation, and is rather slow when performed in software.
		The keys are used within the modular exponentiation operation as part of the process of encrypting or decrypting data.
	www.microsoft.com /technet/security/bulletin/fq00-024.mspx (2090 words)

	Fast server-aided computation system and method for modular exponentiation without revealing client's secret to ... (Site not responding. Last check: 2007-10-14)
		However, in a case where an other system other than the IC card performs the exponentiation, the keys d and n must be sent to a system in which the modular exponentiation is operated, thus causing a possibility that the secret exponent d may be revealed to a third party.
		Moreover, though the modular exponentiation may be operated by a CPU (central processing unit) equipped in the IC card, it is difficult to achieve a practically fast processing time by the CPU equipped with the IC card since a capacity for numeric calculation in the IC card's CPU is very limited.
		Now, in case where the modular exponentiation of RSA cryptosystem is operated, the modulus n is a product of two prime numbers p, q each of which has approximately half of number of digits of n, and the prime numbers p, q are usually known to the client.
	www.patentstorm.us /patents/5369708.html (5966 words)

	Secure modular exponentiation with leak minimization for smartcards and other cryptosystems - Patent 6298442 (Site not responding. Last check: 2007-10-14)
		Modular exponentiation with a large exponent is conventionally performed as a series of squaring (multiplication by self) and multiplication (by other than self) operations.
		Another technique of the invention implements modular exponentiation with fixed memory access patterns, so that (for example) information leaked from the memory bus will not be correlated to the secret key in ways that compromise the key.
		Modular multiplication and squaring operations are particularly vulnerable to external monitoring attacks because they involve a relatively high number of state transitions, resulting in significant fluctuations in power consumption.
	www.freepatentsonline.com /6298442.html (7909 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		Modular exponentiation and modular multiplication are the most used operations in current cryptographic systems.
		This is performed with a series of modular multiplications to the extent of its exponent in a certain fashion depending on the exponentiation algorithm used.
		The modular exponentiation system experimental results are analyzed and compared with software and other hardware implementations.
	islab.oregonstate.edu /papers/02Kurniawan.html (394 words)

	Modular exponentiation - Encyclopedia, History, Geography and Biography
		If b, e, and m are non-negative and b < m, then a unique solution c exists and has the property 0 ≤ c < m.
		On the contrary, computing the discrete logarithm (finding b given c, e, and m) is believed to be difficult.
		Modular exponentiation, Straightforward method, Memory-efficient method, An efficient method: the Right to Left Binary Algorithm, Optimized methods, External links, Cryptographic algorithms and Modular arithmetic.
	www.arikah.net /encyclopedia/Modular_exponentiation (1547 words)

	Circuit and method of modulo multiplication - US Patent 6182104 (Site not responding. Last check: 2007-10-14)
		The modular exponentiation mathematics of the RSA algorithm can be efficiently computed using Montgomery's method for modular reduction based on a hardware multiplier.
		Modular exponentiation of large integers can be efficiently computed with repeated modular multiplications and the efficiency of the overall RSA computation is directly related to the speed of the multiplier.
		Although the mathematics of the RSA algorithm with modular exponentiation are straight forward, efficient hardware implementation is not straight forward.
	www.patentstorm.us /patents/6182104.html (11657 words)

	Comparison of Three Modular Reduction Functions - Bosselaers, Govaerts, Vandewalle (ResearchIndex) (Site not responding. Last check: 2007-10-14)
		Three modular reduction algorithms for large integers are compared with respect to their performance in portable software: the classical algorithm, Barrett's algorithm and Montgomery's algorithm.
		For each of these algorithms their application in the modular exponentiation operation is considered.
		Modular exponentiation constitutes the basis of manywell known and widely used public...
	sherry.ifi.unizh.ch /bosselaers93comparison.html (441 words)

	Math Forum - Ask Dr. Math
		In Maple, the % sign refers to the previous answer and the way to force Maple to do modular exponentiation is by using &^ to indicate exponentiation instead of the usual ^ key.
		Date: 03/26/2005 at 17:50:25 From: Vladimir Subject: modular exponentiation An interesting pattern was pointed out in the reponse to this question.
		Date: 03/27/2005 at 04:54:31 From: Doctor Jacques Subject: Re: modular exponentiation Hi Vladimir, Euler's theorem is indeed the right idea, but we need to improve it a little.
	mathforum.org /library/drmath/view/51625.html (1089 words)

	Modular exponentiation as a one-way function
		However, to calculate b from the value y is to solve the modular logarithm problem.
		In discussing the second of two applications below, we see that modular exponentiation has at least one weakness that makes it not quite suitable; there is a kind of security defect.
		Use of modular exponentiation as a one-way function also has a usability defect: even though we have a reasonably fast algorithm for modular exponentiation, the repeated-squaring algorithm, for some applications even this is not fast enough.
	www.cs.brown.edu /courses/cs007/oneway/node3.html (470 words)

	Method for the modular reduction of numbers (EP0449349B1)
		It is known that the exponentiation can be accelerated by iteratively multiplying and squaring, with a modular reduction after each step.
		The invention provides a method of also accelerating the modular reduction at the same time, as a result of which the exponentiation modulo large number can be further accelerated.
		For this purpose, the invention describes a protocol for a modular reduction of a 2n-digit number x in a number system with base b to obtain an n-digit remainder, in which use is made of a specific modulus p which satisfies p = bn - a, where 0
	www.delphion.com /details?&pn=EP00449349B1 (260 words)

	Title page for ETD etd-06092005-122043
		The required modular exponentiation is computed by a series of modular multiplications.
		In these dissertation algorithms for integer multiplication, modular reduction and modular exponentiation, was developed and implemented in software, as required for public-key cryptography.
		This research shows that a total speed improvement of 13% can be achieved on existing modular exponentiation based public-key cryptosystems, in particular for the RSA cryptosystem.
	upetd.up.ac.za /thesis/available/etd-06092005-122043 (241 words)

	ICS 180: Schedule and Handouts
		We covered some basic modular arithmetic in the "primes" handout, and the extended Euclidean algorithm for computing gcd and modular inverses from chapter 4 of Dana's notes.
		We showed that modular exponentation can be done efficiently (polynomial time), but we posed the inverse of the exponentiation, namely the discrete logarithm problem, as a problem for which no known efficient algorithm is known.
		the assumption that modular exponentiation is a one-way function.
	www.ics.uci.edu /~stasio/fall04/outline268.html (976 words)

	[No title]
		Each one has a reciprocal mod n; more general, modular division is defined on these residues.
		Notice that this hand computation also goves an example of how to do modular computations "by hand".
		Choose gmodular division, contrary to assumption.
	math.boisestate.edu /~holmes/holmes/cryptofiles/Notes1.txt (765 words)

	[No title]
		The topics covered in this lecture are RSA modular exponentiation and Timing Attack.
		In the above algorithm, modular exponentiation is accomplished bit by bit, with one modular multiplication performed, at each iteration and an additional modular multiplication performed for each 1 bit.The attack proceeds bit by bit starting with the leftmost bit, b
		For a few values of y and z, the modular multiplication will be extremely slow, and the attacker knows which these are.
	www.bennetyee.org /ucsd-pages/Courses/cse127.w02/lec23 (454 words)

	Informatica (Site not responding. Last check: 2007-10-14)
		Efficient computation of the modular exponentiations is very important and useful for public-key cryptosystems.
		By using the CMM technique of computing the common part from two modular multiplications, the same common part in two modular multiplications can be computed once rather twice, we can thus improve the efficiency of the binary exponentiation algorithm by decreasing the number of modular multiplications.
		Furthermore, by using the proposed parallel SDF-CMM Montgomery binary exponentiation algorithm, on average the total number of single-precision multiplications can be reduced by about 61.3% and 74.1% as compared with Chang-Kuo-Lin's CMM modular exponentiation algorithm and Ha-Moon's CMM Montgomery modular exponentiation algorithm, respectively.
	www.vtex.lt /informatica/htm/INFO600.htm (240 words)

	Lucas Sequences in Cryptography (Site not responding. Last check: 2007-10-14)
		A Lucas sequence is a sequence of integers characterized by two parameters, P and Q. In practice Q is always 1 and the sequence is taken modulo a large integer.
		It uses m modular multiplies and m modular squarings, where m is the bit length of e.
		Because of the smaller modulus used and depending on your modular multiplication algorithm, they are also 50 to 100 percent faster.
	www.amasci.com /~weidai/lucas.html (449 words)

	Review of CHES 2000, August 17-18, 2000, by Joe Marconis
		One technique presented was a modular residue check that has a high probability of finding a random or intermittent arithmetic fault and can also find the majority of permanent faults (e.g.
		Three requirements were specified: concurrent double modular exponentiation at high speed, small form factor and low power consumption, and resistance to side channel attacks.
		right-to-left binary exponentiation algorithm was chosen and a novel circuit configuration and schedule control method for doing the double modular exponentiation calculations were presented.
	www.ieee-security.org /Cipher/ConfReports/2000/CR2000-CHES2000.html (2605 words)

	DC Fast Cryptoserver
		An example is private information retrieval where the querier to a database wishes to hide his query from the database by, in effect, hiding it among a large number of other queries.
		However, as technologies like the cryptoserver reduce the cost of modular exponentiations, cryptographic algorithms and protocols considered too costly and complex for practical use will be used, and these algorithms do not respect these narrow limits.
		We wrote a microbenchmark that repeatedly performs a modular exponentiation using the Atalla-supplied libraries, and compares the result to the correct value.
	www2.parc.com /csl/projects/dc/cryptoserver.html (3586 words)

	Modular Exponentiation Algorithm Analysis (ResearchIndex)
		Abstract: Modular Exponentiation for very large integers is the core of many modern cryptographic algorithms.
		Figure 0 shows how modular exponentiation is implemented in the essence of most algorithms.
		In this project, how the building blocks build up the modular exponentiation in popular algorithms is studied in terms of their complexity, parallelism and latency.
	citeseer.ist.psu.edu /657073.html (167 words)

	sci.crypt: Re: modular exponentiation and parallel Montgomery multipliers
		In reply to: Giox: "modular exponentiation and parallel Montgomery multipliers"
		Reply: Giox: "Re: modular exponentiation and parallel Montgomery multipliers"
		exponentiation modulo p and modulo q, and then recombine the results
	www.derkeiler.com /Newsgroups/sci.crypt/2005-08/1163.html (429 words)

	The International Arab Journal of Information Technology
		This paper provides fast algorithms for modular multiplication and exponentiation that are suitable for implementation in RSA and DSS public key cryptographic schemes.
		The first is a modified convolution algorithm for modular multiplication while the second is a Tabulated Modular Exponentiation (TME) algorithm based on the modified sign-digit algorithm.
		They are found to give significant overall improvement to modular exponentiation over that of the fastest algorithms studied.
	www.iajit.org /ABSTRACTS.htm (1204 words)

	complex modular exponentiation
		Also, You deleted part of the message so I went an retrieved it, this is from your orignal post on this subject: (or I deleted part of the message, one of us did).
		Subtraction trivially works; dunno > about division, and I'm not familiar with exponentiation in the integer > realm (except what obviously follows from what's above, namely ((a mod > n)^b) mod n = (a^b) mod n.)) You clearly say that you are not familiar with exponentiation in the integer realm.
		The suggestions I gave were for exponentiation in the integer realm.
	lists.canonical.org /pipermail/kragen-discuss/1998-December/000013.html (2429 words)

	Zvon - RFC 2522 [Photuris: Session-Key Management Protocol] - Public Value Exchanges
		Only the public-value result of the modular exponentiation with (x) or (y) is sent as the Initiator and Responder Exchange-Value.
		Since these operations involve several time-consuming modular exponentiations, moving them to the "background" substantially improves the apparent execution speed of the Photuris protocol.
		It also reduces CPU loading sufficiently to allow a single public/private key-pair to be used in several closely spaced Photuris executions, when creating Security Associations with several different nodes over a short period of time.
	www.zvon.org /tmRFC/RFC2522/Output/chapter8.html (1030 words)

Try your search on: Qwika (all wikis)