The factored number was of a special form ; the special number field sieve cannot be used on RSA keys.

Cryptography professor Arjen Lenstra observed that " Last time, it took nine years for us to generalize from a special to a nonspecial, hard-to-factor number " and when asked whether 1024-bit RSA keys are dead, said: " The answer to that question is an unqualified yes.

The large number of operations ( 2 < sup > 128 </ sup >) required to try all possible 128-bit keys is widely considered to be out of reach for conventional digital computing techniques for the foreseeable future.

The large number of combinations available from a small number of keys allows text or commands to be entered with one hand, leaving the other hand free.

Due to the small number of keys required, chording is easily adapted from a desktop to mobile environment.

With eleven keys in a 3 / 4 / 4 arrangement, 43 symbols could be arranged allowing for lowercase text, numbers and a modest number of punctuation symbols to be represented along with a ' shift ' function for accessing uppercase letters.

But it does not scale to support more keys than the number of bits in a single byte or word.

The symbol is then encoded, or converted into a number, based on the status of modifier keys and the desired character encoding.

This led to the addition of the number sign (#, sometimes called ' octothorpe ,' ' pound ' or ' diamond ' in this context-' hash ' or ' gate ' in the UK ) and asterisk or " star " (*) keys as well as a group of keys for menu selection: A, B, C and D. In the end, the lettered keys were dropped from most phones, and it was many years before these keys became widely used for vertical service codes such as * 67 in the United States of America and Canada to suppress caller ID.

It uses an optimized structure to minimize the number of requests to retrieve new data, and allows supporting a repository by securely reuploading most parts of the data without requiring the owner's private keys.

A number of small islands and keys can be found nearby, among them Cayos Zapotillos and Cayos Cochinos.

A universal hashing scheme is a randomized algorithm that selects a hashing function h among a family of such functions, in such a way that the probability of a collision of any two distinct keys is 1 / n, where n is the number of distinct hash values desired — independently of the two keys.

This algorithm has proven to be very fast and of high quality for hashing purposes ( especially hashing of integer number keys ).

" This liminal role is reflected in a number of her cult titles: Apotropaia ( that turns away / protects ); Enodia ( on the way ); Propulaia / Propylaia ( before the gate ); Triodia / Trioditis ( who frequents crossroads ); Klêidouchos ( holding the keys ), etc.

Certain, often more expensive keyboards have a diode between each intersection, allowing the keyboard microcontroller to accurately sense any number of simultaneous keys being pressed, without generating erroneous ghost keys ..

The numbers inside the circle show the number of sharps or flats in the key signature, with the sharp keys going clockwise, and the flat keys counterclockwise from C major ( which has no sharps or flats.

It is one of the ironic quirks of history that the viability and usefulness of nationalism and the territorial state are rapidly dissipating at precisely the time that the nation-state attained its highest number ( approximately 100 ).

It is worth dwelling in some detail on the crisis of this story, because it brings together a number of characteristic elements and makes of them a curious, riddling compound obscurely but centrally significant for Mann's work.

The cyclist, a sufficiently commonplace young fellow, is not named but identified simply as `` Life '' -- that and a license number, which Piepsam uses in addressing him.

Presumably a cocktail party is expected to fulfill the host's desire to get together a number of people who are inadequately acquainted and thereby arrange for bringing the level of acquaintance up to adequacy for future cooperative endeavors.

The number of primitive ideas in systematically-simple theories is reduced to a minimum.

I assume that the number of readers of this anthology who regard themselves as morally perfect is small, and that most readers are willing to consider procedures by which they may gain more insight into themselves and better understanding of others.

That is how the real routine of resistance goes on, and its strength is directly proportionate to the number of insignificant people who can let themselves be taken to pieces, piece by piece, without quitting.

Although the United States and the U.S.S.R. have been arguing whether there shall be four, five or six top assistants, the most important element in the situation is not the number of deputies but the manner in which these deputies are to do their work.

There are a number of other considerations besides this one but it is for the Congress, not the Department of Justice, to balance these various considerations and make a judgment about legislation.

Nonetheless, although few in number they are a stubborn crew, as tenacious of life as the Hardshell Baptists, which suggests that there is some kind of vital principle embodied in their faith.

For that is the one an increasingly large number of prominent Americans are now proposing.

and it is still very far from certain how valid the party's claim is that in `` a growing number of kolkhozes '' the peasants are finding it more profitable, to surrender their private plots to the kolkhoz and to let the latter be turned into something increasingly like a state farm.

Too bad your number is in the directory ''.

The number of countries thus favorably situated is small, but their peoples constitute over half of the population of the underdeveloped world.

The resulting setup, it was declared, `` would be similar to that which is in successful operation in a number of metropolitan counties as large or larger than Rhode Island ''.

The Navy, on April 25, announced it is bringing back the carrier Shangri-La from the Mediterranean, increasing to four the number of attack carriers in the vicinity of Cuba.

There is little doubt that the number of those who wish to serve will be far greater than our capacity to absorb them.

Since broadcast frequencies are very limited in number, these objectives are to some extent inconsistent in that not all of them can be fully realized, and to the extent that each is realized, there is a corresponding reduction of the possibilities for fullest achievement of the others.

During nighttime hours, because of the intense skywave propagation then prevailing, no large number of stations can be permitted to operate on one of these channels, if the wide area service for which these frequencies are assigned is to be rendered satisfactorily by the dominant stations which must be relied upon to render it.

In the daytime, on the other hand, since skywave transmission is relatively inefficient, it is possible to assign a substantially larger number of stations on these channels.

Such additional daytime class 2, assignments are appropriate if optimum use is to be made of these frequencies, and the Commission has over the years made a large number of them.

The Peace Corps can either begin in very low gear, with only preparatory work undertaken between now and when Congress finally appropriates special funds for it -- or it can be launched now and in earnest by executive action, with sufficient funds and made available from existing Mutual Security appropriations to permit a number of substantial projects to start this summer.

Although a large number of coatings systems, particularly at low thicknesses fail cohesively by the cutting mechanism, frequently a second type of cohesive failure may also take place.

The low number of abnormalities compares favourably with the health condition of a Majungasaurus population of which it in 2007 was established that 19 % of individuals showed bone pathologies.

Of the capabilities present in the BBC Micro but absent from the Electron, the teletext-style mode 7 was particularly conspicuous because of the very low memory usage in that mode ( just less than 1 kB ) and the high number of BBC programs that used it.

In physics and optics, the Abbe number, also known as the V-number or constringence of a transparent material, is a measure of the material's dispersion ( variation of refractive index with wavelength ) in relation to the refractive index, with high values of V indicating low dispersion ( low chromatic aberration ).

A packet voice system can produce this low jitter in a number of ways:

This number climbs to 42. 05 main lines / 1000 in Ouagadougou, but still a very low penetration rate.

A number of possible reasons have been proposed, but none have been proven, and none can explain the lack of efficacy in both low TB burden countries ( US ) and high TB burden countries ( India ).

Using this method, scientists and farmers hope to keep the number of resistant genes very low, and use genetic drift to ensure any resistance that does emerge does not spread.

The effects of glaciation include a number of large lakes and a relatively low diversity of freshwater fish.

Cucurbitales form an important component of tropical forests from low to montane forests with greater representation in terms of number of species.

The energy bands each correspond to a large number of discrete quantum states of the electrons, and most of the states with low energy ( closer to the nucleus ) are occupied, up to a particular band called the valence band.

A problem with a low condition number is said to be well-conditioned, while a problem with a high condition number is said to be ill-conditioned.

Though the combination of both companies ' PC manufacturing capacity initially made it the number one, it soon lost the lead and further market share to Dell which squeezed HP on low end PCs.

Conversely, a column with a low reflux ratio must have a large number of stages, thus requiring a taller column.

A small number of bass players tune their strings in fifths, like a cello but an octave lower ( C-G-D-A low to high ).

Statistics on the number of televisions, radios, and computers in use in Eritrea are not available, but it is widely thought that rates of ownership of these items are very low.

This approach means it's always technically possible to succeed with a low step number, yet leaves room for failure on high step numbers.

Classical analog filters are IIR filters, and classical filter theory centers on the determination of transfer functions given by low order rational functions, which can be synthesized using the same small number of reactive components.

The ideal situation, of course, is to have an unbiased estimator with low variance, and also try to limit the number of samples where the error is extreme ( that is, have few outliers ).

This error decreases with frequency, so it is a problem at low frequencies where the number of counts N is small.

The total number of fossil felids that are known to science is low compared to other carnivoran families such as dogs and bears.