# TWINKLE

TWINKLE

TWINKLE is a hypothetical integer factorization device described in 1999 by Adi Shamir and purported to be capable of factoring 512-bit integers. The name is an acronym of "The Weizmann Institute Key Locating Engine". It is also a pun on the twinkling LEDs used in the device.

The goal of TWINKLE is to implement the sieving step of the Number Field Sieve algorithm, which is the fastest known algorithm for factoring large integers. The sieving step, at least for 512-bit and larger integers, is the most time consuming step of NFS. It involves testing a large set of numbers for B-'smoothness', i.e, absence of a prime factor greater than a specified bound B.

What is remarkable about TWINKLE is that it is not a purely digital device. It gets its efficiency by eschewing binary arithmetic for an "optical" adder which can add hundreds of thousands of quantities in a single clock cycle.

The key idea used is "time-space inversion". Conventional NFS sieving is carried out one prime at a time. For each prime, all the numbers to be tested for smoothness in the range under consideration which are divisible by that prime have their counter incremented by the logarithm of the prime (similar to the sieve of Eratosthenes). TWINKLE, on the other hand, works one candidate smooth number (call it X) at a time. There is one LED corresponding to each prime smaller than B. At the time instant corresponding to X, the set of LEDs glowing corresponds to the set of primes that divide X. This can be accomplished by having the LED associated with the prime "p" glow once every "p" time instants. Further, the intensity of each LED is proportional to the logarithm of the corresponding prime. Thus, the total intensity equals the sum of the logarithms of all the prime factors of X smaller than B. This intensity is equal to the logarithm of X if and only if X is B-smooth.

Even in PC-based implementations, it's a common optimization to speed up sieving by adding approximate logarithms of small primes together. Similarly, TWINKLE has much room for error in its light measurements; as long as the intensity is at about the right level, the number is very likely to be smooth enough for the purposes of known factoring algorithms. The existence of even one large factor would imply that the logarithm of a large number is missing, resulting in a very low intensity; because most numbers have this property, the device's output would tend to consist of stretches of low intensity output with brief bursts of high intensity output.

In the above it is assumed that X is square-free, i.e it is not divisible by the square of any prime. This is acceptable since the factoring algorithms only require "sufficiently many" smooth numbers, and the "yield" decreases only by a small constant factor due to the square-freeness assumption. There is also the problem of false positives due to the inaccuracy of the optoelectronic hardware, but this is easily solved by adding a PC-based post-processing step for verifying the smoothness of the numbers identified by TWINKLE.

Shamir has estimated that the cost of TWINKLE could be as low as \$5000 per unit with bulk production.

TWINKLE has a successor named TWIRL which is more efficient.

References

* Adi Shamir. [http://www.infosecsys.com/image-files/twinkle.pdf Factoring Large Numbers with the TWINKLE Device] . Extended Abstract. The Weizmann Institute of Science.

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• Twinkle — may refer to one of the following.* Twinkle, the common name for Scintillation (astronomy), a phenomenon in the viewing [e.g. of stars] * Twinkle (dance step), a dance step in some ballroom dances * Twinkle (singer), a British pop singer in the… …   Wikipedia

• Twinkle — Hauptfenster und Adressbuch Basisdaten Entwickler Michel de Boer Aktuelle Version …   Deutsch Wikipedia

• Twinkle — Twinkle …   Википедия

• Twinkle — Twinkle, twinkle, lit|tle star a ↑nursery rhyme (=an old song or poem for children) . The rhyme goes: Twinkle, twinkle, little star,/How I wonder what you are./Up above the world so high,/Like a diamond in the sky …   Dictionary of contemporary English

• Twinkle — Logiciel de téléphonie par internet …   Wikipédia en Français

• Twinkle — Twin kle, v. i. [imp. & p. p. {Twinkled}; p. pr. & vb. n. {Twinkling}.] [OE. twinklen, AS. twinclian; akin to OE. twinken to blink, wink, G. zwinken, zwinkern, and perhaps to E. twitch.] 1. To open and shut the eye rapidly; to blink; to wink.… …   The Collaborative International Dictionary of English

• Twinkle — Twin kle, n. 1. A closing or opening, or a quick motion, of the eye; a wink or sparkle of the eye. [1913 Webster] Suddenly, with twinkle of her eye, The damsel broke his misintended dart. Spenser. [1913 Webster] 2. A brief flash or gleam, esp.… …   The Collaborative International Dictionary of English

• twinkle — [twiŋ′kəl] vi. twinkled, twinkling [ME twinklen < OE twinclian, freq. of base seen in MHG zwinken, to wink] 1. to shine with quick, intermittent flashes of light, as some stars; sparkle 2. to light up, as with amusement: said of the eyes 3. to …   English World dictionary

• twinkle — (v.) O.E. twinclian, frequentative of twincan to wink, blink; related to M.H.G. zwinken, Ger. zwinkern, and probably somehow imitative. The noun is recorded from 1540s. Related: Twinkled; twinkling. Phrase in the twinkling of an eye is attested… …   Etymology dictionary

• twinkle — vb *flash, gleam, glance, glint, sparkle, glitter, glisten, scintillate, coruscate …   New Dictionary of Synonyms