A. Karatsuba and Y. Ofman. Multiplication of Multidigit Numbers on Automata. Soviet Physics - Doklady, 7 (1963), 595-596.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....will be concerned with the e#cient computation of the first n coe#cients of the product h = fg = h0 h1z . If the first n coe#cients of f and g are known beforehand, then we may use any fast multiplication for polynomials in order to achieve this goal, such as divide and conquer multiplication [6, 7], which has a time complexity K(n) O(n log 3 log 2 ) or F.F.T. multiplication [2, 9, 1, 11] which has a time complexity M(n) O(n log n log log n) For simplicity, time complexity stands for the required number of operations in R. Similarly, space complexity will stand for the ....

Karatsuba, A., and Ofman, J. Multiplication of multidigit numbers on automata. Soviet Physics Doklady 7 (1963), 595--596.

....improvements. In [GH00] the authors noticed that one can reduce the number of operations required to add double divisors by distinguishing between possible cases according to the properties of the input divisors. This technique is combined with the use of the Karatsuba multiplication algorithm [KO63] and the Chinese remainder theorem to further reduce the complexity of the overall group operations. The work of [GH00] was generalized by [KGM 02] to genus 3 curves de ned over odd characteristic elds. In particular, they notice that for genus 3 curves there are 6 possible choices for the ....

A. Karatsuba and Y. Ofman. Multiplication of multidigit numbers on automata. Sov. Phys. Dokl. (English translation), 7(7):595-596, 1963.

....in Table 2. Our improvements of the original algorithm proposed by Cantor are mainly based on following techniques: 1. Chinese remainder theorem 2. Montgomery s trick of simultaneous inversions [Coh93, Algorithm 10.3.4] 3. Reordering of normalization step [Tak02] 4. Karatsuba multiplication [KO63] 5. Using Karatsuba to reduce complexity of polynomial reduction 6. Smart choice of HEC An extensive description on how to apply the given techniques to reduce the complexity of the group operation can be found in [KGM 02,Lan02a,Pel02] In the following we only stress the smart choice of ....

A. Karatsuba and Y. Ofman. Multiplication of Multidigit Numbers on Automata. Sov. Phys. Dokl. (English translation), 7(7):595-596, 1963.

....due to certain constraints on the size of m. In particular, for p 3, there does not exist a suitable M(x) polynomial. The authors in [25] consider multiplier architectures for composite fields of the form GF( 3n) 3) using Multi Value Logic (MVL) and a modified version of the Karatsuba algorithm [18] for polynomial multiplication over GF( 3n)3) Elements of GF( 3) 3) are represented as polynomials of maximum degree 2 with coefficients in GF(3) Multiplication in GF(3 ) is achieved in the obvious way. Karatsuba multiplication is combined with modular reduction over GF( 3) TM) to reduce the ....

A. Karatsuba and Y. Ofman. Multiplication of multidigit numbers on automata. Sov. Phys. Dokl. (English translation), 7(7):595 596, 1963.

....improvements. In [GH00] the authors noticed that one can reduce the number of operations required to add double divisors by distinguishing between possible cases according to the properties of the input divisors. This technique is combined with the use of the Karatsuba multiplication algorithm [KO63] and the Chinese remainder theorem to further reduce the complexity of the overall group operations. The work of [GH00] was generalized by [KGM 02] to genus 3 curves defined over odd characteristic fields. In particular, they notice that for genus3 curves there are 6 possible choices for the ....

A. Karatsuba and Y. Ofman. Multiplication of multidigit numbers on automata. Sov. Phys. Dokl. (English translation), 7(7):595--596, 1963.

....compute fg mod (2 in Z= 2 1) here m is the smallest power of 2 with m n 1. Here an integer is, by convention, represented in two s complement notation: a sequence of bits (f 0 ; f 1 ; f k 1 ; f k ) represents f 0 2f 1 2 f k 1 2 f k . History. Karatsuba in [55] was the rst to point out that integer multiplication can be done in subquadratic time. This result is often (e.g. in [29, page 58] credited to Karatsuba and Ofman, because [55] was written by Karatsuba and Ofman; but [55] explicitly credited the algorithm to Karatsuba alone. Toom in [98] was ....

....of bits (f 0 ; f 1 ; f k 1 ; f k ) represents f 0 2f 1 2 f k 1 2 f k . History. Karatsuba in [55] was the rst to point out that integer multiplication can be done in subquadratic time. This result is often (e.g. in [29, page 58] credited to Karatsuba and Ofman, because [55] was written by Karatsuba and Ofman; but [55] explicitly credited the algorithm to Karatsuba alone. Toom in [98] was the rst to point out that integer multiplication can be done in essentially linear time: more precisely, time n exp(O( log n) Sch onhage in [84] independently published the ....

[Article contains additional citation context not shown here]

Anatoly A. Karatsuba, Y. Ofman, Multiplication of multidigit numbers on automata, Soviet Physics Doklady 7 (

....hashing functions and the famous BBR polynomial of Barrington, Beigel and Rudich. 1 Introduction Surprising ideas sometimes lead to considerable improvements in algorithms even for the simplest computational tasks, let us mention here the integer multiplication algorithm of Karatsuba and Ofman [15] and the matrix multiplication algorithm of Strassen [24] A new field with surprising algorithms is quantum computing. The most famous and celebrated results are Shor s algorithm for integer factorization [21] and Grover s databasesearch algorithm [13] Since realizable quantum computers can ....

A. Karatsuba and Y. Ofman. Multiplication of multidigit numbers on automata. Sov. Phys.-Dokl. (Engl. transl.), 7(7):595--596, 1963.

.... 3 1 Introduction The multiplication by integer constants occurs in several problems, such as: algorithms requiring some kind of matrix calculations, for instance the Toom Cooklike algorithms to multiply large multiple precision integers [7] this is an extension of Karatsuba s algorithm [3]) the fast approximate computation of consecutive values of a polynomial [6] we can use an extension of the finite di#erence method [4] that needs multiplications by constants) the generation of integer multiplications by compilers, where one of the arguments is statically known (some ....

A. Karatsuba and Y. Ofman. Multiplication of multidigit numbers on automata. Soviet Phys. Doklady, 7(7):595--596, January 1963.

....trick for multiplications in R[y] For example, the product of 1 4x x 2 3x 3 and 8 x 7x 2 2x 3 is the same as the product of (1 y) 4 3y)x and (8 7y) 1 2y)x with y = x 2 . Notes. Karatsuba multiplication for integers was rst presented by Karatsuba and Ofman in [54], where it was credited to Karatsuba alone. Karatsuba observed that his method, applied recursively, takes time O(b log 2 3 ) to multiply b bit numbers. 6 DANIEL J. BERNSTEIN This was the rst subquadratic time multiplication method. Karatsuba was unable to generalize his trick; apparently he ....

Anatoly A. Karatsuba, Y. Ofman, Multiplication of multidigit numbers on automata, Soviet Physics Doklady 7 (1963), 595-596.

....in subfields, will be briefly described. An inverter over extension fields will be introduced in Subsection 3.3.3. 3.2. 1 Multiplication in GF (2 k ) using the Karatsuba Ofman Algorithm In [Afa90] 1 a method is introduced which allows the application of the Karatsuba Ofman Algorithm (KOA) KO63] Knu81] to the multiplication of finite field elements from GF (2 k ) The elements are represented in standard base. The architecture optimizes the polynomial multiplication, which is the major part in standard base Galois field multiplication. The KOA allows polynomial multiplication with a ....

....separately in the following sections. The basic arithmetic operations, addition and multiplication, which are required for both steps are actually performed in the ground field GF (2 n ) The basic idea of the multiplier introduced here is the application of the KaratsubaOfman Algorithm (KOA) KO63] for efficient multiplication of polynomials over a field F to step 1. Efficient refers to the fact that the algorithm saves multiplications at the 42 Multipliers over Composite Fields 43 cost of extra additions. Hence, if the algorithm is expected to be an improvement in complexity, ....

[Article contains additional citation context not shown here]

A. Karatsuba and Y. Ofman. Multiplication of multidigit numbers on automata. Sov. Phys.-Dokl. (Engl. transl.), 7(7):595--596, 1963.

....modular exponentiation such RSA [107] and Diffie Hellman [36] in software. A very good summary of these methods as they pertain to implementing RSA, and by extension Diffie Hellman, is given by Koc in [68] Several of these techniques such as Montgomery s method [87] the Karatsuba Ofman algorithm [63], and Comba s Method [29] are used to improve the performance of the software implementation described within this dissertation. In addition, Gordon [51] provides a comprehensive overview of optimizations for the modular exponentiation operation that dominates the performance of these ....

A. Karatsuba and Y. Ofman, "Multiplication of multidigit numbers on automata," Soviet Physics - Koklady, vol. 7, 1963, pp. 595-596.

....number of digit products needed for dividing a 2nword number by an n word number. Each Newton step requires two multiplications that are performed by an asymptotically fast algorithm. The only such algorithm which is useful for integers shorter than 400 words is the multiplication algorithm due to Karatsuba and Ofman (1962). An estimate (along the lines suggested by Knuth) of the number R(n) of digit products required by Newton inversion leads to R(n) 2T (4n) 2T (2n) 2T (n) 2T (n=2) where T (n) digit products are needed for the multiplication of n digit numbers. Using the property T (2n) 3T (n) of ....

Karatsuba, A., Ofman, Yu (1962). Multiplication of multidigit numbers on automata. Sov. Phys. Dokl., 7:595--596.

....power of cancellation to compute efficiently has been a recurrent theme in the history of algebraic algorithm design. An early example of this is integer multiplication. Two n digit integers can be multiplied trivially in O(n 2 ) steps. Making clever usage of cancellation, Karatsuba and Ofman [4] obtained a O(n 1:59 ) divide and conquer algorithm. In a breakthrough result, Strassen [7] made elegant usage of cancellation to obtain a surprising O(n 2:81 ) algorithm for matrix multiplication, improving the obvious O(n 3 ) algorithm. Schnorr later showed that the O(n 3 ) algorithm is ....

A. Karatsuba and Y. Ofman, Multiplication of multi-digit numbers on automata,Dokl.Akad. Nauk SSSR 145 (1962), 293-294.

....formal power series. The corresponding complexity results are summarized in table 1. In the table, M(n) denotes the time complexity for fast multiplication (see section 3. 1) basic references for fast integer and polynomial multiplication algorithms are (Knuth, 1981; Nussbaumer, 1981) and (Karatsuba and Ofman, 1962; Toom, 1963b; Cooley and Tukey, 1965; Cook, 1966; Sch onhage and Strassen, 1971; Cantor and Kaltofen, 1991; Heideman et al. 1984) Most of the remaining results are due to Brent and Kung (Brent and Kung, 1975; Brent and Kung, 1978) The result about general composition in nite characteristic is due to ....

Karatsuba, A., Ofman, J. (1963). Multiplication of multidigit numbers on automata. Soviet Physics Doklady, 7:595-596.

No context found.

A. Karatsuba and Y. Ofman. Multiplication of Multidigit Numbers on Automata. Soviet Physics - Doklady, 7 (1963), 595-596.

....polynomials up to a degree of 127. 1 Introduction Multiplying two polynomials eciently is an important issue in a variety of applications, including signal processing, cryptography and coding theory. The present paper provides a generalization and detailed analysis of the algorithm by Karatsuba [2] to multiply two polynomials which was introduced in 1962. The Karatsuba Algorithm (KA) saves coecient multiplications at the cost of extra additions compared to the schoolbook or ordinary multiplication method. We consider the KA to be ecient if the total cost of using it is less than the cost of ....

.... trick we obtain C(x) A 1 (x)B 1 (x)y ( A 0 A 1 (x) B 0 (x) B 1 (x) A 0 (x)B 0 (x) A 1 (x)B 1 (x) y A 0 (x)B 0 (x) After substituting y = x one can obtain the result in R[x] Further discussion can also be found in [5] The KA for degree 1 polynomials was introduced by Karatsuba in [2]. Rather than deriving it via the CRT, we will now develop the KA through simple algebraic manipulations. Consider two degree 1 polynomials A(x) and B(x) A(x) a 1 x a 0 B(x) b 1 x b 0 Let D 0 ; D 1 ; D 0;1 be auxiliary variables with D 0 = a 0 b 0 D 1 = a 1 b 1 D 0;1 = a 0 a 1 ) b 0 ....

A. Karatsuba and Y. Ofman. Multiplication of Multidigit Numbers on Automata. Soviet Physics - Doklady, 7 (1963), 595-596.

No context found.

A. Karatsuba and Y. Ofman. Multiplication of Multidigit Numbers on Automata. Soviet Phys. Doklady, 7(7):595596, January 1963.

No context found.

A. Karatsuba and Y. Ofman. Multiplication of multidigit numbers on automata. Sov. Phys.-Dokl. (Engl. transl.), 7(7):595--596, 1963.

No context found.

A. Karatsuba and J. Ofman. Multiplication of multidigit numbers on automata. Soviet Physics Doklady , 7:595596, 1963.

No context found.

A. A. Karatsuba and Y. Ofman. Multiplication of multidigit numbers on automata. Soviet Physics Doklady, 7:595--596, 1963.

No context found.

A. A. Karatsuba and Y. Ofman. Multiplication of multidigit numbers on automata. Soviet Physics Doklady, 7:595-596, 1963.

No context found.

A. Karatsuba and Y. Ofman. Multiplication of multidigit numbers on automata. Soviet Physics Doklady, 7:595--596, 1963.

No context found.

Anatoly A.Karatsuba and Y.Ofman. Multiplication of multidigit numbers on automata. Soviet Physics Doklady, January 1963.

No context found.

A. Karatsuba and Y. Ofman. Multiplication of multidigit numbers on automata. Sov. Phys.-Dokl. (Engl. transl.), 7(7):595--596, 1963.

No context found.

Karatsuba, A. and Ofman, Y. (1969). Multiplication of multidigit numbers on automata. Soviet Physics Doklady 7, 595--596.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC