Lines Matching refs:integers
185 integers of significant magnitude to resist known cryptanalytic attacks. For example, at the time of this writing a
187 Java \cite{JAVA} only provide instrinsic support for integers which are relatively small and single precision.
210 extending the range of representable integers while using single precision data types.
219 Another auxiliary use of multiple precision integers is high precision floating point data types.
227 Yet another use for large integers is within arithmetic on polynomials of large characteristic (i.e. $GF(p)[x]$ for large $p$).
234 the product of two $n$-bit integers requires at least $2n$ bits of precision to be represented faithfully. A multiple
238 It is possible to implement algorithms which require large integers with fixed precision algorithms. For example, elliptic
239 curve cryptography (\textit{ECC}) is often implemented on smartcards by fixing the precision of the integers to the maximum
241 integers required even if the host platform cannot natively accomodate them\footnote{For example, the average smartcard
248 inputs. That is, the same algorithms based on multiple precision integers can accomodate any reasonable size input
302 a plain integer with no additional multiple-precision members. That is, algorithms that use integers as opposed to
309 must be of the form $q^p$ for $q, p \in \Z^+$. A single precision variable must be able to represent integers in
310 the range $0 \le x < q \beta$ while a double precision variable must be able to represent integers in the range
527 inability to accomodate multiple precision integers is the problem. Futhermore, the solution must be written
564 to use fixed precision data types to create and manipulate multiple precision integers which may represent values
571 multiple precision arithmetic is essentially the same concept. Larger integers are represented by adjoining fixed
726 would occur when operations are performed on the integers. There is a tradeoff between how many default digits to allocate
768 used to dictate the minimum precision of newly initialized mp\_int integers. Ideally, it is at least equal to the smallest
1057 For example, suppose the product of two integers was $x_n = (0x_{n-1}x_{n-2}...x_0)_{\beta}$. The leading zero digit
1637 In common twos complement fixed precision arithmetic negative numbers are easily represented by subtraction from the modulus. For example, with 32-bit integers
1645 The lower level algorithms will add or subtract integers without regard to the sign flag. That is they will add or subtract the magnitude of
1646 the integers respectively.
1649 An unsigned addition of multiple precision integers is performed with the same long-hand algorithm used to add decimal numbers. That is to add the
2462 & algorithm to multiply two integers in roughly $O(2n^2)$ time for \\
2501 Computing the product of two integers in software can be achieved using a trivial adaptation of the standard $O(n^2)$ long-hand multiplication
2825 the use of polynomial basis representation for two integers $a$ and $b$ as $f(x) = \sum_{i=0}^{n} a_i x^i$ and
2840 by multiplying $f(y)g(y)$ involves multiplying integers that are much smaller than either of the inputs.
2907 grows the complexity grows substantially. Ideally solving the system will only involve addition, subtraction and shifting of integers. This
2922 light algebra \cite{KARAP} that the following polynomial is equivalent to multiplication of the two integers the polynomials represent.
3128 The two inputs $a$ and $b$ are first split into three $k$-digit integers $a_0, a_1, a_2$ and $b_0, b_1, b_2$ respectively. From these smaller
3129 integers the coefficients of the polynomial basis representations $f(x)$ and $g(x)$ are known and can be used to find the relations required.
3632 The trick used to optimize the above equation is based on a technique of emulating floating point data types with fixed precision integers. Fixed
3638 value $2.3125$. To multiply two fixed point numbers the integers are multiplied using traditional arithmetic and subsequently normalized by
3639 moving the implied decimal point back to where it should be. For example, with $q = 4$ to multiply the integers $9$ and $5$ they must be converted
5251 various representations of integers. For example, converting from an mp\_int to a string of character.
5679 simply $f'(x) = nx^{n - 1}$. Of particular importance is that this algorithm will be used over the integers not over the a more continuous domain
5767 the integers from $0$ to $\beta - 1$.
5928 The greatest common divisor of two integers $a$ and $b$, often denoted as $(a, b)$ is the largest integer $k$ that is a proper divisor of
5940 \textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\
5966 \textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\
6000 \textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\
6123 The least common multiple of a pair of integers is their product divided by their greatest common divisor. For two integers $a$ and $b$ the
6344 fields of integers. However, the former will be the matter of discussion.
6363 Where $a$, $b$, $p$ and $q$ are all integers. If such a pair of integers $ \left < b, q \right >$ exist than $b$ is the multiplicative inverse of
6456 since the integers $2 \ldots 6$ do not evenly divide $a$. By contrast, $a = 6$ is not prime since $a = 6 = 2 \cdot 3$.
6468 Trial division means to attempt to evenly divide a candidate integer by small prime integers. If the candidate can be evenly divided it obviously
6482 approximately $80\%$ of all candidate integers. The constant \textbf{PRIME\_SIZE} is equal to the number of primes in the test base. The
6538 integers known as Carmichael numbers will be a pseudo-prime to all valid bases. Fortunately such numbers are extremely rare as $n$ grows
6574 candidate integers. The algorithm is based on the observation that if $n - 1 = 2^kr$ and if $b^r \nequiv \pm 1$ then after upto $k - 1$ squarings the