Lines Matching refs:fact
169 When we think of long-hand arithmetic such as addition or multiplication we rarely consider the fact that we instinctively
214 various cryptographic journals, can render algorithms such as RSA and Diffie-Hellman more efficient. In fact, several
228 In fact the library discussed within this text has already been used to form a polynomial basis library\footnote{See \url{http://poly.libtomcrypt.org} for more details.}.
262 the fact that the two integer inputs may be of differing magnitudes. As a result the implementation is not as simple
1058 will not contribute to the precision of the result. In fact, through subsequent operations more leading zero digits would
1197 fact the alias for $b$ is carried through into the second ``for'' loop to clear the excess digits. This optimization
2056 an algorithm despite the fact it arises often in other algorithms. The algorithm is setup much like the lower level algorithm s\_mp\_add since
2398 algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$.
2485 The importance of the multiplier algorithms is for the most part driven by the fact that certain popular public key algorithms are based on modular
2698 $241 \cdot 576$ is in fact $138816$ and the procedure succeeded. If the algorithm is correct and as will be demonstrated shortly more
2839 fact that $f(y)$ and $g(y)$ are much smaller than either $a$ or $b$ respectively. As a result finding the $2n + 1$ relations required
2850 $\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ for small values of $q$. The term ``mirror point'' stems from the fact that
2942 of this system of equations has made Karatsuba fairly popular. In fact the cutoff point is often fairly low\footnote{With LibTomMath 0.18 it is 70 and 109 digits for the Intel P4 and AMD Athlon respectively.}
3213 operator from the C programming language. Line 48 computes $\delta$ using the fact that $1 << k$ is equal to $2^k$.
3219 available but in fact there is. Consider the multiplication of $576$ against $241$. In total there will be nine single precision multiplications
3245 appear twice hence the name ``double product''. Every odd column is made up entirely of double products. In fact every column is made up of double
3332 propagation operations from the inner loop. However, the inner product must still be doubled $O(n^2)$ times. The solution stems from the simple fact
3577 $\left [ 2 \right ] $ & In section 5.3 the fact that every column of a squaring is made up \\
3645 equivalent to one than $2^q/b$ is equivalent to the fixed point approximation of $1/b$ using real arithmetic. Using this fact dividing an integer
3734 In fact the lower $m - 2$ digits will not affect the upper half of the product at all and do not need to be computed.
3746 The next optimization arises from this very fact. Instead of computing $b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ using a full
3880 this is an application of the fact that if $x$ is evenly divisible by any $k \in \Z$ then division in $\Z$ will be congruent to
4272 This observation was used in the MMB \cite{MMB} block cipher to create a diffusion primitive. It used the fact that if $n = 2^{31}$ and $k=1$ that
4387 modulus'' will refer to a modulus of the form $2^p - k$. The word ``restricted'' in this case refers to the fact that it is based on the
4625 one digit than it will always be of the correct form. Otherwise all of the bits above the first digit must be one. This arises from the fact
5316 $-x_s + 1$ arises from the fact that a truncated integer division will give the same quotient for at most $x_s - 1$ values. Next a series of
6237 By fact five,
6243 Subsequently by fact three since $p \equiv (p \mbox{ mod }a) \mbox{ (mod }a\mbox{)}$ then
6530 The Fermat test is probably one the oldest tests to have a non-trivial probability of success. It is based on the fact that if $n$ is in
6531 fact prime then $a^{n} \equiv a \mbox{ (mod }n\mbox{)}$ for all $0 < a < n$. The reason being that if $n$ is prime than the order of