Lines Matching refs:le
310 the range $0 \le x < q \beta$ while a double precision variable must be able to represent integers in the range
311 $0 \le x < q \beta^2$. The extra radix-$q$ factor allows additions and subtractions to proceed without truncation of the
325 range $0 \le x < 10^3$, while a double precision data type may represent a value in the range $0 \le x < 10^5$. Let
782 a subsequent expression (or body of expressions) are to be evaluated upto $c - b$ times so long as $b \le c$. In each
1714 in $\mu$ and finally the digit of the result $c_n$ is truncated within the range $0 \le c_n < \beta$.
1754 the range $0 \le x < 2\beta$ for the algorithms to work correctly. However, it is allowable that a mp\_digit represent a larger range of values. For
1759 data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma \ge 32$.
2154 1. If $b \le 0$ then return(\textit{MP\_OKAY}). \\
2180 $b \le 0$ since the rest of algorithm is only valid when $b > 0$.
2221 1. If $b \le 0$ then return. \\
2222 2. If $a.used \le b$ then do \\
2317 Essentially the loop is a generic version of algorithm mp\_mul\_2 designed to handle any shift count in the range $1 \le x < lg(\beta)$. The $mask$
2350 1. If $b \le 0$ then do \\
2408 1. If $b \le 0$ then do \\
2441 We first avoid cases of $b \le 0$ by simply mp\_zero()'ing the destination in such cases. Next if $2^b$ is larger
2568 is given the count of digits to read from $b$ inside the nested loop. If $pb \le 1$ then no more output digits can be produced and the algorithm
2732 $256$ digits would allow for numbers in the range of $0 \le x < 2^{7168}$ which, is much larger than most public key cryptographic algorithms require.
3187 \hspace{3mm}5.2 If $digs < MP\_ARRAY$ and min$(a.used, b.used) \le \delta$ then \\
3303 The requirement that a mp\_word be able to represent the range $0 \le x < 2 \beta^2$ arises from this
3304 very algorithm. The product $a_{ix}a_{iy}$ will lie in the range $0 \le x \le \beta^2 - 2\beta + 1$ which is obviously less than $\beta^2$ meaning that
3490 5pn +{{q(n^2 + n)} \over 2} \le pn + qn^2
3547 \hspace{3mm}3.2 If $digs < MP\_ARRAY$ and $a.used \le \delta$ then \\
3615 range $0 \le x < c^2$ which can be taken advantage of to create several efficient algorithms. They have also been used to create redundancy check
3659 larger than the dividend. In effect if $a$ is the dividend then $q$ should allow $0 \le \lfloor a/2^q \rfloor \le 1$ in order for this approach
3674 reduction the value of $a$ is bound by $0 \le a \le (b - 1)^2$ meaning that $2^q \ge b^2$ is sufficient to ensure the reciprocal will have enough
3694 $m - 1$'th digit of $a$ will contribute at most a value of $1$ to the quotient because $\beta^k < b$ for any $0 \le k \le m - 1$. Another way to
3697 is bound by $0 \le {a' \over b} < 1$.
3713 $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. By first subtracting $b$ times the quotient and then conditionally subtracting
3742 multiple of the modulus, that is $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. If $b$ is $m$ digits than the
3760 \textbf{Input}. mp\_int $a$, mp\_int $b$ and $\mu = \lfloor \beta^{2m}/b \rfloor, m = \lceil lg_{\beta}(b) \rceil, (0 \le a < b^2, b > 1)$ \\
3800 a multiple of it would either accomplish nothing or actually enlarge the input. The input $a$ must be in the range $0 \le a < b^2$ in order
3874 is restricted to $0 \le x < n^2$. To begin the description some simple number theory facts must be established.
3908 final result of the Montgomery algorithm. If $k > lg(n)$ and $0 \le x < n^2$ then the final result is limited to
3909 $0 \le r < \lfloor x/2^k \rfloor + n$. As a result at most a single subtraction is required to get the residue desired.
4060 \hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\
4100 restrictions on this algorithm are fairly easy to adapt to. First $0 \le x < n^2$ bounds the input to numbers in the same range as
4148 \hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\
4313 This algorithm will reduce $x$ modulo $n - k$ and return the residue. If $0 \le x < (n - k)^2$ then the algorithm will loop almost always
4317 0 \le x < n^2 + k^2 - 2nk
4320 The true bound is $0 \le x < (n - k - 1)^2$ but this has quite a few more terms. The value of $q$ after step 1 is bounded by the following.
4327 $0 \le x < n$. By step four the sum $x + q$ is bounded by
4330 0 \le q + x < (k + 1)n - 2k^2 - 1
4333 With a second pass $q$ will be loosely bounded by $0 \le q < k^2$ after step 2 while $x$ will still be loosely bounded by $0 \le x < n$ after step 3. After the second pass it is highly unlike that the
4335 range $0 \le x < (n - k - 1)^2$.
4397 an input $x$ within the range $0 \le x < n^2$ using only a couple passes of the algorithm demonstrated in figure~\ref{fig:DR}. The implementation
4408 \hspace{11.5mm}($0 \le x < n^2$, $n > 1$, $0 < k < \beta$) \\
4665 all three algorithms have the restriction that $0 \le x < n^2$ and $n > 1$ those limitations are not included in the table.
4699 & terminate within $1 \le k \le 10$ iterations. \\
4869 However, when it is not the remaining $0 < x \le k - 1$ bits can be handled with algorithm~\ref{fig:LTOR}.
4903 A simple modification to the previous algorithm is only generate the upper half of the table in the range $2^{k-1} \le g < 2^k$. Essentially
4905 algorithm values of $g$ in the range $0 \le g < 2^{k-1}$ must be avoided.
4974 This guarantees that any intermediate result is bounded by $0 \le d \le c^2 - 2c + 1$ and can be reduced modulo $c$ quickly using
5049 2 & \mbox{if }k \le 7 \\
5050 3 & \mbox{if }7 < k \le 36 \\
5051 4 & \mbox{if }36 < k \le 140 \\
5052 5 & \mbox{if }140 < k \le 450 \\
5053 6 & \mbox{if }450 < k \le 1303 \\
5054 7 & \mbox{if }1303 < k \le 3529 \\
5142 table will hold the values of $g^x \mbox{ (mod }p\mbox{)}$ for $2^{winsize - 1} \le x < 2^{winsize}$.
5320 y - \hat k x \le y - \hat k x_s\beta^s
5326 y - \hat k x \le y_t\beta^t + \ldots + y_0 - (y_t\beta^t + y_{t-1}\beta^{t-1} - x_s\beta^t + \beta^s)
5332 y - \hat k x \le y_{t-2}\beta^{t-2} + \ldots + y_0 + x_s\beta^s - \beta^s
5338 y_{t-2}\beta^{t-2} + \ldots + y_0 + x_s\beta^s - \beta^s < x_s\beta^s \le x
5341 Which proves that $y - \hat kx \le x$ and by consequence $\hat k \ge k$ which concludes the proof. \textbf{QED}
5351 {{\beta^2 - 1} \over { \beta / 2}} \le 2\beta - {2 \over \beta}
5481 outside their respective boundaries. For example, if $t = 0$ or $i \le 1$ then the digits would be undefined. In those cases the digits should
5648 after step 7.1 will be limited such that $0 \le \lfloor \hat w / b \rfloor < \beta$.
5681 algorithm the $n$'th root $b$ of an integer $a$ is desired such that $b^n \le a$.
5689 \textbf{Output}. $c^b \le a$ \\
5726 root. Ideally this algorithm is meant to find the $n$'th root of an input where $n$ is bounded by $2 \le n \le 5$.
5750 2. If $b \le 0$ return(\textit{MP\_OKAY}) \\
6378 1. If $b \le 0$ then return(\textit{MP\_VAL}). \\
6407 12. While $C \le 0$ do \\
6423 If $b \le 0$ than the modulus is invalid and MP\_VAL is returned. Similarly if both $a$ and $b$ are even then there cannot be a multiplicative
6435 within $1 \le a^{-1} < b$. Step numbers twelve and thirteen adjust the inverse until it is in range. If the original input $a$ is within $0 < a < p$
6469 cannot be prime. By dividing by all primes $1 < p \le \sqrt{n}$ this test can actually prove whether an integer is prime. However, such a test
6478 $3 \le q \le 100$.
6595 \hspace{3mm}6.2 While $j \le (s - 1)$ and $y \nequiv a'$ \\