From 26a029d407be480d791972afb5975cf62c9360a6 Mon Sep 17 00:00:00 2001 From: Daniel Baumann Date: Fri, 19 Apr 2024 02:47:55 +0200 Subject: Adding upstream version 124.0.1. Signed-off-by: Daniel Baumann --- security/nss/lib/freebl/mpi/doc/square.txt | 72 ++++++++++++++++++++++++++++++ 1 file changed, 72 insertions(+) create mode 100644 security/nss/lib/freebl/mpi/doc/square.txt (limited to 'security/nss/lib/freebl/mpi/doc/square.txt') diff --git a/security/nss/lib/freebl/mpi/doc/square.txt b/security/nss/lib/freebl/mpi/doc/square.txt new file mode 100644 index 0000000000..edbb97882c --- /dev/null +++ b/security/nss/lib/freebl/mpi/doc/square.txt @@ -0,0 +1,72 @@ +Squaring Algorithm + +When you are squaring a value, you can take advantage of the fact that +half the multiplications performed by the more general multiplication +algorithm (see 'mul.txt' for a description) are redundant when the +multiplicand equals the multiplier. + +In particular, the modified algorithm is: + +k = 0 +for j <- 0 to (#a - 1) + w = c[2*j] + (a[j] ^ 2); + k = w div R + + for i <- j+1 to (#a - 1) + w = (2 * a[j] * a[i]) + k + c[i+j] + c[i+j] = w mod R + k = w div R + endfor + c[i+j] = k; + k = 0; +endfor + +On the surface, this looks identical to the multiplication algorithm; +however, note the following differences: + + - precomputation of the leading term in the outer loop + + - i runs from j+1 instead of from zero + + - doubling of a[i] * a[j] in the inner product + +Unfortunately, the construction of the inner product is such that we +need more than two digits to represent the inner product, in some +cases. In a C implementation, this means that some gymnastics must be +performed in order to handle overflow, for which C has no direct +abstraction. We do this by observing the following: + +If we have multiplied a[i] and a[j], and the product is more than half +the maximum value expressible in two digits, then doubling this result +will overflow into a third digit. If this occurs, we take note of the +overflow, and double it anyway -- C integer arithmetic ignores +overflow, so the two digits we get back should still be valid, modulo +the overflow. + +Having doubled this value, we now have to add in the remainders and +the digits already computed by earlier steps. If we did not overflow +in the previous step, we might still cause an overflow here. That +will happen whenever the maximum value expressible in two digits, less +the amount we have to add, is greater than the result of the previous +step. Thus, the overflow computation is: + + + u = 0 + w = a[i] * a[j] + + if(w > (R - 1)/ 2) + u = 1; + + w = w * 2 + v = c[i + j] + k + + if(u == 0 && (R - 1 - v) < w) + u = 1 + +If there is an overflow, u will be 1, otherwise u will be 0. The rest +of the parameters are the same as they are in the above description. + +------------------------------------------------------------------ + This Source Code Form is subject to the terms of the Mozilla Public + # License, v. 2.0. If a copy of the MPL was not distributed with this + # file, You can obtain one at http://mozilla.org/MPL/2.0/. -- cgit v1.2.3