--- /dev/null
+/* Copyright (c) 2015, Google Inc.
+ *
+ * Permission to use, copy, modify, and/or distribute this software for any
+ * purpose with or without fee is hereby granted, provided that the above
+ * copyright notice and this permission notice appear in all copies.
+ *
+ * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
+ * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
+ * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY
+ * SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
+ * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION
+ * OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN
+ * CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */
+
+#include <openssl/base.h>
+
+#include <openssl/ec.h>
+
+#include "internal.h"
+
+// This function looks at 5+1 scalar bits (5 current, 1 adjacent less
+// significant bit), and recodes them into a signed digit for use in fast point
+// multiplication: the use of signed rather than unsigned digits means that
+// fewer points need to be precomputed, given that point inversion is easy (a
+// precomputed point dP makes -dP available as well).
+//
+// BACKGROUND:
+//
+// Signed digits for multiplication were introduced by Booth ("A signed binary
+// multiplication technique", Quart. Journ. Mech. and Applied Math., vol. IV,
+// pt. 2 (1951), pp. 236-240), in that case for multiplication of integers.
+// Booth's original encoding did not generally improve the density of nonzero
+// digits over the binary representation, and was merely meant to simplify the
+// handling of signed factors given in two's complement; but it has since been
+// shown to be the basis of various signed-digit representations that do have
+// further advantages, including the wNAF, using the following general
+// approach:
+//
+// (1) Given a binary representation
+//
+// b_k ... b_2 b_1 b_0,
+//
+// of a nonnegative integer (b_k in {0, 1}), rewrite it in digits 0, 1, -1
+// by using bit-wise subtraction as follows:
+//
+// b_k b_(k-1) ... b_2 b_1 b_0
+// - b_k ... b_3 b_2 b_1 b_0
+// -------------------------------------
+// s_k b_(k-1) ... s_3 s_2 s_1 s_0
+//
+// A left-shift followed by subtraction of the original value yields a new
+// representation of the same value, using signed bits s_i = b_(i+1) - b_i.
+// This representation from Booth's paper has since appeared in the
+// literature under a variety of different names including "reversed binary
+// form", "alternating greedy expansion", "mutual opposite form", and
+// "sign-alternating {+-1}-representation".
+//
+// An interesting property is that among the nonzero bits, values 1 and -1
+// strictly alternate.
+//
+// (2) Various window schemes can be applied to the Booth representation of
+// integers: for example, right-to-left sliding windows yield the wNAF
+// (a signed-digit encoding independently discovered by various researchers
+// in the 1990s), and left-to-right sliding windows yield a left-to-right
+// equivalent of the wNAF (independently discovered by various researchers
+// around 2004).
+//
+// To prevent leaking information through side channels in point multiplication,
+// we need to recode the given integer into a regular pattern: sliding windows
+// as in wNAFs won't do, we need their fixed-window equivalent -- which is a few
+// decades older: we'll be using the so-called "modified Booth encoding" due to
+// MacSorley ("High-speed arithmetic in binary computers", Proc. IRE, vol. 49
+// (1961), pp. 67-91), in a radix-2^5 setting. That is, we always combine five
+// signed bits into a signed digit:
+//
+// s_(4j + 4) s_(4j + 3) s_(4j + 2) s_(4j + 1) s_(4j)
+//
+// The sign-alternating property implies that the resulting digit values are
+// integers from -16 to 16.
+//
+// Of course, we don't actually need to compute the signed digits s_i as an
+// intermediate step (that's just a nice way to see how this scheme relates
+// to the wNAF): a direct computation obtains the recoded digit from the
+// six bits b_(4j + 4) ... b_(4j - 1).
+//
+// This function takes those five bits as an integer (0 .. 63), writing the
+// recoded digit to *sign (0 for positive, 1 for negative) and *digit (absolute
+// value, in the range 0 .. 8). Note that this integer essentially provides the
+// input bits "shifted to the left" by one position: for example, the input to
+// compute the least significant recoded digit, given that there's no bit b_-1,
+// has to be b_4 b_3 b_2 b_1 b_0 0.
+void ec_GFp_nistp_recode_scalar_bits(uint8_t *sign, uint8_t *digit,
+ uint8_t in) {
+ uint8_t s, d;
+
+ s = ~((in >> 5) - 1); /* sets all bits to MSB(in), 'in' seen as
+ * 6-bit value */
+ d = (1 << 6) - in - 1;
+ d = (d & s) | (in & ~s);
+ d = (d >> 1) + (d & 1);
+
+ *sign = s & 1;
+ *digit = d;
+}
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