diff options
Diffstat (limited to 'src/crypto/fipsmodule/ec/util-64.c')
| -rw-r--r-- | src/crypto/fipsmodule/ec/util-64.c | 144 |
1 files changed, 72 insertions, 72 deletions
diff --git a/src/crypto/fipsmodule/ec/util-64.c b/src/crypto/fipsmodule/ec/util-64.c index 40062712..0cb117b4 100644 --- a/src/crypto/fipsmodule/ec/util-64.c +++ b/src/crypto/fipsmodule/ec/util-64.c @@ -21,77 +21,77 @@ #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. */ +// 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; @@ -106,4 +106,4 @@ void ec_GFp_nistp_recode_scalar_bits(uint8_t *sign, uint8_t *digit, *digit = d; } -#endif /* 64_BIT && !WINDOWS */ +#endif // 64_BIT && !WINDOWS |