[PATCH v4 4/6] lib: rsa: generate additional parameters for public key
AKASHI Takahiro
takahiro.akashi at linaro.org
Tue Jan 14 08:15:16 CET 2020
On Wed, Jan 08, 2020 at 07:07:01PM +0100, Heinrich Schuchardt wrote:
>
>
> On 11/21/19 1:11 AM, AKASHI Takahiro wrote:
> >In the current implementation of FIT_SIGNATURE, five parameters for
> >a RSA public key are required while only two of them are essential.
> >(See rsa-mod-exp.h and uImage.FIT/signature.txt)
> >This is a result of considering relatively limited computer power
> >and resources on embedded systems, while such a assumption may not
> >be quite practical for other use cases.
> >
> >In this patch, added is a function, rsa_gen_key_prop(), which will
> >generate additional parameters for other uses, in particular
> >UEFI secure boot, on the fly.
> >
> >Note: the current code uses some "big number" rouKtines from BearSSL
> >for the calculation.
> >
> >Signed-off-by: AKASHI Takahiro <takahiro.akashi at linaro.org>
> >---
> > include/u-boot/rsa-mod-exp.h | 23 ++
> > lib/rsa/Kconfig | 1 +
> > lib/rsa/Makefile | 1 +
> > lib/rsa/rsa-keyprop.c | 725 +++++++++++++++++++++++++++++++++++
> > 4 files changed, 750 insertions(+)
> > create mode 100644 lib/rsa/rsa-keyprop.c
> >
> >diff --git a/include/u-boot/rsa-mod-exp.h b/include/u-boot/rsa-mod-exp.h
> >index 8a428c4b6a1a..1da8af1bb83d 100644
> >--- a/include/u-boot/rsa-mod-exp.h
> >+++ b/include/u-boot/rsa-mod-exp.h
> >@@ -26,6 +26,29 @@ struct key_prop {
> > uint32_t exp_len; /* Exponent length in number of uint8_t */
> > };
> >
> >+/**
> >+ * rsa_gen_key_prop() - Generate key properties of RSA public key
> >+ * @key: Specifies key data in DER format
> >+ * @keylen: Length of @key
> >+ * @prop: Generated key property
> >+ *
> >+ * This function takes a blob of encoded RSA public key data in DER
> >+ * format, parse it and generate all the relevant properties
> >+ * in key_prop structure.
> >+ * Return a pointer to struct key_prop in @prop on success.
> >+ *
> >+ * Return: 0 on success, negative on error
> >+ */
> >+int rsa_gen_key_prop(const void *key, uint32_t keylen, struct key_prop **proc);
> >+
> >+/**
> >+ * rsa_free_key_prop() - Free key properties
> >+ * @prop: Pointer to struct key_prop
> >+ *
> >+ * This function frees all the memories allocated by rsa_gen_key_prop().
> >+ */
> >+void rsa_free_key_prop(struct key_prop *prop);
> >+
> > /**
> > * rsa_mod_exp_sw() - Perform RSA Modular Exponentiation in sw
> > *
> >diff --git a/lib/rsa/Kconfig b/lib/rsa/Kconfig
> >index 71e4c06bf883..d1d6f6cf64a3 100644
> >--- a/lib/rsa/Kconfig
> >+++ b/lib/rsa/Kconfig
> >@@ -33,6 +33,7 @@ config RSA_VERIFY
> > config RSA_VERIFY_WITH_PKEY
> > bool "Execute RSA verification without key parameters from FDT"
> > depends on RSA
> >+ imply RSA_PUBLIC_KEY_PARSER
>
> Do we really need RSA_PUBLIC_KEY_PARSER whenever we use CONFIG_RSA=y?
???
RSA_PUBLIC_KEY_PARSER will be selected only if RSA_VERIFY_WITH_PKEY
is enabled. I avoided to use 'select' here because RSA_PUBLIC_KEY_PARSER
is also 'selectable.'
> E.g. on a system without the UEFI sub-system?
>
> Otherwise simply let RSA_PUBLIC_KEY_PARSER depend on RSA.
Such a dependency sounds odd to me.
>
> > help
> > The standard RSA-signature verification code (FIT_SIGNATURE) uses
> > pre-calculated key properties, that are stored in fdt blob, in
> >diff --git a/lib/rsa/Makefile b/lib/rsa/Makefile
> >index c07305188e0c..14ed3cb4012b 100644
> >--- a/lib/rsa/Makefile
> >+++ b/lib/rsa/Makefile
> >@@ -6,4 +6,5 @@
> > # Wolfgang Denk, DENX Software Engineering, wd at denx.de.
> >
> > obj-$(CONFIG_$(SPL_)RSA_VERIFY) += rsa-verify.o rsa-checksum.o
> >+obj-$(CONFIG_RSA_VERIFY_WITH_PKEY) += rsa-keyprop.o
> > obj-$(CONFIG_RSA_SOFTWARE_EXP) += rsa-mod-exp.o
> >diff --git a/lib/rsa/rsa-keyprop.c b/lib/rsa/rsa-keyprop.c
> >new file mode 100644
> >index 000000000000..9464df009343
> >--- /dev/null
> >+++ b/lib/rsa/rsa-keyprop.c
> >@@ -0,0 +1,725 @@
> >+// SPDX-License-Identifier: GPL-2.0+ and MIT
> >+/*
> >+ * RSA library - generate parameters for a public key
> >+ *
> >+ * Copyright (c) 2019 Linaro Limited
> >+ * Author: AKASHI Takahiro
> >+ *
> >+ * Big number routines in this file come from BearSSL:
> >+ * Copyright (c) 2016 Thomas Pornin <pornin at bolet.org>
> >+ */
> >+
> >+#include <common.h>
> >+#include <image.h>
> >+#include <malloc.h>
> >+#include <asm/byteorder.h>
> >+#include <crypto/internal/rsa.h>
> >+#include <u-boot/rsa-mod-exp.h>
> >+
> >+/**
> >+ * br_dec16be() - Convert 16-bit big-endian integer to native
> >+ * @src: Pointer to data
> >+ * Return: Native-endian integer
> >+ */
> >+static unsigned br_dec16be(const void *src)
> >+{
> >+ return be16_to_cpup(src);
> >+}
> >+
> >+/**
> >+ * br_dec32be() - Convert 32-bit big-endian integer to native
> >+ * @src: Pointer to data
> >+ * Return: Native-endian integer
> >+ */
> >+static uint32_t br_dec32be(const void *src)
> >+{
> >+ return be32_to_cpup(src);
> >+}
> >+
> >+/**
> >+ * br_enc32be() - Convert native 32-bit integer to big-endian
> >+ * @dst: Pointer to buffer to store big-endian integer in
> >+ * @x: Native 32-bit integer
> >+ */
> >+static void br_enc32be(void *dst, uint32_t x)
> >+{
> >+ __be32 tmp;
> >+
> >+ tmp = cpu_to_be32(x);
> >+ memcpy(dst, &tmp, sizeof(tmp));
> >+}
> >+
> >+/* from BearSSL's src/inner.h */
> >+
> >+/*
> >+ * Negate a boolean.
> >+ */
> >+static uint32_t NOT(uint32_t ctl)
> >+{
> >+ return ctl ^ 1;
> >+}
> >+
> >+/*
> >+ * Multiplexer: returns x if ctl == 1, y if ctl == 0.
> >+ */
> >+static uint32_t MUX(uint32_t ctl, uint32_t x, uint32_t y)
> >+{
> >+ return y ^ (-ctl & (x ^ y));
> >+}
> >+
> >+/*
> >+ * Equality check: returns 1 if x == y, 0 otherwise.
> >+ */
> >+static uint32_t EQ(uint32_t x, uint32_t y)
> >+{
> >+ uint32_t q;
> >+
> >+ q = x ^ y;
> >+ return NOT((q | -q) >> 31);
> >+}
> >+
> >+/*
> >+ * Inequality check: returns 1 if x != y, 0 otherwise.
> >+ */
> >+static uint32_t NEQ(uint32_t x, uint32_t y)
> >+{
> >+ uint32_t q;
> >+
> >+ q = x ^ y;
> >+ return (q | -q) >> 31;
>
> We want to minimize the code size of U-Boot.
I don't get your point. Please elaborate your concern.
> So, please, review this code and remove all of this bogus.
Which part of the code do you mention as 'bogus'?
Thanks,
-Takahiro Akashi
>
> Best regards
>
> Heinrich
>
> >+}
> >+
> >+/*
> >+ * Comparison: returns 1 if x > y, 0 otherwise.
> >+ */
> >+static uint32_t GT(uint32_t x, uint32_t y)
> >+{
> >+ /*
> >+ * If both x < 2^31 and y < 2^31, then y-x will have its high
> >+ * bit set if x > y, cleared otherwise.
> >+ *
> >+ * If either x >= 2^31 or y >= 2^31 (but not both), then the
> >+ * result is the high bit of x.
> >+ *
> >+ * If both x >= 2^31 and y >= 2^31, then we can virtually
> >+ * subtract 2^31 from both, and we are back to the first case.
> >+ * Since (y-2^31)-(x-2^31) = y-x, the subtraction is already
> >+ * fine.
> >+ */
> >+ uint32_t z;
> >+
> >+ z = y - x;
> >+ return (z ^ ((x ^ y) & (x ^ z))) >> 31;
> >+}
> >+
> >+/*
> >+ * Compute the bit length of a 32-bit integer. Returned value is between 0
> >+ * and 32 (inclusive).
> >+ */
> >+static uint32_t BIT_LENGTH(uint32_t x)
> >+{
> >+ uint32_t k, c;
> >+
> >+ k = NEQ(x, 0);
> >+ c = GT(x, 0xFFFF); x = MUX(c, x >> 16, x); k += c << 4;
> >+ c = GT(x, 0x00FF); x = MUX(c, x >> 8, x); k += c << 3;
> >+ c = GT(x, 0x000F); x = MUX(c, x >> 4, x); k += c << 2;
> >+ c = GT(x, 0x0003); x = MUX(c, x >> 2, x); k += c << 1;
> >+ k += GT(x, 0x0001);
> >+ return k;
> >+}
> >+
> >+#define GE(x, y) NOT(GT(y, x))
> >+#define LT(x, y) GT(y, x)
> >+#define MUL(x, y) ((uint64_t)(x) * (uint64_t)(y))
> >+
> >+/*
> >+ * Integers 'i32'
> >+ * --------------
> >+ *
> >+ * The 'i32' functions implement computations on big integers using
> >+ * an internal representation as an array of 32-bit integers. For
> >+ * an array x[]:
> >+ * -- x[0] contains the "announced bit length" of the integer
> >+ * -- x[1], x[2]... contain the value in little-endian order (x[1]
> >+ * contains the least significant 32 bits)
> >+ *
> >+ * Multiplications rely on the elementary 32x32->64 multiplication.
> >+ *
> >+ * The announced bit length specifies the number of bits that are
> >+ * significant in the subsequent 32-bit words. Unused bits in the
> >+ * last (most significant) word are set to 0; subsequent words are
> >+ * uninitialized and need not exist at all.
> >+ *
> >+ * The execution time and memory access patterns of all computations
> >+ * depend on the announced bit length, but not on the actual word
> >+ * values. For modular integers, the announced bit length of any integer
> >+ * modulo n is equal to the actual bit length of n; thus, computations
> >+ * on modular integers are "constant-time" (only the modulus length may
> >+ * leak).
> >+ */
> >+
> >+/*
> >+ * Extract one word from an integer. The offset is counted in bits.
> >+ * The word MUST entirely fit within the word elements corresponding
> >+ * to the announced bit length of a[].
> >+ */
> >+static uint32_t br_i32_word(const uint32_t *a, uint32_t off)
> >+{
> >+ size_t u;
> >+ unsigned j;
> >+
> >+ u = (size_t)(off >> 5) + 1;
> >+ j = (unsigned)off & 31;
> >+ if (j == 0) {
> >+ return a[u];
> >+ } else {
> >+ return (a[u] >> j) | (a[u + 1] << (32 - j));
> >+ }
> >+}
> >+
> >+/* from BearSSL's src/int/i32_bitlen.c */
> >+
> >+/*
> >+ * Compute the actual bit length of an integer. The argument x should
> >+ * point to the first (least significant) value word of the integer.
> >+ * The len 'xlen' contains the number of 32-bit words to access.
> >+ *
> >+ * CT: value or length of x does not leak.
> >+ */
> >+static uint32_t br_i32_bit_length(uint32_t *x, size_t xlen)
> >+{
> >+ uint32_t tw, twk;
> >+
> >+ tw = 0;
> >+ twk = 0;
> >+ while (xlen -- > 0) {
> >+ uint32_t w, c;
> >+
> >+ c = EQ(tw, 0);
> >+ w = x[xlen];
> >+ tw = MUX(c, w, tw);
> >+ twk = MUX(c, (uint32_t)xlen, twk);
> >+ }
> >+ return (twk << 5) + BIT_LENGTH(tw);
> >+}
> >+
> >+/* from BearSSL's src/int/i32_decode.c */
> >+
> >+/*
> >+ * Decode an integer from its big-endian unsigned representation. The
> >+ * "true" bit length of the integer is computed, but all words of x[]
> >+ * corresponding to the full 'len' bytes of the source are set.
> >+ *
> >+ * CT: value or length of x does not leak.
> >+ */
> >+static void br_i32_decode(uint32_t *x, const void *src, size_t len)
> >+{
> >+ const unsigned char *buf;
> >+ size_t u, v;
> >+
> >+ buf = src;
> >+ u = len;
> >+ v = 1;
> >+ for (;;) {
> >+ if (u < 4) {
> >+ uint32_t w;
> >+
> >+ if (u < 2) {
> >+ if (u == 0) {
> >+ break;
> >+ } else {
> >+ w = buf[0];
> >+ }
> >+ } else {
> >+ if (u == 2) {
> >+ w = br_dec16be(buf);
> >+ } else {
> >+ w = ((uint32_t)buf[0] << 16)
> >+ | br_dec16be(buf + 1);
> >+ }
> >+ }
> >+ x[v ++] = w;
> >+ break;
> >+ } else {
> >+ u -= 4;
> >+ x[v ++] = br_dec32be(buf + u);
> >+ }
> >+ }
> >+ x[0] = br_i32_bit_length(x + 1, v - 1);
> >+}
> >+
> >+/* from BearSSL's src/int/i32_encode.c */
> >+
> >+/*
> >+ * Encode an integer into its big-endian unsigned representation. The
> >+ * output length in bytes is provided (parameter 'len'); if the length
> >+ * is too short then the integer is appropriately truncated; if it is
> >+ * too long then the extra bytes are set to 0.
> >+ */
> >+static void br_i32_encode(void *dst, size_t len, const uint32_t *x)
> >+{
> >+ unsigned char *buf;
> >+ size_t k;
> >+
> >+ buf = dst;
> >+
> >+ /*
> >+ * Compute the announced size of x in bytes; extra bytes are
> >+ * filled with zeros.
> >+ */
> >+ k = (x[0] + 7) >> 3;
> >+ while (len > k) {
> >+ *buf ++ = 0;
> >+ len --;
> >+ }
> >+
> >+ /*
> >+ * Now we use k as index within x[]. That index starts at 1;
> >+ * we initialize it to the topmost complete word, and process
> >+ * any remaining incomplete word.
> >+ */
> >+ k = (len + 3) >> 2;
> >+ switch (len & 3) {
> >+ case 3:
> >+ *buf ++ = x[k] >> 16;
> >+ /* fall through */
> >+ case 2:
> >+ *buf ++ = x[k] >> 8;
> >+ /* fall through */
> >+ case 1:
> >+ *buf ++ = x[k];
> >+ k --;
> >+ }
> >+
> >+ /*
> >+ * Encode all complete words.
> >+ */
> >+ while (k > 0) {
> >+ br_enc32be(buf, x[k]);
> >+ k --;
> >+ buf += 4;
> >+ }
> >+}
> >+
> >+/* from BearSSL's src/int/i32_ninv32.c */
> >+
> >+/*
> >+ * Compute -(1/x) mod 2^32. If x is even, then this function returns 0.
> >+ */
> >+static uint32_t br_i32_ninv32(uint32_t x)
> >+{
> >+ uint32_t y;
> >+
> >+ y = 2 - x;
> >+ y *= 2 - y * x;
> >+ y *= 2 - y * x;
> >+ y *= 2 - y * x;
> >+ y *= 2 - y * x;
> >+ return MUX(x & 1, -y, 0);
> >+}
> >+
> >+/* from BearSSL's src/int/i32_add.c */
> >+
> >+/*
> >+ * Add b[] to a[] and return the carry (0 or 1). If ctl is 0, then a[]
> >+ * is unmodified, but the carry is still computed and returned. The
> >+ * arrays a[] and b[] MUST have the same announced bit length.
> >+ *
> >+ * a[] and b[] MAY be the same array, but partial overlap is not allowed.
> >+ */
> >+static uint32_t br_i32_add(uint32_t *a, const uint32_t *b, uint32_t ctl)
> >+{
> >+ uint32_t cc;
> >+ size_t u, m;
> >+
> >+ cc = 0;
> >+ m = (a[0] + 63) >> 5;
> >+ for (u = 1; u < m; u ++) {
> >+ uint32_t aw, bw, naw;
> >+
> >+ aw = a[u];
> >+ bw = b[u];
> >+ naw = aw + bw + cc;
> >+
> >+ /*
> >+ * Carry is 1 if naw < aw. Carry is also 1 if naw == aw
> >+ * AND the carry was already 1.
> >+ */
> >+ cc = (cc & EQ(naw, aw)) | LT(naw, aw);
> >+ a[u] = MUX(ctl, naw, aw);
> >+ }
> >+ return cc;
> >+}
> >+
> >+/* from BearSSL's src/int/i32_sub.c */
> >+
> >+/*
> >+ * Subtract b[] from a[] and return the carry (0 or 1). If ctl is 0,
> >+ * then a[] is unmodified, but the carry is still computed and returned.
> >+ * The arrays a[] and b[] MUST have the same announced bit length.
> >+ *
> >+ * a[] and b[] MAY be the same array, but partial overlap is not allowed.
> >+ */
> >+static uint32_t br_i32_sub(uint32_t *a, const uint32_t *b, uint32_t ctl)
> >+{
> >+ uint32_t cc;
> >+ size_t u, m;
> >+
> >+ cc = 0;
> >+ m = (a[0] + 63) >> 5;
> >+ for (u = 1; u < m; u ++) {
> >+ uint32_t aw, bw, naw;
> >+
> >+ aw = a[u];
> >+ bw = b[u];
> >+ naw = aw - bw - cc;
> >+
> >+ /*
> >+ * Carry is 1 if naw > aw. Carry is 1 also if naw == aw
> >+ * AND the carry was already 1.
> >+ */
> >+ cc = (cc & EQ(naw, aw)) | GT(naw, aw);
> >+ a[u] = MUX(ctl, naw, aw);
> >+ }
> >+ return cc;
> >+}
> >+
> >+/* from BearSSL's src/int/i32_div32.c */
> >+
> >+/*
> >+ * Constant-time division. The dividend hi:lo is divided by the
> >+ * divisor d; the quotient is returned and the remainder is written
> >+ * in *r. If hi == d, then the quotient does not fit on 32 bits;
> >+ * returned value is thus truncated. If hi > d, returned values are
> >+ * indeterminate.
> >+ */
> >+static uint32_t br_divrem(uint32_t hi, uint32_t lo, uint32_t d, uint32_t *r)
> >+{
> >+ /* TODO: optimize this */
> >+ uint32_t q;
> >+ uint32_t ch, cf;
> >+ int k;
> >+
> >+ q = 0;
> >+ ch = EQ(hi, d);
> >+ hi = MUX(ch, 0, hi);
> >+ for (k = 31; k > 0; k --) {
> >+ int j;
> >+ uint32_t w, ctl, hi2, lo2;
> >+
> >+ j = 32 - k;
> >+ w = (hi << j) | (lo >> k);
> >+ ctl = GE(w, d) | (hi >> k);
> >+ hi2 = (w - d) >> j;
> >+ lo2 = lo - (d << k);
> >+ hi = MUX(ctl, hi2, hi);
> >+ lo = MUX(ctl, lo2, lo);
> >+ q |= ctl << k;
> >+ }
> >+ cf = GE(lo, d) | hi;
> >+ q |= cf;
> >+ *r = MUX(cf, lo - d, lo);
> >+ return q;
> >+}
> >+
> >+/*
> >+ * Wrapper for br_divrem(); the remainder is returned, and the quotient
> >+ * is discarded.
> >+ */
> >+static uint32_t br_rem(uint32_t hi, uint32_t lo, uint32_t d)
> >+{
> >+ uint32_t r;
> >+
> >+ br_divrem(hi, lo, d, &r);
> >+ return r;
> >+}
> >+
> >+/*
> >+ * Wrapper for br_divrem(); the quotient is returned, and the remainder
> >+ * is discarded.
> >+ */
> >+static uint32_t br_div(uint32_t hi, uint32_t lo, uint32_t d)
> >+{
> >+ uint32_t r;
> >+
> >+ return br_divrem(hi, lo, d, &r);
> >+}
> >+
> >+/* from BearSSL's src/int/i32_muladd.c */
> >+
> >+/*
> >+ * Multiply x[] by 2^32 and then add integer z, modulo m[]. This
> >+ * function assumes that x[] and m[] have the same announced bit
> >+ * length, and the announced bit length of m[] matches its true
> >+ * bit length.
> >+ *
> >+ * x[] and m[] MUST be distinct arrays.
> >+ *
> >+ * CT: only the common announced bit length of x and m leaks, not
> >+ * the values of x, z or m.
> >+ */
> >+static void br_i32_muladd_small(uint32_t *x, uint32_t z, const uint32_t *m)
> >+{
> >+ uint32_t m_bitlen;
> >+ size_t u, mlen;
> >+ uint32_t a0, a1, b0, hi, g, q, tb;
> >+ uint32_t chf, clow, under, over;
> >+ uint64_t cc;
> >+
> >+ /*
> >+ * We can test on the modulus bit length since we accept to
> >+ * leak that length.
> >+ */
> >+ m_bitlen = m[0];
> >+ if (m_bitlen == 0) {
> >+ return;
> >+ }
> >+ if (m_bitlen <= 32) {
> >+ x[1] = br_rem(x[1], z, m[1]);
> >+ return;
> >+ }
> >+ mlen = (m_bitlen + 31) >> 5;
> >+
> >+ /*
> >+ * Principle: we estimate the quotient (x*2^32+z)/m by
> >+ * doing a 64/32 division with the high words.
> >+ *
> >+ * Let:
> >+ * w = 2^32
> >+ * a = (w*a0 + a1) * w^N + a2
> >+ * b = b0 * w^N + b2
> >+ * such that:
> >+ * 0 <= a0 < w
> >+ * 0 <= a1 < w
> >+ * 0 <= a2 < w^N
> >+ * w/2 <= b0 < w
> >+ * 0 <= b2 < w^N
> >+ * a < w*b
> >+ * I.e. the two top words of a are a0:a1, the top word of b is
> >+ * b0, we ensured that b0 is "full" (high bit set), and a is
> >+ * such that the quotient q = a/b fits on one word (0 <= q < w).
> >+ *
> >+ * If a = b*q + r (with 0 <= r < q), we can estimate q by
> >+ * doing an Euclidean division on the top words:
> >+ * a0*w+a1 = b0*u + v (with 0 <= v < w)
> >+ * Then the following holds:
> >+ * 0 <= u <= w
> >+ * u-2 <= q <= u
> >+ */
> >+ a0 = br_i32_word(x, m_bitlen - 32);
> >+ hi = x[mlen];
> >+ memmove(x + 2, x + 1, (mlen - 1) * sizeof *x);
> >+ x[1] = z;
> >+ a1 = br_i32_word(x, m_bitlen - 32);
> >+ b0 = br_i32_word(m, m_bitlen - 32);
> >+
> >+ /*
> >+ * We estimate a divisor q. If the quotient returned by br_div()
> >+ * is g:
> >+ * -- If a0 == b0 then g == 0; we want q = 0xFFFFFFFF.
> >+ * -- Otherwise:
> >+ * -- if g == 0 then we set q = 0;
> >+ * -- otherwise, we set q = g - 1.
> >+ * The properties described above then ensure that the true
> >+ * quotient is q-1, q or q+1.
> >+ */
> >+ g = br_div(a0, a1, b0);
> >+ q = MUX(EQ(a0, b0), 0xFFFFFFFF, MUX(EQ(g, 0), 0, g - 1));
> >+
> >+ /*
> >+ * We subtract q*m from x (with the extra high word of value 'hi').
> >+ * Since q may be off by 1 (in either direction), we may have to
> >+ * add or subtract m afterwards.
> >+ *
> >+ * The 'tb' flag will be true (1) at the end of the loop if the
> >+ * result is greater than or equal to the modulus (not counting
> >+ * 'hi' or the carry).
> >+ */
> >+ cc = 0;
> >+ tb = 1;
> >+ for (u = 1; u <= mlen; u ++) {
> >+ uint32_t mw, zw, xw, nxw;
> >+ uint64_t zl;
> >+
> >+ mw = m[u];
> >+ zl = MUL(mw, q) + cc;
> >+ cc = (uint32_t)(zl >> 32);
> >+ zw = (uint32_t)zl;
> >+ xw = x[u];
> >+ nxw = xw - zw;
> >+ cc += (uint64_t)GT(nxw, xw);
> >+ x[u] = nxw;
> >+ tb = MUX(EQ(nxw, mw), tb, GT(nxw, mw));
> >+ }
> >+
> >+ /*
> >+ * If we underestimated q, then either cc < hi (one extra bit
> >+ * beyond the top array word), or cc == hi and tb is true (no
> >+ * extra bit, but the result is not lower than the modulus). In
> >+ * these cases we must subtract m once.
> >+ *
> >+ * Otherwise, we may have overestimated, which will show as
> >+ * cc > hi (thus a negative result). Correction is adding m once.
> >+ */
> >+ chf = (uint32_t)(cc >> 32);
> >+ clow = (uint32_t)cc;
> >+ over = chf | GT(clow, hi);
> >+ under = ~over & (tb | (~chf & LT(clow, hi)));
> >+ br_i32_add(x, m, over);
> >+ br_i32_sub(x, m, under);
> >+}
> >+
> >+/* from BearSSL's src/int/i32_reduce.c */
> >+
> >+/*
> >+ * Reduce an integer (a[]) modulo another (m[]). The result is written
> >+ * in x[] and its announced bit length is set to be equal to that of m[].
> >+ *
> >+ * x[] MUST be distinct from a[] and m[].
> >+ *
> >+ * CT: only announced bit lengths leak, not values of x, a or m.
> >+ */
> >+static void br_i32_reduce(uint32_t *x, const uint32_t *a, const uint32_t *m)
> >+{
> >+ uint32_t m_bitlen, a_bitlen;
> >+ size_t mlen, alen, u;
> >+
> >+ m_bitlen = m[0];
> >+ mlen = (m_bitlen + 31) >> 5;
> >+
> >+ x[0] = m_bitlen;
> >+ if (m_bitlen == 0) {
> >+ return;
> >+ }
> >+
> >+ /*
> >+ * If the source is shorter, then simply copy all words from a[]
> >+ * and zero out the upper words.
> >+ */
> >+ a_bitlen = a[0];
> >+ alen = (a_bitlen + 31) >> 5;
> >+ if (a_bitlen < m_bitlen) {
> >+ memcpy(x + 1, a + 1, alen * sizeof *a);
> >+ for (u = alen; u < mlen; u ++) {
> >+ x[u + 1] = 0;
> >+ }
> >+ return;
> >+ }
> >+
> >+ /*
> >+ * The source length is at least equal to that of the modulus.
> >+ * We must thus copy N-1 words, and input the remaining words
> >+ * one by one.
> >+ */
> >+ memcpy(x + 1, a + 2 + (alen - mlen), (mlen - 1) * sizeof *a);
> >+ x[mlen] = 0;
> >+ for (u = 1 + alen - mlen; u > 0; u --) {
> >+ br_i32_muladd_small(x, a[u], m);
> >+ }
> >+}
> >+
> >+/**
> >+ * rsa_free_key_prop() - Free key properties
> >+ * @prop: Pointer to struct key_prop
> >+ *
> >+ * This function frees all the memories allocated by rsa_gen_key_prop().
> >+ */
> >+void rsa_free_key_prop(struct key_prop *prop)
> >+{
> >+ if (!prop)
> >+ return;
> >+
> >+ free((void *)prop->modulus);
> >+ free((void *)prop->public_exponent);
> >+ free((void *)prop->rr);
> >+
> >+ free(prop);
> >+}
> >+
> >+/**
> >+ * rsa_gen_key_prop() - Generate key properties of RSA public key
> >+ * @key: Specifies key data in DER format
> >+ * @keylen: Length of @key
> >+ * @prop: Generated key property
> >+ *
> >+ * This function takes a blob of encoded RSA public key data in DER
> >+ * format, parse it and generate all the relevant properties
> >+ * in key_prop structure.
> >+ * Return a pointer to struct key_prop in @prop on success.
> >+ *
> >+ * Return: 0 on success, negative on error
> >+ */
> >+int rsa_gen_key_prop(const void *key, uint32_t keylen, struct key_prop **prop)
> >+{
> >+ struct rsa_key rsa_key;
> >+ uint32_t *n = NULL, *rr = NULL, *rrtmp = NULL;
> >+ const int max_rsa_size = 4096;
> >+ int rlen, i, ret;
> >+
> >+ *prop = calloc(sizeof(**prop), 1);
> >+ n = calloc(sizeof(uint32_t), 1 + (max_rsa_size >> 5));
> >+ rr = calloc(sizeof(uint32_t), 1 + (max_rsa_size >> 5));
> >+ rrtmp = calloc(sizeof(uint32_t), 1 + (max_rsa_size >> 5));
> >+ if (!(*prop) || !n || !rr || !rrtmp) {
> >+ ret = -ENOMEM;
> >+ goto err;
> >+ }
> >+
> >+ ret = rsa_parse_pub_key(&rsa_key, key, keylen);
> >+ if (ret)
> >+ goto err;
> >+
> >+ /* modulus */
> >+ /* removing leading 0's */
> >+ for (i = 0; i < rsa_key.n_sz && !rsa_key.n[i]; i++)
> >+ ;
> >+ (*prop)->num_bits = (rsa_key.n_sz - i) * 8;
> >+ (*prop)->modulus = malloc(rsa_key.n_sz - i);
> >+ if (!(*prop)->modulus) {
> >+ ret = -ENOMEM;
> >+ goto err;
> >+ }
> >+ memcpy((void *)(*prop)->modulus, &rsa_key.n[i], rsa_key.n_sz - i);
> >+
> >+ /* exponent */
> >+ (*prop)->public_exponent = calloc(1, sizeof(uint64_t));
> >+ if (!(*prop)->public_exponent) {
> >+ ret = -ENOMEM;
> >+ goto err;
> >+ }
> >+ memcpy((void *)(*prop)->public_exponent + sizeof(uint64_t)
> >+ - rsa_key.e_sz,
> >+ rsa_key.e, rsa_key.e_sz);
> >+ (*prop)->exp_len = rsa_key.e_sz;
> >+
> >+ /* n0 inverse */
> >+ br_i32_decode(n, &rsa_key.n[i], rsa_key.n_sz - i);
> >+ (*prop)->n0inv = br_i32_ninv32(n[1]);
> >+
> >+ /* R^2 mod n; R = 2^(num_bits) */
> >+ rlen = (*prop)->num_bits * 2; /* #bits of R^2 = (2^num_bits)^2 */
> >+ rr[0] = 0;
> >+ *(uint8_t *)&rr[0] = (1 << (rlen % 8));
> >+ for (i = 1; i < (((rlen + 31) >> 5) + 1); i++)
> >+ rr[i] = 0;
> >+ br_i32_decode(rrtmp, rr, ((rlen + 7) >> 3) + 1);
> >+ br_i32_reduce(rr, rrtmp, n);
> >+
> >+ rlen = ((*prop)->num_bits + 7) >> 3; /* #bytes of R^2 mod n */
> >+ (*prop)->rr = malloc(rlen);
> >+ if (!(*prop)->rr) {
> >+ ret = -ENOMEM;
> >+ goto err;
> >+ }
> >+ br_i32_encode((void *)(*prop)->rr, rlen, rr);
> >+
> >+ return 0;
> >+
> >+err:
> >+ free(n);
> >+ free(rr);
> >+ free(rrtmp);
> >+ rsa_free_key_prop(*prop);
> >+ return ret;
> >+}
> >
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