diff --git a/crypto/fipsmodule/bn/asm/x86_64-mont5.pl b/crypto/fipsmodule/bn/asm/x86_64-mont5.pl
index 54335cc..012e7aa 100755
--- a/crypto/fipsmodule/bn/asm/x86_64-mont5.pl
+++ b/crypto/fipsmodule/bn/asm/x86_64-mont5.pl
@@ -2088,194 +2088,6 @@
 .size	__bn_post4x_internal,.-__bn_post4x_internal
 ___
 }
-{
-$code.=<<___;
-.globl	bn_from_montgomery
-.type	bn_from_montgomery,\@abi-omnipotent
-.align	32
-bn_from_montgomery:
-.cfi_startproc
-	testl	\$7,`($win64?"48(%rsp)":"%r9d")`
-	jz	bn_from_mont8x
-	xor	%eax,%eax
-	ret
-.cfi_endproc
-.size	bn_from_montgomery,.-bn_from_montgomery
-
-.type	bn_from_mont8x,\@function,6
-.align	32
-bn_from_mont8x:
-.cfi_startproc
-	.byte	0x67
-	mov	%rsp,%rax
-.cfi_def_cfa_register	%rax
-	push	%rbx
-.cfi_push	%rbx
-	push	%rbp
-.cfi_push	%rbp
-	push	%r12
-.cfi_push	%r12
-	push	%r13
-.cfi_push	%r13
-	push	%r14
-.cfi_push	%r14
-	push	%r15
-.cfi_push	%r15
-.Lfrom_prologue:
-
-	shl	\$3,${num}d		# convert $num to bytes
-	lea	($num,$num,2),%r10	# 3*$num in bytes
-	neg	$num
-	mov	($n0),$n0		# *n0
-
-	##############################################################
-	# Ensure that stack frame doesn't alias with $rptr+3*$num
-	# modulo 4096, which covers ret[num], am[num] and n[num]
-	# (see bn_exp.c). The stack is allocated to aligned with
-	# bn_power5's frame, and as bn_from_montgomery happens to be
-	# last operation, we use the opportunity to cleanse it.
-	#
-	lea	-320(%rsp,$num,2),%r11
-	mov	%rsp,%rbp
-	sub	$rptr,%r11
-	and	\$4095,%r11
-	cmp	%r11,%r10
-	jb	.Lfrom_sp_alt
-	sub	%r11,%rbp		# align with $aptr
-	lea	-320(%rbp,$num,2),%rbp	# future alloca(frame+2*$num*8+256)
-	jmp	.Lfrom_sp_done
-
-.align	32
-.Lfrom_sp_alt:
-	lea	4096-320(,$num,2),%r10
-	lea	-320(%rbp,$num,2),%rbp	# future alloca(frame+2*$num*8+256)
-	sub	%r10,%r11
-	mov	\$0,%r10
-	cmovc	%r10,%r11
-	sub	%r11,%rbp
-.Lfrom_sp_done:
-	and	\$-64,%rbp
-	mov	%rsp,%r11
-	sub	%rbp,%r11
-	and	\$-4096,%r11
-	lea	(%rbp,%r11),%rsp
-	mov	(%rsp),%r10
-	cmp	%rbp,%rsp
-	ja	.Lfrom_page_walk
-	jmp	.Lfrom_page_walk_done
-
-.Lfrom_page_walk:
-	lea	-4096(%rsp),%rsp
-	mov	(%rsp),%r10
-	cmp	%rbp,%rsp
-	ja	.Lfrom_page_walk
-.Lfrom_page_walk_done:
-
-	mov	$num,%r10
-	neg	$num
-
-	##############################################################
-	# Stack layout
-	#
-	# +0	saved $num, used in reduction section
-	# +8	&t[2*$num], used in reduction section
-	# +32	saved *n0
-	# +40	saved %rsp
-	# +48	t[2*$num]
-	#
-	mov	$n0,  32(%rsp)
-	mov	%rax, 40(%rsp)		# save original %rsp
-.cfi_cfa_expression	%rsp+40,deref,+8
-.Lfrom_body:
-	mov	$num,%r11
-	lea	48(%rsp),%rax
-	pxor	%xmm0,%xmm0
-	jmp	.Lmul_by_1
-
-.align	32
-.Lmul_by_1:
-	movdqu	($aptr),%xmm1
-	movdqu	16($aptr),%xmm2
-	movdqu	32($aptr),%xmm3
-	movdqa	%xmm0,(%rax,$num)
-	movdqu	48($aptr),%xmm4
-	movdqa	%xmm0,16(%rax,$num)
-	.byte	0x48,0x8d,0xb6,0x40,0x00,0x00,0x00	# lea	64($aptr),$aptr
-	movdqa	%xmm1,(%rax)
-	movdqa	%xmm0,32(%rax,$num)
-	movdqa	%xmm2,16(%rax)
-	movdqa	%xmm0,48(%rax,$num)
-	movdqa	%xmm3,32(%rax)
-	movdqa	%xmm4,48(%rax)
-	lea	64(%rax),%rax
-	sub	\$64,%r11
-	jnz	.Lmul_by_1
-
-	movq	$rptr,%xmm1
-	movq	$nptr,%xmm2
-	.byte	0x67
-	mov	$nptr,%rbp
-	movq	%r10, %xmm3		# -num
-___
-$code.=<<___ if ($addx);
-	leaq	OPENSSL_ia32cap_P(%rip),%r11
-	mov	8(%r11),%r11d
-	and	\$0x80108,%r11d
-	cmp	\$0x80108,%r11d		# check for AD*X+BMI2+BMI1
-	jne	.Lfrom_mont_nox
-
-	lea	(%rax,$num),$rptr
-	call	__bn_sqrx8x_reduction
-	call	__bn_postx4x_internal
-
-	pxor	%xmm0,%xmm0
-	lea	48(%rsp),%rax
-	jmp	.Lfrom_mont_zero
-
-.align	32
-.Lfrom_mont_nox:
-___
-$code.=<<___;
-	call	__bn_sqr8x_reduction
-	call	__bn_post4x_internal
-
-	pxor	%xmm0,%xmm0
-	lea	48(%rsp),%rax
-	jmp	.Lfrom_mont_zero
-
-.align	32
-.Lfrom_mont_zero:
-	mov	40(%rsp),%rsi		# restore %rsp
-.cfi_def_cfa	%rsi,8
-	movdqa	%xmm0,16*0(%rax)
-	movdqa	%xmm0,16*1(%rax)
-	movdqa	%xmm0,16*2(%rax)
-	movdqa	%xmm0,16*3(%rax)
-	lea	16*4(%rax),%rax
-	sub	\$32,$num
-	jnz	.Lfrom_mont_zero
-
-	mov	\$1,%rax
-	mov	-48(%rsi),%r15
-.cfi_restore	%r15
-	mov	-40(%rsi),%r14
-.cfi_restore	%r14
-	mov	-32(%rsi),%r13
-.cfi_restore	%r13
-	mov	-24(%rsi),%r12
-.cfi_restore	%r12
-	mov	-16(%rsi),%rbp
-.cfi_restore	%rbp
-	mov	-8(%rsi),%rbx
-.cfi_restore	%rbx
-	lea	(%rsi),%rsp
-.cfi_def_cfa_register	%rsp
-.Lfrom_epilogue:
-	ret
-.cfi_endproc
-.size	bn_from_mont8x,.-bn_from_mont8x
-___
-}
 }}}
 
 if ($addx) {{{
@@ -3864,10 +3676,6 @@
 	.rva	.LSEH_begin_bn_power5
 	.rva	.LSEH_end_bn_power5
 	.rva	.LSEH_info_bn_power5
-
-	.rva	.LSEH_begin_bn_from_mont8x
-	.rva	.LSEH_end_bn_from_mont8x
-	.rva	.LSEH_info_bn_from_mont8x
 ___
 $code.=<<___ if ($addx);
 	.rva	.LSEH_begin_bn_mulx4x_mont_gather5
@@ -3899,11 +3707,6 @@
 	.byte	9,0,0,0
 	.rva	mul_handler
 	.rva	.Lpower5_prologue,.Lpower5_body,.Lpower5_epilogue	# HandlerData[]
-.align	8
-.LSEH_info_bn_from_mont8x:
-	.byte	9,0,0,0
-	.rva	mul_handler
-	.rva	.Lfrom_prologue,.Lfrom_body,.Lfrom_epilogue		# HandlerData[]
 ___
 $code.=<<___ if ($addx);
 .align	8
diff --git a/crypto/fipsmodule/bn/bn_test.cc b/crypto/fipsmodule/bn/bn_test.cc
index 7d57802..3bcdd63 100644
--- a/crypto/fipsmodule/bn/bn_test.cc
+++ b/crypto/fipsmodule/bn/bn_test.cc
@@ -2788,15 +2788,6 @@
                 words, 13);
       CHECK_ABI(bn_power5, r.data(), a.data(), table.data(), m->d, mont->n0,
                 words, 13);
-      EXPECT_EQ(1, CHECK_ABI(bn_from_montgomery, r.data(), r.data(), nullptr,
-                             m->d, mont->n0, words));
-      EXPECT_EQ(1, CHECK_ABI(bn_from_montgomery, r.data(), a.data(), nullptr,
-                             m->d, mont->n0, words));
-    } else {
-      EXPECT_EQ(0, CHECK_ABI(bn_from_montgomery, r.data(), r.data(), nullptr,
-                             m->d, mont->n0, words));
-      EXPECT_EQ(0, CHECK_ABI(bn_from_montgomery, r.data(), a.data(), nullptr,
-                             m->d, mont->n0, words));
     }
   }
 }
diff --git a/crypto/fipsmodule/bn/bn_tests.txt b/crypto/fipsmodule/bn/bn_tests.txt
index 334ef8b..4bc1fa2 100644
--- a/crypto/fipsmodule/bn/bn_tests.txt
+++ b/crypto/fipsmodule/bn/bn_tests.txt
@@ -10604,6 +10604,76 @@
 E = 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
 M = fef15d5ce4625f1bccfbba49fc8439c72bf8202af039a2259678941b60bb4a8f2987e965d58fd8cf86a856674d519763d0e1211cc9f8596971050d56d9b35db3785866cfbca17cfdbed6060be3629d894f924a89fdc1efc624f80d41a22f19009503fcc3824ef62ccb9208430c26f2d8ceb2c63488ec4c07437aa4c96c43dd8b9289ed00a712ff66ee195dc71f5e4ead02172b63c543d69baf495f5fd63ba7bcc633bd309c016e37736da92129d0b053d4ab28d21ad7d8b6fab2a8bbdc8ee647d2fbcf2cf426cf892e6f5639e0252993965dfb73ccd277407014ea784aaa280cb7b03972bc8b0baa72360bdb44b82415b86b2f260f877791cd33ba8f2d65229b
 
+# The following inputs trigger an edge case between Montgomery reduction and the
+# "almost" reduction variant from https://eprint.iacr.org/2011/239
+
+ModExp = 00
+A = 19c7bc9b97c6083cd7b8d1cd001452c9b67983247169c6532047eb7fc8933014dbf69fee7a358769f1429802c8ea89d4f9ca6ba6f368fbdb1fa5717b4a00
+E = 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
+M = 519b6e57781d40d897ec0c1b648d195526726b295438c9a70928ac25979563d72db91c8c42298a33b572edecdf40904c68a23337aa5341b56e92b0da5041
+
+# To fully exercise BN_mod_exp_mont_consttime codepaths, we generate inputs at
+# different bitwidths. rsaz-avx2.pl only runs at 1024-bit moduli, and
+# x86_64-mont5.pl unrolls 8 64-bit words at a time, so we want to capture both
+# multiples of 512- and non-multiples. Also include moduli that are not quite a
+# full word.
+
+# 512-bit
+ModExp = 00
+A = 8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e
+E = ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
+M = 8f42c9e9e351ba9b32ab0cf69da43f4acf7028d19cff6e5059ea0e3fcc97c97f36a31470044737d4c0c933ac441ecb29e32c81401523afdac7de9c3fd8493c97
+
+# 1024-bit
+# TODO(davidben): This test breaks the RSAZ implementation. Fix it and enable
+# this test.
+# ModExp = 00
+# A = 800000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002f
+# E = ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
+# M = 9da8dc26fdf4d2e49833b240ee552beb7a6e251caa91bfb5d6cafaf8ed9461877fda8f6ac299036d35806bc1ae7872e54eaac1ec6bee6d02c6621a9cf8883b3abc33c49b3e601203e0e86ef8f0562412cc689ee2670704583909ca6d7774c9f9f9f4d77d37fedef9cb51d207cb629ec02fa03b526fd6594bfa8f2da71238a0b7
+
+# 1025-bit
+ModExp = 00
+A = 010000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011
+E = ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
+M = 010223abfdda02e84e11cec8ee7fc784fa135733935f7b9054bb70f1f06d234d76dcf3beed55c7f39e955dc1fef2b65009240fd02f7a1b27a78fc2867144bf666efb929856db9f671c356c4c67a068a70fe83c52eebda03668872fd270d0794f0771d217fb6b93b12529a944f7f0496a9158757c55b8ee14f803f1d2d887e2f561
+
+# 1088-bit
+ModExp = 00
+A = 8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003d
+E = ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
+M = e91f6d748773cb212a23aa348125615123b1800c9ea222c9374c757702ae4140fa333790ed8f6bf60a1d7dda65c2767cc5f33e32e333d19fbfb5a2b85795757c9ca070268763a618e9d33873d28a89bf88acd209efbb15b80cd33b92a6b3a682e1c91782fc24fb86ddff4f809219c977b54b99359094bbcc51dfe17b992ab24b74a17950ad754281
+
+# 1472-bit
+ModExp = 00
+A = 8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001d
+E = ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
+M = a8770362f4bfe4fc1ab0e52705c11a9b6ba235d5a5f22197c2d68e27ed18426ede3316af706aa79bcf943dbd51459eb15ae1f9386216b3f3a847f94440a65b97659bc5ba2adb67173714ecaa886c0b926d7a64ea45576f9d2171784ce7e801724d5b0abfd93357d538ea7ad3ad89a74f4660bdb66dfb5f684dcf00402e3cdf0ab58afd867c943c8f47b80268a789456aa7c50a619dd2f9f5e3f74b5d810f0f8dadbf4ad5b917cdcb156c4c132611c8b3b035118a9e03551f
+
+# 1536-bit
+ModExp = 00
+A = 800000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002
+E = ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
+M = 878cd000778f927b2f1a4b8bac86efd282079a7ac0d25e09ffd2f72fbc282e65e233929d2457c7b1d63c56fb706cdfa04fb87e654c578c98d7cf59c2293dc5641086b68db4867105981daaf147a0ee91f6932ef064deae4142c19e58d50c0686f0eaf778be72450f89a98b4680bbc5ffab942195e44dd20616150fd1deca058068ca31ab2f861e99082588f17a2025bf5e536150142fca3187a259c791fc721430f24d7e338f8dc02e693a7e694d42775e80f7f7c03600b6ae86b4aba2b0e991
+
+# 2048-bit
+ModExp = 00
+A = 8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000f
+E = ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
+M = 9f40a7535c561208ecb38e17c9336d9bc8484d335901b2cd42759cf03689227f6992f10cb6b586d767fbcdf30e9d82a0eda60d2694ccd0194fa96b50b56e0cdeec1951ea9e58b07e334a7f108841a0ab28256917fecea561388807ed124a17386a7a7b501f9cbf3404247a76948d0561e48137d3f9669e36f175731796aeaf78851f7d866917f661422186a4814aa35c066b5a90b9cfc918af769a9f0bb30c12581027df64ac328a0f07dbd20adb704479f6d0f233a131828c71bab19c3c34795ea4fb68aa632c6f688e5b3b84413c9031d8dc251003a590dec0dd09bfa6109ed4570701439b6f265b84ac2170c317357b5fbe5535e2bbdd93c1aacfdaa28c85
+
+# 3072-bit
+ModExp = 00
+A = 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001d
+E = ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
+M = 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
+
+# 4096-bit
+ModExp = 00
+A = 8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001
+E = ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
+M = 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
+
 
 # RSAZ 512-bit.
 #
diff --git a/crypto/fipsmodule/bn/exponentiation.c b/crypto/fipsmodule/bn/exponentiation.c
index f5f9dc2..38013ed 100644
--- a/crypto/fipsmodule/bn/exponentiation.c
+++ b/crypto/fipsmodule/bn/exponentiation.c
@@ -979,13 +979,13 @@
 #if defined(OPENSSL_BN_ASM_MONT5)
   if (window >= 5) {
     window = 5;  // ~5% improvement for RSA2048 sign, and even for RSA4096
-    // reserve space for mont->N.d[] copy
+    // Reserve space for the |mont->N| copy.
     powerbufLen += top * sizeof(mont->N.d[0]);
   }
 #endif
 
   // Allocate a buffer large enough to hold all of the pre-computed
-  // powers of am, am itself and tmp.
+  // powers of |am|, |am| itself, and |tmp|.
   numPowers = 1 << window;
   powerbufLen +=
       sizeof(m->d[0]) *
@@ -1008,7 +1008,7 @@
   }
   OPENSSL_memset(powerbuf, 0, powerbufLen);
 
-  // lay down tmp and am right after powers table
+  // Place |tmp| and |am| right after powers table.
   tmp.d = powerbuf + top * numPowers;
   am.d = tmp.d + top;
   tmp.width = am.width = 0;
@@ -1028,18 +1028,26 @@
   }
 
 #if defined(OPENSSL_BN_ASM_MONT5)
-  // This optimization uses ideas from http://eprint.iacr.org/2011/239,
-  // specifically optimization of cache-timing attack countermeasures
-  // and pre-computation optimization.
-
-  // Dedicated window==4 case improves 512-bit RSA sign by ~15%, but as
-  // 512-bit RSA is hardly relevant, we omit it to spare size...
+  // This optimization uses ideas from https://eprint.iacr.org/2011/239,
+  // specifically optimization of cache-timing attack countermeasures,
+  // pre-computation optimization, and Almost Montgomery Multiplication.
+  //
+  // The paper discusses a 4-bit window to optimize 512-bit modular
+  // exponentiation, used in RSA-1024 with CRT, but RSA-1024 is no longer
+  // important.
+  //
+  // |bn_mul_mont_gather5| and |bn_power5| implement the "almost" reduction
+  // variant, so the values here may not be fully reduced. They are bounded by R
+  // (i.e. they fit in |top| words), not |m|. Additionally, we pass these
+  // "almost" reduced inputs into |bn_mul_mont|, which implements the normal
+  // reduction variant. Given those inputs, |bn_mul_mont| may not give reduced
+  // output, but it will still produce "almost" reduced output.
+  //
+  // TODO(davidben): Using "almost" reduction complicates analysis of this code,
+  // and its interaction with other parts of the project. Determine whether this
+  // is actually necessary for performance.
   if (window == 5 && top > 1) {
-    const BN_ULONG *n0 = mont->n0;
-    BN_ULONG *np;
-
-    // BN_to_montgomery can contaminate words above .top
-    // [in BN_DEBUG[_DEBUG] build]...
+    // Ensure |am| and |tmp| are padded to the right width.
     for (i = am.width; i < top; i++) {
       am.d[i] = 0;
     }
@@ -1047,30 +1055,34 @@
       tmp.d[i] = 0;
     }
 
-    // copy mont->N.d[] to improve cache locality
-    for (np = am.d + top, i = 0; i < top; i++) {
+    // Copy |mont->N| to improve cache locality.
+    BN_ULONG *np = am.d + top;
+    for (i = 0; i < top; i++) {
       np[i] = mont->N.d[i];
     }
 
+    // Fill |powerbuf| with the first 32 powers of |am|.
+    const BN_ULONG *n0 = mont->n0;
     bn_scatter5(tmp.d, top, powerbuf, 0);
     bn_scatter5(am.d, am.width, powerbuf, 1);
     bn_mul_mont(tmp.d, am.d, am.d, np, n0, top);
     bn_scatter5(tmp.d, top, powerbuf, 2);
 
-    // same as above, but uses squaring for 1/2 of operations
+    // Square to compute powers of two.
     for (i = 4; i < 32; i *= 2) {
       bn_mul_mont(tmp.d, tmp.d, tmp.d, np, n0, top);
       bn_scatter5(tmp.d, top, powerbuf, i);
     }
+    // Compute odd powers |i| based on |i - 1|, then all powers |i * 2^j|.
     for (i = 3; i < 8; i += 2) {
-      int j;
       bn_mul_mont_gather5(tmp.d, am.d, powerbuf, np, n0, top, i - 1);
       bn_scatter5(tmp.d, top, powerbuf, i);
-      for (j = 2 * i; j < 32; j *= 2) {
+      for (int j = 2 * i; j < 32; j *= 2) {
         bn_mul_mont(tmp.d, tmp.d, tmp.d, np, n0, top);
         bn_scatter5(tmp.d, top, powerbuf, j);
       }
     }
+    // These two loops are the above with the inner loop unrolled.
     for (; i < 16; i += 2) {
       bn_mul_mont_gather5(tmp.d, am.d, powerbuf, np, n0, top, i - 1);
       bn_scatter5(tmp.d, top, powerbuf, i);
@@ -1137,15 +1149,15 @@
         bn_power5(tmp.d, tmp.d, powerbuf, np, n0, top, val);
       }
     }
-
-    ret = bn_from_montgomery(tmp.d, tmp.d, NULL, np, n0, top);
-    tmp.width = top;
-    if (ret) {
-      if (!BN_copy(rr, &tmp)) {
-        ret = 0;
-      }
-      goto err;  // non-zero ret means it's not error
-    }
+    // The result is now in |tmp| in Montgomery form, but it may not be fully
+    // reduced. This is within bounds for |BN_from_montgomery| (tmp < R <= m*R)
+    // so it will, when converting from Montgomery form, produce a fully reduced
+    // result.
+    //
+    // This differs from Figure 2 of the paper, which uses AMM(h, 1) to convert
+    // from Montgomery form with unreduced output, followed by an extra
+    // reduction step. In the paper's terminology, we replace steps 9 and 10
+    // with MM(h, 1).
   } else
 #endif
   {
@@ -1206,7 +1218,11 @@
     }
   }
 
-  // Convert the final result from montgomery to standard format
+  // Convert the final result from Montgomery to standard format. If we used the
+  // |OPENSSL_BN_ASM_MONT5| codepath, |tmp| may not be fully reduced. It is only
+  // bounded by R rather than |m|. However, that is still within bounds for
+  // |BN_from_montgomery|, which implements full Montgomery reduction, not
+  // "almost" Montgomery reduction.
   if (!BN_from_montgomery(rr, &tmp, mont, ctx)) {
     goto err;
   }
diff --git a/crypto/fipsmodule/bn/internal.h b/crypto/fipsmodule/bn/internal.h
index d6eb684..79b25b04 100644
--- a/crypto/fipsmodule/bn/internal.h
+++ b/crypto/fipsmodule/bn/internal.h
@@ -353,6 +353,9 @@
 // corresponding field in |BN_MONT_CTX|. It returns one if |bn_mul_mont| handles
 // inputs of this size and zero otherwise.
 //
+// If at least one of |ap| or |bp| is fully reduced, |rp| will be fully reduced.
+// If neither is fully-reduced, the output may not be either.
+//
 // TODO(davidben): The x86_64 implementation expects a 32-bit input and masks
 // off upper bits. The aarch64 implementation expects a 64-bit input and does
 // not. |size_t| is the safer option but not strictly correct for x86_64. But
@@ -372,6 +375,10 @@
 // by |ap| modulo |np|, and stores the result in |rp|. The values are |num|
 // words long and represented in Montgomery form. |n0| is a pointer to the
 // corresponding field in |BN_MONT_CTX|.
+//
+// WARNING: This function implements Almost Montgomery Multiplication from
+// https://eprint.iacr.org/2011/239. The inputs do not need to be fully reduced.
+// However, even if they are fully reduced, the output may not be.
 void bn_mul_mont_gather5(BN_ULONG *rp, const BN_ULONG *ap,
                          const BN_ULONG *table, const BN_ULONG *np,
                          const BN_ULONG *n0, int num, int power);
@@ -391,16 +398,12 @@
 // values are |num| words long and represented in Montgomery form. |n0| is a
 // pointer to the corresponding field in |BN_MONT_CTX|. |num| must be divisible
 // by 8.
+//
+// WARNING: This function implements Almost Montgomery Multiplication from
+// https://eprint.iacr.org/2011/239. The inputs do not need to be fully reduced.
+// However, even if they are fully reduced, the output may not be.
 void bn_power5(BN_ULONG *rp, const BN_ULONG *ap, const BN_ULONG *table,
                const BN_ULONG *np, const BN_ULONG *n0, int num, int power);
-
-// bn_from_montgomery converts |ap| from Montgomery form modulo |np| and writes
-// the result in |rp|, each of which is |num| words long. It returns one on
-// success and zero if it cannot handle inputs of length |num|. |n0| is a
-// pointer to the corresponding field in |BN_MONT_CTX|.
-int bn_from_montgomery(BN_ULONG *rp, const BN_ULONG *ap,
-                       const BN_ULONG *not_used, const BN_ULONG *np,
-                       const BN_ULONG *n0, int num);
 #endif  // !OPENSSL_NO_ASM && OPENSSL_X86_64
 
 uint64_t bn_mont_n0(const BIGNUM *n);