diff --git a/include/decaf.h b/include/decaf.h
index f9a62b2..2647aca 100644
--- a/include/decaf.h
+++ b/include/decaf.h
@@ -16,7 +16,8 @@
  * decaf_448_decode can fail because not every sequence of bytes is a valid group
  * element.
  *
- * The formulas contain no data-dependent branches, timing or memory accesses.
+ * The formulas contain no data-dependent branches, timing or memory accesses,
+ * except for decaf_448_base_double_scalarmul_non_secret.
  *
  * This library may support multiple curves eventually.  The Ed448-Goldilocks
  * specific identifiers are prefixed with DECAF_448 or decaf_448.
@@ -116,8 +117,6 @@ extern const struct decaf_448_precomputed_s *decaf_448_precomputed_base API_VIS;
 #ifdef __cplusplus
 extern "C" {
 #endif
-    
-/* TODO: scalar invert? */
 
 /**
  * @brief Read a scalar from wire format or from bytes.
@@ -207,6 +206,17 @@ void decaf_448_scalar_mul (
     const decaf_448_scalar_t b
 ) API_VIS NONNULL3 NOINLINE;
 
+/**
+ * @brief Invert a scalar.  When passed zero, return 0.  The input and output may alias.
+ * @param [in] a A scalar.
+ * @param [out] out 1/a.
+ * @return DECAF_TRUE The input is nonzero.
+ */  
+decaf_bool_t decaf_448_scalar_invert (
+    decaf_448_scalar_t out,
+    const decaf_448_scalar_t a
+) API_VIS NONNULL2 NOINLINE;
+
 /**
  * @brief Copy a scalar.  The scalars may use the same memory, in which
  * case this function does nothing.
diff --git a/src/decaf.c b/src/decaf.c
index 7de3f56..e586c48 100644
--- a/src/decaf.c
+++ b/src/decaf.c
@@ -11,6 +11,7 @@
 #define __STDC_WANT_LIB_EXT1__ 1 /* for memset_s */
 #include "decaf.h"
 #include <string.h>
+#include <assert.h>
 
 #define WBITS DECAF_WORD_BITS
 
@@ -363,6 +364,33 @@ void decaf_448_scalar_mul (
     decaf_448_montmul(out,out,decaf_448_scalar_r2,decaf_448_scalar_p,DECAF_MONTGOMERY_FACTOR);
 }
 
+decaf_bool_t decaf_448_scalar_invert (
+    decaf_448_scalar_t out,
+    const decaf_448_scalar_t a
+) {
+    decaf_448_scalar_t b, ma;
+    int i;
+    decaf_448_montmul(b,decaf_448_scalar_one,decaf_448_scalar_r2,decaf_448_scalar_p,DECAF_MONTGOMERY_FACTOR);
+    decaf_448_montmul(ma,a,decaf_448_scalar_r2,decaf_448_scalar_p,DECAF_MONTGOMERY_FACTOR);
+    for (i=DECAF_448_SCALAR_BITS-1; i>=0; i--) {
+        decaf_448_montmul(b,b,b,decaf_448_scalar_p,DECAF_MONTGOMERY_FACTOR);
+            
+        decaf_word_t w = decaf_448_scalar_p->limb[i/WBITS];
+        if (i<WBITS) {
+            assert(w >= 2);
+            w-=2;
+        }
+        if (1 & w>>(i%WBITS)) {
+            decaf_448_montmul(b,b,ma,decaf_448_scalar_p,DECAF_MONTGOMERY_FACTOR);
+        }
+    }
+
+    decaf_448_montmul(out,b,decaf_448_scalar_one,decaf_448_scalar_p,DECAF_MONTGOMERY_FACTOR);
+    decaf_448_scalar_destroy(b);
+    decaf_448_scalar_destroy(ma);
+    return ~decaf_448_scalar_eq(out,decaf_448_scalar_zero);
+}
+
 void decaf_448_scalar_sub (
     decaf_448_scalar_t out,
     const decaf_448_scalar_t a,
diff --git a/src/decaf_fast.c b/src/decaf_fast.c
index 00fa6de..62857b2 100644
--- a/src/decaf_fast.c
+++ b/src/decaf_fast.c
@@ -340,6 +340,66 @@ void decaf_448_scalar_mul (
     decaf_448_montmul(out,out,decaf_448_scalar_r2,decaf_448_scalar_p,DECAF_MONTGOMERY_FACTOR);
 }
 
+/* PERF: could implement this */
+siv decaf_448_montsqr (
+    decaf_448_scalar_t out,
+    const decaf_448_scalar_t a
+) {
+    decaf_448_montmul(out,a,a,decaf_448_scalar_p,DECAF_MONTGOMERY_FACTOR);
+}
+
+decaf_bool_t decaf_448_scalar_invert (
+    decaf_448_scalar_t out,
+    const decaf_448_scalar_t a
+) {
+    decaf_448_scalar_t chain[7], tmp;
+    decaf_448_montmul(chain[0],a,decaf_448_scalar_r2,decaf_448_scalar_p,DECAF_MONTGOMERY_FACTOR);
+    
+    unsigned int i,j;
+    /* Addition chain generated by a not-too-clever SAGE script.  First part: compute a^(2^222-1) */
+    struct { uint8_t widx, sidx, sct, midx; } muls [] = {
+        {2,0,1,0}, {3,2,1,0}, {4,3,1,0}, {5,4,1,0}, /* 0x3,7,f,1f */
+        {1,5,1,0}, {1,1,3,3}, {6,1,9,1}, {1,6,1,0}, {6,1,18,6}, /* a^(2^37-1) */
+        {1,6,37,6}, {1,1,37,6}, {1,1,111,1} /* a^(2^222-1) */
+    };
+    /* Second part: sliding window */
+    struct { uint8_t sct, midx; } muls1 [] = {
+        {6, 5}, {4, 2}, {3, 0}, {2, 0}, {4, 0}, {8, 5},
+        {2, 0}, {5, 3}, {4, 0}, {4, 0}, {5, 3}, {3, 2},
+        {3, 2}, {3, 2}, {2, 0}, {3, 0}, {4, 2}, {2, 0},
+        {4, 3}, {3, 2}, {2, 0}, {3, 2}, {5, 2}, {3, 2},
+        {2, 0}, {3, 0}, {7, 0}, {5, 0}, {3, 2}, {3, 2},
+        {4, 2}, {5, 0}, {5, 3}, {3, 0}, {2, 0}, {5, 2},
+        {4, 3}, {4, 0}, {3, 2}, {7, 4}, {2, 0}, {2, 0},
+        {2, 0}, {2, 0}, {3, 0}, {5, 2}, {5, 4}, {5, 2},
+        {5, 0}, {2, 0}, {3, 0}, {3, 0}, {2, 0}, {2, 0},
+        {2, 0}, {3, 2}, {2, 0}, {3, 2}, {5, 0}, {4, 0},
+        {6, 4}, {4, 0}
+    };
+    
+    for (i=0; i<sizeof(muls)/sizeof(muls[0]); i++) {
+        decaf_448_montsqr(tmp, chain[muls[i].sidx]);
+        for (j=1; j<muls[i].sct; j++) {
+            decaf_448_montsqr(tmp, tmp);
+        }
+        decaf_448_montmul(chain[muls[i].widx], tmp, chain[muls[i].midx], decaf_448_scalar_p,DECAF_MONTGOMERY_FACTOR);
+    }
+    
+    for (i=0; i<sizeof(muls1)/sizeof(muls1[0]); i++) {
+        decaf_448_montsqr(tmp, chain[1]);
+        for (j=1; j<muls1[i].sct; j++) {
+            decaf_448_montsqr(tmp, tmp);
+        }
+        decaf_448_montmul(chain[1], tmp, chain[muls1[i].midx], decaf_448_scalar_p,DECAF_MONTGOMERY_FACTOR);
+    }
+    
+    decaf_448_montmul(out,chain[1],decaf_448_scalar_one,decaf_448_scalar_p,DECAF_MONTGOMERY_FACTOR);
+    for (i=0; i<sizeof(chain)/sizeof(chain[0]); i++) {
+        decaf_448_scalar_destroy(chain[i]);
+    }
+    return ~decaf_448_scalar_eq(out,decaf_448_scalar_zero);
+}
+
 void decaf_448_scalar_sub (
     decaf_448_scalar_t out,
     const decaf_448_scalar_t a,
diff --git a/test/bench.c b/test/bench.c
index 6a1bed2..167dbc0 100644
--- a/test/bench.c
+++ b/test/bench.c
@@ -276,6 +276,13 @@ int main(int argc, char **argv) {
     when = now() - when;
     printf("decaf mulsc: %5.1fns\n", when * 1e9 / i);
     
+    when = now();
+    for (i=0; i<nbase/10; i++) {
+        decaf_448_scalar_invert(csc,bsc);
+    }
+    when = now() - when;
+    printf("decaf invsc: %5.1fµs\n", when * 1e6 / i);
+    
     memset(&ext,0,sizeof(ext));
     memset(&niels,0,sizeof(niels)); /* avoid assertions in p521 even though this isn't a valid ext or niels */
 
diff --git a/test/test_arithmetic.c b/test/test_arithmetic.c
index 98e8ba8..20acc49 100644
--- a/test/test_arithmetic.c
+++ b/test/test_arithmetic.c
@@ -215,7 +215,8 @@ static mask_t test_add_sub_RAW (
 
 static mask_t test_scalar (
     const mpz_t x,
-    const mpz_t y
+    const mpz_t y,
+    int inv
 ) {
     decaf_448_scalar_t xx,yy,tt;
     mpz_t t;
@@ -236,6 +237,18 @@ static mask_t test_scalar (
     mpz_mul(t,x,y);
     succ &= scalar_assert_eq_gmp("scalar mul",xx,yy,tt,x,y,t);
     
+    if (inv) {
+        decaf_bool_t ret = decaf_448_scalar_invert(tt,xx);
+        if (!mpz_cmp_ui(x,0)) {
+            mpz_set_ui(t,0);
+            succ &= (ret == 0) ? MASK_SUCCESS : MASK_FAILURE;
+        } else {
+            mpz_invert(t,x,mp_scalar_field);
+            succ &= (ret == MASK_SUCCESS) ? MASK_SUCCESS : MASK_FAILURE;
+        }
+        succ &= scalar_assert_eq_gmp("scalar inv",xx,yy,tt,x,y,t);
+    }
+    
     mpz_clear(t);
     
     return succ;
@@ -361,7 +374,7 @@ int test_arithmetic (void) {
         
         succ &= test_add_sub_RAW(x,y,word);
         succ &= test_mul_sqr(x,y,word);
-        succ &= test_scalar(x,y);
+        succ &= test_scalar(x,y,(j%20==0));
         
         if (j < 1000)
             succ &= test_isr(x);