function to add new shares to a set of existing ones

Author: gryphius

SSSA/sssa.py

@@ -69,3 +69,30 @@ def combine(self, shares):
                 secret[part_index] = (secret[part_index] + working) % self.util.prime
 
         return self.util.merge_ints(secret)
+
+    def add_share(self, shares):
+        # The "raw" secret is padded with \x00 and split into 32 char (64 hex) chunks: for each part there will be a polynomial.
+        # The result of 'create' still has "shares" entries, but each is a \sum_i^parts x_i+y_i.
+        # Reconstruct points on the polynomial(s) from the shares
+        x,y=self.util.shares_to_x_y(shares)
+
+        # Initialize known numbers
+        numbers = []
+        for parts in x:
+            numbers.extend(parts)
+
+        # For each polynomial, interpolate another point
+        new_share = ""
+        for pol_nr in range(len(x)):
+            value = self.util.random()
+            while value in numbers:
+                value = self.util.random()
+            numbers.append(value)
+
+            y_interpolated = self.util.lagrange_interpolate(value,x[pol_nr],y[pol_nr],self.util.prime)
+
+            new_share += self.util.to_base64(value)
+            new_share += self.util.to_base64(y_interpolated)
+
+        shares.append(new_share)
+        return shares

SSSA/utils.py

@@ -107,3 +107,72 @@ def mod_inverse(self, number):
             if number < 0:
                 remainder *= -1
             return (self.prime + remainder) % self.prime
+
+
+    def shares_to_x_y(self,shares):
+        x = []
+        y = []
+        # must be: len(shares[0])//88 == len(shares[0])/88
+        for i in range(len(shares[0])//88):
+            part_x = []
+            part_y = []
+            for share in shares:
+                part_x.append(self.from_base64(share[88*i+0:88*i+44]))
+                part_y.append(self.from_base64(share[88*i+44:88*i+88]))
+            x.append(tuple(part_x))
+            y.append(tuple(part_y))
+        return x,y
+
+    # extended_gcd, divmod, lagrange_interpolate are copied from https://en.wikipedia.org/wiki/Shamir%27s_Secret_Sharing
+
+    def extended_gcd(self, a, b):
+        """
+        Division in integers modulus p means finding the inverse of the
+        denominator modulo p and then multiplying the numerator by this
+        inverse (Note: inverse of A is B such that A*B % p == 1) this can
+        be computed via extended Euclidean algorithm
+        http://en.wikipedia.org/wiki/Modular_multiplicative_inverse#Computation
+        """
+        x = 0
+        last_x = 1
+        y = 1
+        last_y = 0
+        while b != 0:
+            quot = a // b
+            a, b = b, a % b
+            x, last_x = last_x - quot * x, x
+            y, last_y = last_y - quot * y, y
+        return last_x, last_y
+
+    def divmod(self, num, den, p):
+        """Compute num / den modulo prime p
+
+        To explain what this means, the return value will be such that
+        the following is true: den * divmod(num, den, p) % p == num
+        """
+        inv, _ = self.extended_gcd(den, p)
+        return num * inv
+
+    def lagrange_interpolate(self, x, x_s, y_s, p):
+        """
+        Find the y-value for the given x, given n (x, y) points;
+        k points will define a polynomial of up to kth order.
+        """
+        k = len(x_s)
+        assert k == len(set(x_s)), "points must be distinct"
+        def PI(vals):  # upper-case PI -- product of inputs
+            accum = 1
+            for v in vals:
+                accum *= v
+            return accum
+        nums = []  # avoid inexact division
+        dens = []
+        for i in range(k):
+            others = list(x_s)
+            cur = others.pop(i)
+            nums.append(PI(x - o for o in others))
+            dens.append(PI(cur - o for o in others))
+        den = PI(dens)
+        num = sum([self.divmod(nums[i] * den * y_s[i] % p, dens[i], p)
+                for i in range(k)])
+        return (self.divmod(num, den, p) + p) % p

sssa_tests.py

@@ -18,5 +18,17 @@ def test_library_combine(self):
 
         self.assertEqual(sss.combine(shares), "test-pass")
 
+    def test_add_share(self):
+        secret='the cake is a lie'
+        sss = sssa()
+        shares = sss.create(3,5,secret)
+        self.assertEqual(len(shares),5)
+        # restore secret using original shares
+        self.assertEqual(sss.combine(shares[:3]),secret)
+        # add new share and restore secret with a mix of original and new share
+        new_shares = sss.add_share(shares)
+        self.assertEqual(len(new_shares),6)
+        self.assertEqual(sss.combine(new_shares[-3:]),secret)
+
 if __name__ == '__main__':
     unittest.main()