--- /dev/null
+#!python
+# MIT or any Tahoe license.
+
+# range of hash output lengths
+range_L_hash = [128]
+
+lg_M = 53 # lg(required number of signatures before losing security)
+
+limit_bytes = 480000 # limit on signature length
+limit_cost = 500 # limit on Mcycles_Sig + weight_ver*Mcycles_ver
+weight_ver = 1 # how important verification cost is relative to signature cost
+ # (note: setting this too high will just exclude useful candidates)
+
+L_block = 512 # bitlength of hash input blocks
+L_pad = 64 # bitlength of hash padding overhead (for M-D hashes)
+L_label = 80 # bitlength of hash position label
+L_prf = 256 # bitlength of hash output when used as a PRF
+cycles_per_byte = 15.8 # cost of hash
+
+Mcycles_per_block = cycles_per_byte * L_block / (8 * 1000000.0)
+
+
+from math import floor, ceil, log, log1p, pow, e, sqrt
+from sys import stderr
+from gc import collect
+
+def lg(x):
+ return log(x, 2)
+def ln(x):
+ return log(x, e)
+def ceil_log(x, B):
+ return int(ceil(log(x, B)))
+def ceil_div(x, y):
+ return int(ceil(float(x) / float(y)))
+def floor_div(x, y):
+ return int(floor(float(x) / float(y)))
+
+# number of compression function evaluations to hash k hash-outputs
+# we assume that there is a label in each block
+def compressions(k):
+ return ceil_div(k + L_pad, L_block - L_label)
+
+# sum of power series sum([pow(p, i) for i in range(n)])
+def sum_powers(p, n):
+ if p == 1: return n
+ return (pow(p, n) - 1)/(p - 1)
+
+
+def make_candidate(B, K, K1, K2, q, T, T_min, L_hash, lg_N, sig_bytes, c_sign, c_ver, c_ver_pm):
+ Mcycles_sign = c_sign * Mcycles_per_block
+ Mcycles_ver = c_ver * Mcycles_per_block
+ Mcycles_ver_pm = c_ver_pm * Mcycles_per_block
+ cost = Mcycles_sign + weight_ver*Mcycles_ver
+
+ if sig_bytes >= limit_bytes or cost > limit_cost:
+ return []
+
+ return [{
+ 'B': B, 'K': K, 'K1': K1, 'K2': K2, 'q': q, 'T': T,
+ 'T_min': T_min,
+ 'L_hash': L_hash,
+ 'lg_N': lg_N,
+ 'sig_bytes': sig_bytes,
+ 'c_sign': c_sign,
+ 'Mcycles_sign': Mcycles_sign,
+ 'c_ver': c_ver,
+ 'c_ver_pm': c_ver_pm,
+ 'Mcycles_ver': Mcycles_ver,
+ 'Mcycles_ver_pm': Mcycles_ver_pm,
+ 'cost': cost,
+ }]
+
+
+# K1 = size of root Merkle tree
+# K = size of middle Merkle trees
+# K2 = size of leaf Merkle trees
+# q = number of revealed private keys per signed message
+
+# Winternitz with B < 4 is never optimal. For example, going from B=4 to B=2 halves the
+# chain depth, but that is cancelled out by doubling (roughly) the number of digits.
+range_B = xrange(4, 33)
+
+M = pow(2, lg_M)
+
+def calculate(K, K1, K2, q_max, L_hash, trees):
+ candidates = []
+ lg_K = lg(K)
+ lg_K1 = lg(K1)
+ lg_K2 = lg(K2)
+
+ # We want the optimal combination of q and T. That takes too much time and memory
+ # to search for directly, so we start by calculating the lowest possible value of T
+ # for any q. Then for potential values of T, we calculate the smallest q such that we
+ # will have at least L_hash bits of security against forgery using revealed private keys
+ # (i.e. this method of forgery is no easier than finding a hash preimage), provided
+ # that fewer than 2^lg_S_min messages are signed.
+
+ # min height of certification tree (excluding root and bottom layer)
+ T_min = ceil_div(lg_M - lg_K1, lg_K)
+
+ last_q = None
+ for T in xrange(T_min, T_min+21):
+ # lg(total number of leaf private keys)
+ lg_S = lg_K1 + lg_K*T
+ lg_N = lg_S + lg_K2
+
+ # Suppose that m signatures have been made. The number of times X that a given bucket has
+ # been chosen follows a binomial distribution B(m, p) where p = 1/S and S is the number of
+ # buckets. I.e. Pr(X = x) = C(m, x) * p^x * (1-p)^(m-x).
+ #
+ # If an attacker picks a random seed and message that falls into a bucket that has been
+ # chosen x times, then at most q*x private values in that bucket have been revealed, so
+ # (ignoring the possibility of guessing private keys, which is negligable) the attacker's
+ # success probability for a forgery using the revealed values is at most min(1, q*x / K2)^q.
+ #
+ # Let j = floor(K2/q). Conditioning on x, we have
+ #
+ # Pr(forgery) = sum_{x = 0..j}(Pr(X = x) * (q*x / K2)^q) + Pr(x > j)
+ # = sum_{x = 1..j}(Pr(X = x) * (q*x / K2)^q) + Pr(x > j)
+ #
+ # We lose nothing by approximating (q*x / K2)^q as 1 for x > 4, i.e. ignoring the resistance
+ # of the HORS scheme to forgery when a bucket has been chosen 5 or more times.
+ #
+ # Pr(forgery) < sum_{x = 1..4}(Pr(X = x) * (q*x / K2)^q) + Pr(x > 4)
+ #
+ # where Pr(x > 4) = 1 - sum_{x = 0..4}(Pr(X = x))
+ #
+ # We use log arithmetic here because values very close to 1 cannot be represented accurately
+ # in floating point, but their logarithms can (provided we use appropriate functions such as
+ # log1p).
+
+ lg_p = -lg_S
+ lg_1_p = log1p(-pow(2, lg_p))/ln(2) # lg(1-p), computed accurately
+ j = 5
+ lg_px = [lg_1_p * M]*j
+
+ # We approximate lg(M-x) as lg(M)
+ lg_px_step = lg_M + lg_p - lg_1_p
+ for x in xrange(1, j):
+ lg_px[x] = lg_px[x-1] - lg(x) + lg_px_step
+
+ def find_min_q():
+ for q in xrange(1, q_max+1):
+ lg_q = lg(q)
+ lg_pforge = [lg_px[x] + (lg_q*x - lg_K2)*q for x in xrange(1, j)]
+ if max(lg_pforge) < -L_hash + lg(j) and lg_px[j-1] + 1.0 < -L_hash:
+ #print "K = %d, K1 = %d, K2 = %d, L_hash = %d, lg_K2 = %.3f, q = %d, lg_pforge_1 = %.3f, lg_pforge_2 = %.3f, lg_pforge_3 = %.3f" \
+ # % (K, K1, K2, L_hash, lg_K2, q, lg_pforge_1, lg_pforge_2, lg_pforge_3)
+ return q
+ return None
+
+ q = find_min_q()
+ if q is None or q == last_q:
+ # if q hasn't decreased, this will be strictly worse than the previous candidate
+ continue
+ last_q = q
+
+ # number of compressions to compute the Merkle hashes
+ (h_M, c_M, _) = trees[K]
+ (h_M1, c_M1, _) = trees[K1]
+ (h_M2, c_M2, (dau, tri)) = trees[K2]
+
+ # B = generalized Winternitz base
+ for B in range_B:
+ # n is the number of digits needed to sign the message representative and checksum.
+ # The representation is base-B, except that we allow the most significant digit
+ # to be up to 2B-1.
+ n_L = ceil_div(L_hash-1, lg(B))
+ firstL_max = floor_div(pow(2, L_hash)-1, pow(B, n_L-1))
+ C_max = firstL_max + (n_L-1)*(B-1)
+ n_C = ceil_log(ceil_div(C_max, 2), B)
+ n = n_L + n_C
+ firstC_max = floor_div(C_max, pow(B, n_C-1))
+
+ # Total depth of Winternitz hash chains. The chains for the most significant
+ # digit of the message representative and of the checksum may be a different
+ # length to those for the other digits.
+ c_D = (n-2)*(B-1) + firstL_max + firstC_max
+
+ # number of compressions to hash a Winternitz public key
+ c_W = compressions(n*L_hash + L_label)
+
+ # bitlength of a single Winternitz signature and authentication path
+ L_MW = (n + h_M ) * L_hash
+ L_MW1 = (n + h_M1) * L_hash
+
+ # bitlength of the HORS signature and authentication paths
+ # For all but one of the q authentication paths, one of the sibling elements in
+ # another path is made redundant where they intersect. This cancels out the hash
+ # that would otherwise be needed at the bottom of the path, making the total
+ # length of the signature q*h_M2 + 1 hashes, rather than q*(h_M2 + 1).
+ L_leaf = (q*h_M2 + 1) * L_hash
+
+ # length of the overall GMSS+HORS signature and seeds
+ sig_bytes = ceil_div(L_MW1 + T*L_MW + L_leaf + L_prf + ceil(lg_N), 8)
+
+ c_MW = K *(c_D + c_W) + c_M + ceil_div(K *n*L_hash, L_prf)
+ c_MW1 = K1*(c_D + c_W) + c_M1 + ceil_div(K1*n*L_hash, L_prf)
+
+ # For simplicity, c_sign and c_ver don't take into account compressions saved
+ # as a result of intersecting authentication paths in the HORS signature, so
+ # are slight overestimates.
+
+ c_sign = c_MW1 + T*c_MW + q*(c_M2 + 1) + ceil_div(K2*L_hash, L_prf)
+
+ # *expected* number of compressions to verify a signature
+ c_ver = c_D/2.0 + c_W + c_M1 + T*(c_D/2.0 + c_W + c_M) + q*(c_M2 + 1)
+ c_ver_pm = (1 + T)*c_D/2.0
+
+ candidates += make_candidate(B, K, K1, K2, q, T, T_min, L_hash, lg_N, sig_bytes, c_sign, c_ver, c_ver_pm)
+
+ return candidates
+
+def search():
+ for L_hash in range_L_hash:
+ print >>stderr, "collecting... \r",
+ collect()
+
+ print >>stderr, "precomputing... \r",
+
+ """
+ # d/dq (lg(q+1) + L_hash/q) = 1/(ln(2)*(q+1)) - L_hash/q^2
+ # Therefore lg(q+1) + L_hash/q is at a minimum when 1/(ln(2)*(q+1)) = L_hash/q^2.
+ # Let alpha = L_hash*ln(2), then from the quadratic formula, the integer q that
+ # minimizes lg(q+1) + L_hash/q is the floor or ceiling of (alpha + sqrt(alpha^2 - 4*alpha))/2.
+ # (We don't want the other solution near 0.)
+
+ alpha = floor(L_hash*ln(2)) # float
+ q = floor((alpha + sqrt(alpha*(alpha-4)))/2)
+ if lg(q+2) + L_hash/(q+1) < lg(q+1) + L_hash/q:
+ q += 1
+ lg_S_margin = lg(q+1) + L_hash/q
+ q_max = int(q)
+
+ q = floor(L_hash*ln(2)) # float
+ if lg(q+1) + L_hash/(q+1) < lg(q) + L_hash/q:
+ q += 1
+ lg_S_margin = lg(q) + L_hash/q
+ q_max = int(q)
+ """
+ q_max = 4000
+
+ # find optimal Merkle tree shapes for this L_hash and each K
+ trees = {}
+ K_max = 50
+ c2 = compressions(2*L_hash + L_label)
+ c3 = compressions(3*L_hash + L_label)
+ for dau in xrange(0, 10):
+ a = pow(2, dau)
+ for tri in xrange(0, ceil_log(30-dau, 3)):
+ x = int(a*pow(3, tri))
+ h = dau + 2*tri
+ c_x = int(sum_powers(2, dau)*c2 + a*sum_powers(3, tri)*c3)
+ for y in xrange(1, x+1):
+ if tri > 0:
+ # If the bottom level has arity 3, then for every 2 nodes by which the tree is
+ # imperfect, we can save c3 compressions by pruning 3 leaves back to their parent.
+ # If the tree is imperfect by an odd number of nodes, we can prune one extra leaf,
+ # possibly saving a compression if c2 < c3.
+ c_y = c_x - floor_div(x-y, 2)*c3 - ((x-y) % 2)*(c3-c2)
+ else:
+ # If the bottom level has arity 2, then for each node by which the tree is
+ # imperfect, we can save c2 compressions by pruning 2 leaves back to their parent.
+ c_y = c_x - (x-y)*c2
+
+ if y not in trees or (h, c_y, (dau, tri)) < trees[y]:
+ trees[y] = (h, c_y, (dau, tri))
+
+ #for x in xrange(1, K_max+1):
+ # print x, trees[x]
+
+ candidates = []
+ progress = 0
+ fuzz = 0
+ complete = (K_max-1)*(2200-200)/100
+ for K in xrange(2, K_max+1):
+ for K2 in xrange(200, 2200, 100):
+ for K1 in xrange(max(2, K-fuzz), min(K_max, K+fuzz)+1):
+ candidates += calculate(K, K1, K2, q_max, L_hash, trees)
+ progress += 1
+ print >>stderr, "searching: %3d %% \r" % (100.0 * progress / complete,),
+
+ print >>stderr, "filtering... \r",
+ step = 2.0
+ bins = {}
+ limit = floor_div(limit_cost, step)
+ for bin in xrange(0, limit+2):
+ bins[bin] = []
+
+ for c in candidates:
+ bin = floor_div(c['cost'], step)
+ bins[bin] += [c]
+
+ del candidates
+
+ # For each in a range of signing times, find the best candidate.
+ best = []
+ for bin in xrange(0, limit):
+ candidates = bins[bin] + bins[bin+1] + bins[bin+2]
+ if len(candidates) > 0:
+ best += [min(candidates, key=lambda c: c['sig_bytes'])]
+
+ def format_candidate(candidate):
+ return ("%(B)3d %(K)3d %(K1)3d %(K2)5d %(q)4d %(T)4d "
+ "%(L_hash)4d %(lg_N)5.1f %(sig_bytes)7d "
+ "%(c_sign)7d (%(Mcycles_sign)7.2f) "
+ "%(c_ver)7d +/-%(c_ver_pm)5d (%(Mcycles_ver)5.2f +/-%(Mcycles_ver_pm)5.2f) "
+ ) % candidate
+
+ print >>stderr, " \r",
+ if len(best) > 0:
+ print " B K K1 K2 q T L_hash lg_N sig_bytes c_sign (Mcycles) c_ver ( Mcycles )"
+ print "---- ---- ---- ------ ---- ---- ------ ------ --------- ------------------ --------------------------------"
+
+ best.sort(key=lambda c: (c['sig_bytes'], c['cost']))
+ last_sign = None
+ last_ver = None
+ for c in best:
+ if last_sign is None or c['c_sign'] < last_sign or c['c_ver'] < last_ver:
+ print format_candidate(c)
+ last_sign = c['c_sign']
+ last_ver = c['c_ver']
+
+ print
+ else:
+ print "No candidates found for L_hash = %d or higher." % (L_hash)
+ return
+
+ del bins
+ del best
+
+print "Maximum signature size: %d bytes" % (limit_bytes,)
+print "Maximum (signing + %d*verification) cost: %.1f Mcycles" % (weight_ver, limit_cost)
+print "Hash parameters: %d-bit blocks with %d-bit padding and %d-bit labels, %.2f cycles per byte" \
+ % (L_block, L_pad, L_label, cycles_per_byte)
+print "PRF output size: %d bits" % (L_prf,)
+print "Security level given by L_hash is maintained for up to 2^%d signatures.\n" % (lg_M,)
+
+search()
projects / tahoe-lafs / tahoe-lafs.git / commitdiff