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