129 lines
3.2 KiB
Python
129 lines
3.2 KiB
Python
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#!/usr/bin/env sage
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#coding: utf8
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proof.all(False)
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# parameters.
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ls = list(primes(3, 374)) + [587] # Elkies primes
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#ls = list(primes(3, 47)) + [97] # (a smaller example)
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p = 4 * prod(ls) - 1
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assert is_prime(p)
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print "\nElkies primes:", " ".join(map(str, ls))
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max_exp = ceil((sqrt(p) ** (1/len(ls)) - 1) / 2)
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assert (2 * max_exp + 1) ** len(ls) >= sqrt(p)
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print "exponents are chosen in the range {}..{}.".format(-max_exp, max_exp)
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base = GF(p)(0) # Montgomery coefficient of starting curve
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# helper functions.
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# NB: all the operations can be computed entirely over the prime field,
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# but for simplicity of this implementation we will make use of curves
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# defined over GF(p^2). note this slows everything down quite a bit.
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Fp2.<i> = GF(p**2, modulus = x**2 + 1)
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def montgomery_curve(A):
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return EllipticCurve(Fp2, [0, A, 0, 1, 0])
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# sage's isogeny formulas return Weierstraß curves, hence we need this...
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def montgomery_coefficient(E):
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Ew = E.change_ring(GF(p)).short_weierstrass_model()
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_, _, _, a, b = Ew.a_invariants()
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R.<z> = GF(p)[]
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r = (z**3 + a*z + b).roots(multiplicities=False)[0]
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s = sqrt(3 * r**2 + a)
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if not is_square(s): s = -s
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A = 3 * r / s
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assert montgomery_curve(A).change_ring(GF(p)).is_isomorphic(Ew)
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return GF(p)(A)
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# actual implementation.
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def private():
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return [randrange(-max_exp, max_exp + 1) for _ in range(len(ls))]
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def validate(A):
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while True:
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k = 1
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P = montgomery_curve(A).lift_x(GF(p).random_element())
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for l in ls:
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Q = (p + 1) // l * P
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if not Q: continue
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if l * Q: return False
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k *= l
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if k > 4 * sqrt(p): return True
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def action(pub, priv):
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E = montgomery_curve(pub)
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es = priv[:]
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while any(es):
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E._order = (p + 1)**2 # else sage computes this
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P = E.lift_x(GF(p).random_element())
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s = +1 if P.xy()[1] in GF(p) else -1
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k = prod(l for l, e in zip(ls, es) if sign(e) == s)
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P *= (p + 1) // k
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for i, (l, e) in enumerate(zip(ls, es)):
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if sign(e) != s: continue
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Q = k // l * P
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if not Q: continue
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Q._order = l # else sage computes this
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phi = E.isogeny(Q)
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E, P = phi.codomain(), phi(P)
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es[i] -= s
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k //= l
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return montgomery_coefficient(E)
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# example.
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print
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print "testing public-key validation on random ordinary curves (should be all 0s):\n ",
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for _ in range(16):
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while True:
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A = GF(p).random_element()
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if montgomery_curve(A).is_ordinary(): break
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print int(validate(A)),
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print
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privA = private()
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print "\nAlice's private key:\n ", " ".join(map('{:2d}'.format, privA))
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pubA = action(base, privA)
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print "\nAlice's public key:\n ", pubA,
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print " (valid: {})".format(int(validate(pubA)))
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privB = private()
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print "\nBob's private key:\n ", " ".join(map('{:2d}'.format, privB))
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pubB = action(base, privB)
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print "\nBob's public key:\n ", pubB,
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print " (valid: {})".format(int(validate(pubB)))
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sharedA = action(pubB, privA)
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print "\nAlice's shared secret:\n ", sharedA
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sharedB = action(pubA, privB)
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print "\nBob's shared secret:\n ", sharedB
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if sharedA == sharedB:
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print "\n--> equal!\n"
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else:
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print "\n--> NOT EQUAL?!\n"
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