ECDSA kurvasi manbalari: Claude AI yordamida yuzaga chiqdi

ECDSA (Elliptic Curve Digital Signature Algorithm) — bu eng keng tarqalgan elliptik kurvali kriptografik standartlardan biri. Bu standart 2000-yilda NIST tomonidan FIPS 186-2 Digital Signature Standardning bir qismi sifatida qabul qilingan. Bu standart turli oʻlchamlar va xavfsizlik darajalariga ega standart elliptik kurvalarni roʻyxatga oladi.

ECDSA kurvasi manbalari haqidagi sir

ECDSA kurvasi manbalari haqidagi sir yillar davomida kriptografik jamiyatda muhokama qilinib kelgan. Bu sirning asosi NIST tomonidan taqdim etilgan kurvalarning “seed” parametrlari edi. Bu parametrlar NSA tomonidan berilgan, ammo ular haqida hech qanday tushuntirish yoki asos berilmagan.

Masalan, NIST P-256 kurvasi uchun seed parametri SEED = C49D3608 86E70493 6A6678E1 139D26B7 819F7E90 edi, ammo bu seedning kelib chiqishi haqida hech qanday maʼlumot yoʻq edi.

Claude AI yordamida sir ochildi

Anthropic kompaniyasining Claude AI modeli bu sirni ochishga muvaffaq boʻldi. Claude AI Unix timestamp va lrand48 random-number generator ishlatilganligini aniqladi. Bu yordamida kurvalarning seed parametrlari aniqlangan.

from datetime import datetime, timezone

def unix(when):
    return int(datetime.fromisoformat(when).replace(tzinfo=timezone.utc).timestamp())

def candidate(t, n, per, shift, width):
    x, out = (t << 16) | 0x330E, ""                      # the state srand48(t) sets
    for i in range(per * n):
        x = (0x5DEECE66D * x + 0xB) % 2**48              # one step of lrand48
        if i >= per * (n - 1):
            out += "%0*X" % (width, x >> shift)
    return out

PRIME = {  # srand48(time); 20 bytes per candidate: lrand48() >> 23
    "prime239v1": ("1997-05-15 15:49:03", 2185),
    "prime239v2": ("1997-05-15 15:49:03", 2320),
    "prime239v3": ("1997-05-15 15:49:03", 3048),
    "P-192":      ("1997-05-18 14:45:34", 139),
    "prime192v3": ("1997-05-18 14:45:34", 5228),
    "prime192v2": ("1997-05-18 14:45:34", 6013),
    "P-521":      ("1997-08-08 01:12:04", 1136),
    "P-256":      ("1997-08-16 01:52:34", 1842),
    "P-384":      ("1997-08-30 14:50:08", 2688),
    "P-224":      ("1998-04-03 12:52:07", 7026),
}
BINARY = {  # srand48(time / 2), if that reading is right; 5 words per candidate: lrand48()
    "B-233": ("1998-04-17 15:24:18", 155),
    "B-283": ("1998-04-17 15:27:44", 32),
    "B-409": ("1998-04-17 15:30:18", 60),
    "B-571": ("1998-04-17 15:32:08", 73),
}
for name, (when, n) in PRIME.items():
    print(f"{name:10} {when} {n:5} {candidate(unix(when), n, 20, 40, 2)}")
for name, (when, n) in BINARY.items():
    print(f"{name:10} {when} {n:5} {candidate(unix(when) // 2, n, 5, 17, 8)}")

Claude AI bu ma'lumotlarni topish uchun bir necha soat ichida ishladi. U lrand48 random-number generator ishlatilganligini aniqladi va bu generator Unix timestamp bilan boshlanganligini aniqladi. Bu ma'lumotlar ECDSA kurvalarining xavfsizligini tasdiqlaydi.

prime239v1 1997-05-15 15:49:03  2185 E43BB460F0B80CC0C0B075798E948060F8321B7D
prime239v2 1997-05-15 15:49:03  2320 E8B4011604095303CA3B8099982BE09FCB9AE616
prime239v3 1997-05-15 15:49:03  3048 7D7374168FFE3471B60A857686A19475D3BFA2FF
P-192      1997-05-18 14:45:34   139 3045AE6FC8422F64ED579528D38120EAE12196D5
prime192v3 1997-05-18 14:45:34  5228 C469684435DEB378C4B65CA9591E2A5763059A2E
prime192v2 1997-05-18 14:45:34  6013 31A92EE2029FD10D901B113E990710F0D21AC6B6
P-521      1997-08-08 01:12:04  1136 D09E8800291CB85396CC6717393284AAA0DA64BA
P-256      1997-08-16 01:52:34  1842 C49D360886E704936A6678E1139D26B7819F7E90
P-384      1997-08-30 14:50:08  2688 A335926AA319A27A1D00896A6773A4827ACDAC73
P-224      1998-04-03 12:52:07  7026 BD71344799D5C7FCDC45B59FA3B9AB8F6A948BC5
B-233      1998-04-17 15:24:18   155 74D59FF07F6B413D0EA14B344B20A2DB049B50C3
B-283      1998-04-17 15:27:44    32 77E2B07370EB0F832A6DD5B62DFC88CD06BB84BE
B-409      1998-04-17 15:30:18    60 4099B5A457F9D69F79213D094C4BCD4D4262210B
B-571      1998-04-17 15:32:08    73 2AA058F73A0E33AB486B0F610410C53A7F132310

ECDSA kurvalari xavfsizligi

Bu topilmalar ECDSA kurvalarining xavfsizligini tasdiqlaydi. Kurvalarning seed parametrlari oddiy pseudorandom number generator (PRNG) bilan ishlab chiqarilganligi aniqlangan. Bu ma'lumotlar kurvalarning xavfsizligini tasdiqlaydi va ular orasida hech qanday orqa qapqali yoki maqsadli zaifliklar yoʻqligini isbotlaydi.

M, A = 2**107 - 1, pow(3, 121, 2**107 - 1)       # zrandom: state = state * 3^121 mod (2^107 - 1)
P192 = 2**192 - 2**64 - 1
P239 = 2**239 - 2**143 - 2**95 + 2**47 - 1

CURVES = [  # name, LIP seed, p, b, published Gx   (a = -3 throughout)
    ("prime192v1", 20000, P192,
     0x64210519e59c80e70fa7e9ab72243049feb8deecc146b9b1,
     0x188da80eb03090f67cbf20eb43a18800f4ff0afd82ff1012),
    ("prime239v1", 20420, P239,
     0x6b016c3bdcf18941d0d654921475ca71a9db2fb27d1d37796185c2942c0a,
     0xffa963cdca8816ccc33b8642bedf905c3d358573d3f27fbbd3b3cb9aaaf),
    ("prime239v2", 20507, P239,
     0x617fab6832576cbbfed50d99f0249c3fee58b94ba0038c7ae84c8c832f2c,
     0x38af09d98727705120c921bb5e9e26296a3cdcf2f35757a0eafd87b830e7),
    ("prime239v3", 20509, P239,
     0x255705fa2a306654b1f4cb03d6a750a30c250102d4988717d9ba15ab6d3e,
     0x6768ae8e18bb92cfcf005c949aa2c6d94853d0e660bbf854b1c9505fe95a),
    ("prime192v2", 20512, P192,
     0xcc22d6dfb95c6b25e49c0d6364a4e5980c393aa21668d953,
     0xeea2bae7e1497842f2de7769cfe9c989c072ad696f48034a),
    ("prime192v3", 20517, P192,
     0x22123dc2395a05caa7423daeccc94760a7d462256bd56916,
     0x7d29778100c65a1da1783716588dce2b8b4aee8e228f1896),
]

def base_point_x(seed, p, b):
    state = seed                         # zrstarts(seed)
    def limb():                          # zrandom(2^26): low 26 bits of the new state
        nonlocal state
        state = state * A % M
        return state % 2**26
    for _ in range(7):                   # seven single-limb draws come first
        limb()
    limbs = -(-p.bit_length() // 26) + 1 # zrandomb(p): limbs of p plus one, top limb first
    tries = 0
    while True:
        x = 0
        for _ in range(limbs):
            x = x << 26 | limb()
        x %= p
        tries += 1
        if pow(x**3 - 3*x + b, (p - 1) // 2, p) == 1:   # is x on the curve?
            return x, tries

for name, seed, p, b, gx in CURVES:
    x, tries = base_point_x(seed, p, b)
    print(f"{name:20} seed {seed}  tries {tries}  matches published Gx: {x == gx}")

ECDSA kurvalari keng tarqalganligi sababli, bu topilmalar kriptografik jamiyat uchun juda muhimdir. Bu topilmalar ECDSA kurvalarining xavfsizligini tasdiqlaydi va ular orasida hech qanday orqa qapqali yoki maqsadli zaifliklar yoʻqligini isbotlaydi.

prime192v1           seed 20000  tries 2  matches published Gx: True
prime239v1           seed 20420  tries 3  matches published Gx: True
prime239v2           seed 20507  tries 2  matches published Gx: True
prime239v3           seed 20509  tries 1  matches published Gx: True
prime192v2           seed 20512  tries 5  matches published Gx: True
prime192v3           seed 20517  tries 1  matches published Gx: True

Xulosa

ECDSA kurvasi manbalari haqidagi sirni Claude AI yordamida ochib berildi. Bu sirni hal qilishda Unix timestamp va lrand48 random-number generator ishlatilganligi aniqlangan. Bu topilmalar ECDSA kurvalarining xavfsizligini tasdiqlaydi va ular orasida hech qanday orqa qapqali yoki maqsadli zaifliklar yoʻqligini isbotlaydi.

Asl manba: anthropic.com