[MPU - Primality Certificate]
Version 1.0
# We should allow base 10, base 16, and base 62.  Base 10 is the default.
Base 10

# This is the N value we're proving.
Proof for:
N 8087094497428743437627091507362881


# The following types allow a simple chain:
#    Pocklington
#    BLS3
#    BLS15
#    ECPP
#    Small
# These could result in a tree:
#    Lucas
#    BLS5
#    BLS7
#
# Types ECPP3 and ECPP4 are from Primo, and can be easily translated into
# a type ECPP.  We include them here rather than convert because they save
# quite a bit of space and we may want to generate them ourselves someday.
# I remain dubious about including them, so it is possible they will go away
# in the format and we'll just have the converter/verifier do the conversion.
#
# MPU will generate types: BLS3, BLS15, ECPP, BLS5, Small.
# Primo (converted) will generate types: Pocklington, BLS15, ECPP3, ECPP4.
# Lucas is included for completeness.
# I no longer use BLS7 so will not include it.

# BLS## stands for theorem ## in the paper:
#   "New Primality Criteria and Factorizations of 2^m +/- 1" by
#   Brillhart, Lehmer, and Selfridge, Mathematics of Computation, 1975.
# which includes 21 theorems related to N-1, N+1, and hybrid primality proofs.
# The paper is often referred to as BLS75, and is highly recommended reading.

# I allow N or any Q smaller than 2^64 to implicitly construct a "Small"
# certificate.  So once we have a Q of <= 2^64, we can run a deterministic
# test to prove its primality.  One of the many BPSW variants works for this,
# or 7 M-R tests with bases 2, 325, 9375, 28178, 450775, 9780504, 1795265022.
# Hence if we have Q values <= 2^64, the verifier needs to do its test, and
# the certificate can leave out an explicit proof for the Q.  This is done
# to prevent a few Lucas, BLS5, etc. type tests from creating a swarm of
# "Small" certificates for each little factor.

Type BLS15
N  8087094497428743437627091507362881
Q  175806402118016161687545467551367
LP 1
LQ 22

# Note: Primo type 2 can map to this, though this allows Q to be
#       smaller.  ( prevR->N, R->Q, S->M, Q->(LP,LQ) )
#       Primo condition a is implied by Q odd
#       Primo condition c is stricter than required by BLS15.
#       Primo conditions e and f relate to the Lucas code
#       Primo condition g is not required
# Verify: Q is odd
# Verify: Q > 2
# Verify: Q divides N+1
# Let: M = (N+1)/Q
# Verify: MQ-1 = N                             # Primo d
# Verify: M > 0                                # Primo b
# Verify: 2Q-1 > sqrt(N)                       # Primo c (less strict)
# Let: D = LP*LP - 4*LQ
# Verify: D != 0
# Verify Jacobi(D,N) = -1                      # Primo h
# Verify V_{m/2} mod N != 0                    # Primo j
# Verify V_{(N+1)/2} mod N == 0                # Primo i
# Then N is prime if Q is prime.


Type ECPP
N  175806402118016161687545467551367
A  96642115784172626892568853507766
B  111378324928567743759166231879523
M  175806402118016177622955224562171
Q  2297612322987260054928384863
X  3273750212
Y  82061726986387565872737368000504

# Generic ECPP / AKGM block
# A and/or B can be -1, so mod them.
# Let A = A % N
# Let B = B % N
# Verify: N > 0                                # Primo b
# Verify: gcd(N, 6) = 1                        # Primo a
# Verify: gcd(4*a^3 + 27*b^2, N) = 1           # Primo i
# Verify: Y^2 mod N = X^3 + A*X + B mod N      # Primo j
# Verify: M >= N + 1 - 2*sqrt(N)               # Primo g
# Verify: M <= N + 1 + 2*sqrt(N)               # Primo h
# Verify: Q > (N^(1/4)+1)^2                    # Primo f
# Verify: Q < N                                # Primo e
# Verify: M != Q
# Verify: Q divides M
# Note:  EC(A,B,N,X,Y) defines the elliptic curve Y^2 = X^3 + A*X + B, mod N
#        with operations defined in affine coordinates.
# Let POINT = (M/Q) * EC(A,B,N,X,Y)
# Verify: POINT is not the identity            # Primo k
# Let POINT = Q * POINT   (or M * EC(A,B,N,X,Y))
# Verify: POINT is the identity                # Primo l
# Then N is prime if Q is prime.


Type BLS3
N  2297612322987260054928384863
Q  16501461106821092981
A  5

# Note: This is similar to Pocklington, but Q can be smaller.
# Verify: Q odd
# Verify: Q > 2
# Verify: Q divides N-1
# Let: M = (N-1)/Q
# Verify: MQ+1 = N
# Verify: M > 0
# Verify: 2Q+1 > sqrt(N)
# Verify A^((N-1)/2) mod N = N-1 
# Verify A^(M/2) mod N != N-1
# Then N is prime if Q is prime.


Type BLS5
N  8087094497428743437627091507362881
Q[1]  98277749
Q[2]  3631
A[0]  11
----

# Note: This also covers generalized Pocklington
# Note: We have to have N-1 factored to (N/2)^1/3
# Note: A line starting with - is required at the end.
# Verify: N > 2, N odd
# For each i (0-max):
#   Q[0] = 2                       # 2 is always a factor of n-1
#   A[i] = 2 unless specified
#   Verify: Q[i] > 1, Q[i] < N-1
#   Verify: A[i] > 1, A[i] < N-1
#   Verify: Q[i] divides N-1
# Let: F = N-1 divided by each Q[i] with multiplicity
#      (i.e. if Q[i] evenly divides N-1 3 times, then divide it out 3 times)
# Let: R = (N-1)/F
# Verify: F is even
# Verify: gcd(F, R) = 1
# Let: s = integer    part of R / 2*F
# Let: r = fractional part of R / 2*F
# Let: P = (F+1) * (2*F*F + (r-1)*F + 1)
# Verify: n < P
# Note: The next condition is trivially met if F >= R,
#       as is the case with Pocklington.
# Let: rt = r^2 - 8s
# Verify: s = 0   OR  rt not a perfect square [e.g. floor(sqrt(rt))^2 != rt]
# For each i:
#   Verify: A[i]^(N-1) mod N = 1
#   Verify: gcd(A[i]^((N-1)/Q[i])-1, N) = 1
# Then N is prime if each Q is prime.


Type Lucas
N     10384593717069655257060992658440473
Q[1]  2
Q[2]  3
Q[3]  13
Q[4]  379
Q[5]  87820459687010818424506060639
A     41

# Note: All factors of N-1 are listed
# Verify: A > 1 and A < N
# Verify: N-1 has only factors Q (to some multiplicity).
# Verify: A^(N-1) mod N = 1
# Verify for each Q:
#   Q > 0
#   Q < N-1
#   Q divides N-1
#   A^((N-1)/Q) mod N != 1
# Then N is prime if each Q is prime.


Type Pocklington
N  2297612322987260054928384863
Q  16501461106821092981
A  5

# Note: This is Primo type 1 ( prevR->N, R->Q, S->M, B->A )
# verify: Q divides N-1
# let: M = (N-1)/Q
# Verify: M is even                            # Primo a
# Verify: M > 0                                # Primo b
# Verify: M < Q                                # Primo c
# Verify: MQ+1 = N                             # Primo d
# Verify: A > 1                                # Primo e
# Verify: A^(N-1) mod N = 1                    # Primo f
# Verify: gcd(A^M - 1, N) = 1                  # Primo g
# Then N is prime if Q is prime.



Type ECPP3
N 33863876771064627047864880693347
S 8929168182
R 3792500721324706215857
A -30
B 56
T 0

# From Primo.  Experimental, may go away.
# Verify: |A| <= N/2
# Verify: |B| <= N/2
# Verify: T >= 0
# Verify: T < N
# Let: L = (T^3 + A*T + B) mod N
# Let: A = (A * L^2) mod N
# Let: B = (B * L^3) mod N
# Let: X = (T*L) mod N
# Let: Y = (L^2) mod N
# Let: Q = R
# Let: M = R*S
# Continue as type ECPP.


Type ECPP4
N 346908375519289784739191985209489924762236002832827279935279239073873837063
S 26591618
R 13045779144363828659812726897983038292936490633310834005307717308699
J -4092776160830678382137043215242735918658074999545950247507675196218505248
T 4

# From Primo.  Experimental, may go away.
# Verify: |J| <= N/2
# Verify: T >= 0
# Verify: T < N
# Let: A = 3 * J * (1728 - J)
# Let: B = 2 * J * (1728 - J)^2
# Let: L = (T^3 + A*T + B) mod N
# Let: A = (A * L^2) mod N
# Let: B = (B * L^3) mod N
# Let: X = (T*L) mod N
# Let: Y = (L^2) mod N
# Let: Q = R
# Let: M = R*S
# Continue as type ECPP.


Type Small
N  5791

# Verify: N < 2^64
# Verify: N is prime using BPSW or deterministic M-R tests


# Experimental, not used:
#Type BLS7
#N  10384593717069655257060992658440473
#Q[1]  87820459687010818424506060639
#Q[2]  379
#Q[3]  13
#Q[4]  3
#Q[5]  2
#A[1]  2
#A[2]  2
#A[3]  2
#A[4]  19
#A[5]  5
#B  10000
#AR 2

# Verify for each i:
#   Q[i] > 1, Q[i] < N
#   A[i] > 1, A[i] < N
#   Q[i] divides n-1
# Let: F = product of all Qs
# Let: R = N/F
# Verify: F is even
# Verify: gcd(F, R) = 1
# Verify: F has no factors smaller than B (this may be time consuming)
# Let: s = integer    part of R / 2*F
# Let: r = fractional part of R / 2*F
# Let: P = (F*B+1) * (2*F*F + (r-B)*F + 1)
# Verify: n < P
# Let: rt = r^2 - 8s
# Verify: s = 0   OR  rt not a perfect square [e.g. floor(sqrt(rt))^2 != rt]
# For each i:
#   Verify: A[i]^(N-1) mod N = 1
#   Verify: gcd(A[i]^((N-1)/Q[i])-1, N) = 1
# Verify: AR^(N-1) mod N = 1
# Verify: gcd(AR^((N-1)/R)-1, N) = 1
# Then N is prime if each Q is prime.


