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attack.mzn
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attack.mzn
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/*
MIT License
Copyright (c) 2024 Hosein Hadipour
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.
*/
% Strategy:
% Divide E into 4 parts: EU, EMU0, EMU1
% EML0, EML1, EL
% Use DDT (LAT) to encode the propagation of differential (resp. linear) trails through EU (resp. EL).
% Use *-DDT (*-LAT) to encode the propagation of differential (resp. linear) trails through EMU0 (resp. EML1).
% Enode the propagation of differential (resp. linear) trails through EMU1 (resp. EML0) with probability one.
% Minimize the probability of transtions through EU, EL as well as the number of common active S-boxes trhough
% the middle part, i.e., EMU0, EMU1, EML0, EML1.
int: RU;
int: RMU;
int: RM;
int: RML;
int: RL;
int: offset;
constraint assert(RU >= 0, "Invalid value for RU: " ++
"RU must be greater than or equal to 0");
constraint assert(RMU >= 0, "Invalid value for RMU: " ++
"RMU must be greater than or equal to 0");
constraint assert(RM >= 0, "Invalid value for RM: " ++
"RM must be greater than or equal to 0");
constraint assert(RML >= 0, "Invalid value for RML: " ++
"RML must be greater than or equal to 0");
constraint assert(RL >= 0, "Invalid value for RL: " ++
"RL must be greater than or equal to 0");
constraint assert(RMU <= RM, "Invalid value for RMU: " ++
"RMU must be less than RM");
constraint assert(RML <= RM, "Invalid value for RML: " ++
"RML must be less than RM");
% #############################################################################################################################################
% #############################################################################################################################################
% #############################################################################################################################################
% ____ __ _ _ _ _____ _ _
% | _ \ ___ / _|(_) _ __ ___ | | ___ ___ | | __ _ _ _ __ |_ _|__ _ | |__ | | ___ ___
% | | | | / _ \| |_ | || '_ \ / _ \ | | / _ \ / _ \ | |/ /| | | || '_ \ | | / _` || '_ \ | | / _ \/ __|
% | |_| || __/| _|| || | | || __/ | |___| (_) || (_) || < | |_| || |_) | | || (_| || |_) || || __/\__ \
% |____/ \___||_| |_||_| |_| \___| |_____|\___/ \___/ |_|\_\ \__,_|| .__/ |_| \__,_||_.__/ |_| \___||___/
% |_|
% define lookup tables
array[0..4, 0..1] of int: rotation_dictionary = array2d(0..4, 0..1,
[
19, 28,
61, 39,
1, 6,
10, 17,
7, 41
]);
% #############################################################################################################################################
% #############################################################################################################################################
% #############################################################################################################################################
% ____ _ _ _ __ _____ _ _
% / ___| ___ _ __ ___ | |_ _ __ __ _ (_) _ __ | |_ ___ / _| ___ _ __ | ____|| | | |
% | | / _ \ | '_ \ / __|| __|| '__|/ _` || || '_ \ | __|/ __| | |_ / _ \ | '__| | _| | | | |
% | |___| (_) || | | |\__ \| |_ | | | (_| || || | | || |_ \__ \ | _|| (_) || | | |___ | |_| |
% \____|\___/ |_| |_||___/ \__||_| \__,_||_||_| |_| \__||___/ |_| \___/ |_| |_____| \___/
% Constraints for EU
array[0..RU, 0..4, 0..63] of var 0..1: xu;
array[0..(RU - 1), 0..4, 0..63] of var 0..1: yu;
array[0..(RU - 1), 0..2, 0..63] of var 0..1: pu;
% S-box layer
constraint forall(round in 0..(RU - 1), column in 0..63)
(
sbox_diff(xu[round, 0, column], xu[round, 1, column], xu[round, 2, column], xu[round, 3, column], xu[round, 4, column],
yu[round, 0, column], yu[round, 1, column], yu[round, 2, column], yu[round, 3, column], yu[round, 4, column],
pu[round, 0, column], pu[round, 1, column], pu[round, 2, column])
% sbox_diff(xu[round, 4, column], xu[round, 3, column], xu[round, 2, column], xu[round, 1, column], xu[round, 0, column],
% yu[round, 4, column], yu[round, 3, column], yu[round, 2, column], yu[round, 1, column], yu[round, 0, column],
% pu[round, 0, column], pu[round, 1, column], pu[round, 2, column])
);
% Diffusion layer
constraint forall(round in 0..(RU - 1), row in 0..4, column in 0..63)
(
xor3(yu[round, row, column], yu[round, row, (column - rotation_dictionary[row, 0] + 64) mod 64], yu[round, row, (column - rotation_dictionary[row, 1] + 64) mod 64],
xu[round + 1, row, column])
);
constraint sum(i in 0..4, j in 0..63) (xu[0, i, j]) != 0;
% #############################################################################################################################################
% #############################################################################################################################################
% #############################################################################################################################################
% ____ _ _ _ __ _____ __ __ _ _
% / ___| ___ _ __ ___ | |_ _ __ __ _ (_) _ __ | |_ ___ / _| ___ _ __ | ____|| \/ || | | |
% | | / _ \ | '_ \ / __|| __|| '__|/ _` || || '_ \ | __|/ __| | |_ / _ \ | '__| | _| | |\/| || | | |
% | |___| (_) || | | |\__ \| |_ | | | (_| || || | | || |_ \__ \ | _|| (_) || | | |___ | | | || |_| |
% \____|\___/ |_| |_||___/ \__||_| \__,_||_||_| |_| \__||___/ |_| \___/ |_| |_____||_| |_| \___/
% Constraints for EMU
array[0..RM, 0..4, 0..63] of var -1..1: xmu;
array[0..(RM - 1), 0..4, 0..63] of var -1..1: ymu;
array[0..(RM - 1), 0..63] of var 0..1: dmu;
% link the input of EM to the output of EU
constraint forall(row in 0..4, column in 0..63)(xmu[0, row, column] = xu[RU, row, column]);
% S-box layer
constraint forall(round in 0..(RM - 1), column in 0..63)
(
if round < RMU then
(
sbox_diff_star(xmu[round, 0, column], xmu[round, 1, column], xmu[round, 2, column], xmu[round, 3, column], xmu[round, 4, column],
ymu[round, 0, column], ymu[round, 1, column], ymu[round, 2, column], ymu[round, 3, column], ymu[round, 4, column],
dmu[round, column])
% sbox_diff_star(xmu[round, 4, column], xmu[round, 3, column], xmu[round, 2, column], xmu[round, 1, column], xmu[round, 0, column],
% ymu[round, 4, column], ymu[round, 3, column], ymu[round, 2, column], ymu[round, 1, column], ymu[round, 0, column],
% dmu[round, column])
) else
(
sbox_diff_m(xmu[round, 0, column], xmu[round, 1, column], xmu[round, 2, column], xmu[round, 3, column], xmu[round, 4, column],
ymu[round, 0, column], ymu[round, 1, column], ymu[round, 2, column], ymu[round, 3, column], ymu[round, 4, column],
dmu[round, column])
% sbox_diff_m(xmu[round, 4, column], xmu[round, 3, column], xmu[round, 2, column], xmu[round, 1, column], xmu[round, 0, column],
% ymu[round, 4, column], ymu[round, 3, column], ymu[round, 2, column], ymu[round, 1, column], ymu[round, 0, column],
% dmu[round, column])
) endif
);
% Diffusion layer
constraint forall(round in 0..(RM - 1), row in 0..4, column in 0..63)
(
if round < RMU then
(
xor3(ymu[round, row, column], ymu[round, row, ((column - rotation_dictionary[row, 0] + 64) mod 64)], ymu[round, row, ((column - rotation_dictionary[row, 1] + 64) mod 64)],
xmu[round + 1, row, column])
) else
(
xor3_pr1(ymu[round, row, column], ymu[round, row, ((column - rotation_dictionary[row, 0] + 64) mod 64)], ymu[round, row, ((column - rotation_dictionary[row, 1] + 64) mod 64)],
xmu[round + 1, row, column])
) endif
);
% #############################################################################################################################################
% #############################################################################################################################################
% #############################################################################################################################################
% ____ _ _ _ __ _____ __ __ _
% / ___| ___ _ __ ___ | |_ _ __ __ _ (_) _ __ | |_ ___ / _| ___ _ __ | ____|| \/ || |
% | | / _ \ | '_ \ / __|| __|| '__|/ _` || || '_ \ | __|/ __| | |_ / _ \ | '__| | _| | |\/| || |
% | |___| (_) || | | |\__ \| |_ | | | (_| || || | | || |_ \__ \ | _|| (_) || | | |___ | | | || |___
% \____|\___/ |_| |_||___/ \__||_| \__,_||_||_| |_| \__||___/ |_| \___/ |_| |_____||_| |_||_____|
% Constraints for EML
array[0..RM, 0..4, 0..63] of var -1..1: xml;
array[0..(RM - 1), 0..4, 0..63] of var -1..1: yml;
array[0..(RM - 1), 0..63] of var 0..1: dml;
% S-box layer
constraint forall(round in 0..(RM - 1), column in 0..63)
(
if round < (RM - RML) then
(
sbox_lin_m(yml[round, 0, column], yml[round, 1, column], yml[round, 2, column], yml[round, 3, column], yml[round, 4, column],
xml[round, 0, column], xml[round, 1, column], xml[round, 2, column], xml[round, 3, column], xml[round, 4, column],
dml[round, column])
% sbox_lin_m(yml[round, 4, column], yml[round, 3, column], yml[round, 2, column], yml[round, 1, column], yml[round, 0, column],
% xml[round, 4, column], xml[round, 3, column], xml[round, 2, column], xml[round, 1, column], xml[round, 0, column],
% dml[round, column])
) else
(
sbox_lin_star(yml[round, 0, column], yml[round, 1, column], yml[round, 2, column], yml[round, 3, column], yml[round, 4, column],
xml[round, 0, column], xml[round, 1, column], xml[round, 2, column], xml[round, 3, column], xml[round, 4, column],
dml[round, column])
% sbox_lin_star(yml[round, 4, column], yml[round, 3, column], yml[round, 2, column], yml[round, 1, column], yml[round, 0, column],
% xml[round, 4, column], xml[round, 3, column], xml[round, 2, column], xml[round, 1, column], xml[round, 0, column],
% dml[round, column])
) endif
);
% Diffusion layer
constraint forall(round in 0..(RM - 1), row in 0..4, column in 0..63)
(
if round < (RM - RML) then
(
xor3_pr1(xml[round + 1, row, column], xml[round + 1, row, ((column + rotation_dictionary[row, 0] + 64) mod 64)], xml[round + 1, row, ((column + rotation_dictionary[row, 1] + 64) mod 64)],
yml[round, row, column])
) else
(
xor3(xml[round + 1, row, column], xml[round + 1, row, ((column + rotation_dictionary[row, 0] + 64) mod 64)], xml[round + 1, row, ((column + rotation_dictionary[row, 1] + 64) mod 64)],
yml[round, row, column])
) endif
);
% #############################################################################################################################################
% #############################################################################################################################################
% #############################################################################################################################################
% ____ _ _ _ __ _____ _
% / ___| ___ _ __ ___ | |_ _ __ __ _ (_) _ __ | |_ ___ / _| ___ _ __ | ____|| |
% | | / _ \ | '_ \ / __|| __|| '__|/ _` || || '_ \ | __|/ __| | |_ / _ \ | '__| | _| | |
% | |___| (_) || | | |\__ \| |_ | | | (_| || || | | || |_ \__ \ | _|| (_) || | | |___ | |___
% \____|\___/ |_| |_||___/ \__||_| \__,_||_||_| |_| \__||___/ |_| \___/ |_| |_____||_____|
% constraints for EL
array[0..RL, 0..4, 0..63] of var 0..1: xl;
array[0..(RL - 1), 0..4, 0..63] of var 0..1: yl;
array[0..(RL - 1), 0..1, 0..63] of var 0..1: pl;
% link the output of EML to the input of EL
constraint forall(row in 0..4, column in 0..63)(xl[0, row, column] = xml[RM, row, column]);
% S-box layer
constraint forall(round in 0..(RL - 1), column in 0..63)
(
sbox_lin(yl[round, 0, column], yl[round, 1, column], yl[round, 2, column], yl[round, 3, column], yl[round, 4, column],
xl[round, 0, column], xl[round, 1, column], xl[round, 2, column], xl[round, 3, column], xl[round, 4, column],
pl[round, 0, column], pl[round, 1, column])
% sbox_lin(yl[round, 4, column], yl[round, 3, column], yl[round, 2, column], yl[round, 1, column], yl[round, 0, column],
% xl[round, 4, column], xl[round, 3, column], xl[round, 2, column], xl[round, 1, column], xl[round, 0, column],
% pl[round, 0, column], pl[round, 1, column])
);
% Diffusion layer
constraint forall(round in 0..(RL - 1), row in 0..4, column in 0..63)
(
xor3(xl[round + 1, row, column], xl[round + 1, row, ((column + rotation_dictionary[row, 0] + 64) mod 64)], xl[round + 1, row, ((column + rotation_dictionary[row, 1] + 64) mod 64)],
yl[round, row, column])
);
constraint sum(row in 0..4, column in 0..63)(xl[RL, row, column]) != 0;
% #############################################################################################################################################
% #############################################################################################################################################
var 0..320: PU;
var 0..(320 + 4*320): CM;
var 0..320: CL;
var 0..320: NASM;
constraint PU = 4*sum(r in offset..(RU - 1), column in 0..63)(pu[r, 0, column]) +
3*sum(r in offset..(RU - 1), column in 0..63)(pu[r, 1, column]) +
2*sum(r in offset..(RU - 1), column in 0..63)(pu[r, 2, column]);
constraint CM =
1*sum(r in 0..(RM - 1), row in 0..4, column in 0..63)(
bool2int(xmu[r, row, column] = -1 /\ xml[r, row, column] = -1)
+
bool2int(xmu[r, row, column] = -1 /\ xml[r, row, column] = 1)
+
bool2int(xmu[r, row, column] = 1 /\ xml[r, row, column] = -1));
% +
% 1*sum(r in 0..(RM - 1), row in 0..4, column in 0..63)(
% bool2int(ymu[r, row, column] = -1 /\ yml[r, row, column] = -1))
% +
% 1*sum(r in 0..(RM - 1), row in 0..4, column in 0..63)(
% bool2int(ymu[r, row, column] = -1 /\ yml[r, row, column] = 1))
% +
% 1*sum(r in 0..(RM - 1), row in 0..4, column in 0..63)(
% bool2int(ymu[r, row, column] = 1 /\ yml[r, row, column] = -1));
% constraint CM = sum(r in 0..(RM - 1), row in 0..4, column in 0..63)(
% bool2int((ymu[r, row, column] != 0 /\ yml[r, row, column] != 0)));
% constraint CM = sum(r in 0..(RM - 1), row in 0..4, column in 0..63)(
% bool2int((xmu[r, row, column] = -1 /\ xml[r, row, column] = -1)
% \/
% (xmu[r, row, column] = -1 /\ xml[r, row, column] = 1)
% \/
% (xmu[r, row, column] = 1 /\ xml[r, row, column] = -1)
% \/
% (xmu[r, row, column] = 1 /\ xml[r, row, column] = 1)));
constraint NASM = sum(r in 0..(RM - 1), column in 0..63)(bool2int(dmu[r, column] + dml[r, column] == 2));
constraint CL = 4*sum(r in 0..(RL - 1), column in 0..63)(pl[r, 0, column]) +
2*sum(r in 0..(RL - 1), column in 0..63)(pl[r, 1, column]);
% #############################################################################################################################################
% #############################################################################################################################################
% #############################################################################################################################################
% ____ _ ____ _ _
% / ___| ___ __ _ _ __ ___ | |__ / ___| | |_ _ __ __ _ | |_ ___ __ _ _ _
% \___ \ / _ \ / _` || '__|/ __|| '_ \ \___ \ | __|| '__|/ _` || __|/ _ \ / _` || | | |
% ___) || __/| (_| || | | (__ | | | | ___) || |_ | | | (_| || |_| __/| (_| || |_| |
% |____/ \___| \__,_||_| \___||_| |_| |____/ \__||_| \__,_| \__|\___| \__, | \__, |
% |___/ |___/
% Search strategy
ann: search_ann;
% search_ann = seq_search([
% int_search([yl[RL - 1, 0, i] | i in 0..63], indomain_min, complete),
% int_search([yl[RL - 1, 1, i] | i in 0..63], indomain_min, complete),
% int_search([yl[RL - 1, 2, i] | i in 0..63], indomain_min, complete),
% int_search([yl[RL - 1, 3, i] | i in 0..63], indomain_min, complete),
% int_search([yl[RL - 1, 3, i] | i in 0..63], indomain_min, complete)]);
% #############################################################################################################################################
% #############################################################################################################################################
% #############################################################################################################################################
% ___ _ _ _ _ _____ _ _
% / _ \ | |__ (_) ___ ___ | |_ (_)__ __ ___ | ___|_ _ _ __ ___ | |_ (_) ___ _ __
% | | | || '_ \ | | / _ \ / __|| __|| |\ \ / // _ \ | |_ | | | || '_ \ / __|| __|| | / _ \ | '_ \
% | |_| || |_) || || __/| (__ | |_ | | \ V /| __/ | _| | |_| || | | || (__ | |_ | || (_) || | | |
% \___/ |_.__/_/ | \___| \___| \__||_| \_/ \___| |_| \__,_||_| |_| \___| \__||_| \___/ |_| |_|
% |__/
% Objective function
% constraint forall(row in 4..4, column in 0..63)(xu[0, row, column] = 0);
% constraint forall(column in 0..63)((xu[0, 0, column] = 1 /\ xu[0, 4, column] = 1) \/ (xu[0, 0, column] = 0 /\ xu[0, 4, column] = 0));
% constraint forall(row in 2..4, column in 0..63)(xl[RL, row, column] = 0);
% constraint PU <= 45;
% constraint CM <= 20;
% constraint CL <= 25;
% solve :: restart_constant(10000)
% :: search_ann minimize (PU + NASM + CL);
% constraint forall(row in 0..0, column in 0..0)(xu[0, row, column] = 1);
% constraint forall(row in 0..0, column in 1..1)(xu[0, row, column] = 0);
% constraint forall(row in 0..0, column in 2..63)(xu[0, row, column] = 0);
% constraint forall(row in 1..4, column in 0..63)(xu[0, row, column] = 0);
% constraint forall(row in 0..0, column in 0..53)(yml[3, row, column] = 0);
% constraint forall(row in 0..0, column in 54..54)(yml[3, row, column] = 1);
% constraint forall(row in 0..0, column in 55..63)(yml[3, row, column] = 0);
% constraint forall(row in 1..4, column in 0..63)(yml[3, row, column] = 0);
% constraint forall(row in 0..0, column in 0..63)(xu[0, row, column] = 0);
% constraint sum(column in 0..63)(xl[RL, 0, column]) != 0;
% constraint sum(column in 0..63)(xl[RL, 1, column]) != 0;
% constraint forall(row in 1..4, column in 0..63)(xl[RL, row, column] = 0);
solve minimize (PU + CM + CL);
% solve minimize (PU + NASM + CL);
% #############################################################################################################################################
% Constraints to find a DL distinguisher usable in the mode of operation
% Ascon-AEAD
% ####################################################
% Difference can be in X[3] and X[4]
% Output mask should be zero in X[1], X[2], X[3], X[4]
% constraint forall(row in 0..2, column in 0..63)(xu[0, row, column] = 0);
% constraint forall(row in 1..4, column in 0..63)(xl[RL, row, column] = 0);
% ####################################################
% Ascon-MAC
% ####################################################
% Difference can be in X[0], X[1], X[2], X[3]
% Output linear mask should be zero in X[2], X[3], X[4]
% constraint forall(column in 0..63)(xu[0, 4, column] = 0);
% constraint forall(row in 2..4, column in 0..63)(xl[RL, row, column] = 0);
% #############################################################################################################################################
% #############################################################################################################################################
% Artificial constraints to reproduce the 4-round DL distinguisher of Ascon from the paper: https://eprint.iacr.org/2015/030.pdf
% constraint forall(row in 0..2, column in 0..<64)(xu[0, row, column] = 0);
% constraint forall(row in 3..4, column in 1..<64)(xu[0, row, column] = 0);
% constraint forall(row in 3..4, column in 0..0)(xu[0, row, column] = 1);
% constraint forall(row in 1..4, column in 0..<64)(yml[1, row, column] = 0);
% constraint forall(row in 0..0, column in 0..<53)(yml[1, row, column] = 0);
% constraint forall(row in 0..0, column in 55..<64)(yml[1, row, column] = 0);
% constraint yml[1, 0, 54] = 1;
% #############################################################################################################################################
% #############################################################################################################################################
% #############################################################################################################################################
% #############################################################################################################################################
% output [
% if i = 0 then "\n\n" else "" endif ++
% "xu_" ++ show(r) ++ "_" ++ show(i) ++ ": \"" ++
% join("", [if fix(xu[r, i, j]) = 1 then "1" else show(fix(xu[r, i, j])) endif | j in 0..63]) ++ "\"\n"
% | r in 0..0, i in 0..4];
% output [
% if i = 0 then "\n\n" else "" endif ++
% "yu_" ++ show(r) ++ "_" ++ show(i) ++ ": \"" ++
% join("", [if fix(yu[r, i, j]) = 1 then "1" else show(fix(yu[r, i, j])) endif | j in 0..63]) ++ "\"\n"
% | r in 0..0 where RU > 0, i in 0..4];
% output [
% if i = 0 then "\n\n" else "" endif ++
% "xu_" ++ show(r) ++ "_" ++ show(i) ++ ": \"" ++
% join("", [if fix(xu[r, i, j]) = 1 then "1" else show(fix(xu[r, i, j])) endif | j in 0..63]) ++ "\"\n"
% | r in 1..RU, i in 0..4];
% output ["\n"];
% output [
% if i = 0 then "\n\n" else "" endif ++
% "xmu_" ++ show(r) ++ "_" ++ show(i) ++ ": \"" ++
% join("", [if fix(xmu[r, i, j]) = -1 then "?" else show(fix(xmu[r, i, j])) endif | j in 0..63]) ++ "\"\n"
% | r in 0..RM, i in 0..4];
% output ["\n"];
% output [
% if i = 0 then "\n\n" else "" endif ++
% "xml_" ++ show(r) ++ "_" ++ show(i) ++ ": \"" ++
% join("", [if fix(xml[r, i, j]) = -1 then "?" else show(fix(xml[r, i, j])) endif | j in 0..63]) ++ "\"\n"
% | r in 0..RM, i in 0..4];
% output ["\n"];
% output [
% if i = 0 then "\n\n" else "" endif ++
% "xl_" ++ show(r) ++ "_" ++ show(i) ++ ": \"" ++
% join("", [if fix(xl[r, i, j]) = 1 then "1" else show(fix(xl[r, i, j])) endif | j in 0..63]) ++ "\"\n"
% | r in 0..RL, i in 0..4];
% output ["\n\nCost of EU: " ++ show(4*sum(r in 0..(RU - 1), column in 0..63)(pu[r, 0, column]) +
% 3*sum(r in 0..(RU - 1), column in 0..63)(pu[r, 1, column]) +
% 2*sum(r in 0..(RU - 1), column in 0..63)(pu[r, 2, column]))];
% output ["\n\nCost of EM: " ++ show(sum(r in 0..(RM - 1), column in 0..63)(bool2int(dmu[r, column] + dml[r, column] == 2)))];
% output ["\n\nCost of EM: " ++ show(sum(r in 0..(RM - 1), row in 0..4, column in 0..63)(
% bool2int((xmu[r, row, column] = -1 /\ xml[r, row, column] = -1)
% \/
% (xmu[r, row, column] = -1 /\ xml[r, row, column] = 1)
% \/
% (xmu[r, row, column] = 1 /\ xml[r, row, column] = -1))))];
% output ["\n\nCost of EL: " ++ show(sum(r in 0..(RL - 1), column in 0..63)(4*pl[r, 0, column]) +
% sum(r in 0..(RL - 1), column in 0..63)(2*pl[r, 1, column]))];
% #############################################################################################################################################
% #############################################################################################################################################
% #############################################################################################################################################
% _ _ _ _ _____ _ _
% / \ _ _ __ __(_)| |(_) __ _ _ __ _ _ | ___|_ _ _ __ ___ | |_ (_) ___ _ __ ___
% / _ \ | | | |\ \/ /| || || | / _` || '__|| | | | | |_ | | | || '_ \ / __|| __|| | / _ \ | '_ \ / __|
% / ___ \| |_| | > < | || || || (_| || | | |_| | | _| | |_| || | | || (__ | |_ | || (_) || | | |\__ \
% /_/ \_\\__,_|/_/\_\|_||_||_| \__,_||_| \__, | |_| \__,_||_| |_| \___| \__||_| \___/ |_| |_||___/
% |___/
% auxiliary functions
predicate xor3(var 0..1: a0, var 0..1: a1, var 0..1: a2, var 0..1: b) =
b - a2 - a1 - a0 >= -2
/\
- b + a2 - a1 - a0 >= -2
/\
- b - a2 + a1 - a0 >= -2
/\
b + a2 + a1 - a0 >= 0
/\
- b - a2 - a1 + a0 >= -2
/\
b + a2 - a1 + a0 >= 0
/\
b - a2 + a1 + a0 >= 0
/\
- b + a2 + a1 + a0 >= 0
;
predicate xor3_pr1(var -1..1: x0, var -1..1: x1, var -1..1: x2, var -1..1: y) =
if exists([x0 == -1, x1 == -1, x2 == -1]) then y = -1
else y = (sum([x0, x1, x2]) mod 2) endif
;
predicate sbox_diff(var 0..1: a0, var 0..1: a1, var 0..1: a2, var 0..1: a3, var 0..1: a4,
var 0..1: b0, var 0..1: b1, var 0..1: b2, var 0..1: b3, var 0..1: b4,
var 0..1: p0, var 0..1: p1, var 0..1: p2) =
- p0 - p1 >= -1
/\
- p0 - p2 >= -1
/\
- p1 - p2 >= -1
/\
a0 + a2 - p2 >= 0
/\
- a1 + a2 - a3 + p0 >= -1
/\
- a0 - a2 + a3 + p1 >= -1
/\
- a0 - a2 - a4 + p1 >= -2
/\
- a2 - a4 + p0 + p1 >= -1
/\
- a3 - b0 + p0 + p1 >= -1
/\
a1 - a4 + p1 + p2 >= 0
/\
- a1 - a2 + a3 + a4 - b2 >= -2
/\
- a1 + a2 + a3 + a4 + b2 >= 0
/\
- a0 - a3 - a4 - b2 - b3 >= -4
/\
a0 + a3 + a4 - b2 + b3 >= 0
/\
- a0 - a3 - a4 + b2 + b3 >= -2
/\
a0 + a1 + a3 + a4 - b4 >= 0
/\
a0 - a1 + a3 - a4 + p0 >= -1
/\
a1 - a3 + a4 + b0 + p0 >= 0
/\
a0 - a1 - a4 + b1 + p0 >= -1
/\
a1 - a2 + a4 + b2 + p0 >= 0
/\
a0 - a1 - a2 - a3 + p1 >= -2
/\
- a0 - a1 + a3 - a4 + p1 >= -2
/\
a0 + a1 + a2 - b4 + p1 >= 0
/\
- a4 + b0 - b4 + p0 + p1 >= -1
/\
a0 - a2 + a3 + a4 + p2 >= 0
/\
- a0 + a1 + a2 + p0 + p2 >= 0
/\
a1 + a3 - b1 + p1 + p2 >= 0
/\
- a0 - a1 - a2 - a3 + a4 + b1 >= -3
/\
a0 - a1 - a2 + a3 + a4 - b3 >= -2
/\
a0 - a1 + a2 - a3 - a4 + b3 >= -2
/\
- a2 + a3 - b0 + b1 - b2 - b4 >= -3
/\
a1 + a3 - b0 - b1 + b2 - b4 >= -2
/\
- a1 + a3 + b0 - b1 + b2 - b4 >= -2
/\
- a1 + a3 - b0 + b1 + b2 - b4 >= -2
/\
a1 + a3 + b0 + b1 + b2 - b4 >= 0
/\
a0 - a1 + a4 + b2 - b3 - b4 >= -2
/\
a0 - a1 + a4 - b2 + b3 - b4 >= -2
/\
- a1 - a2 + a3 + b0 + b1 + b4 >= -1
/\
- a1 + a3 - b0 - b1 + b2 + b4 >= -2
/\
a1 + a3 - b0 + b1 + b2 + b4 >= 0
/\
a3 - a4 + b0 + b1 + b2 + b4 >= 0
/\
a0 - a3 + a4 - b2 - b3 + b4 >= -2
/\
a0 + a1 - a2 - a3 - b0 + p0 >= -2
/\
- a0 - a1 - a2 - a3 - b1 + p0 >= -4
/\
- a0 + a2 + a3 + a4 - b2 + p0 >= -1
/\
a0 - a1 - a2 - b1 - b3 + p0 >= -3
/\
a1 - a2 - a3 - b0 - b4 + p0 >= -3
/\
a0 + a2 - a4 - b0 - b4 + p0 >= -2
/\
- a0 - a1 - b2 - b3 - b4 + p0 >= -4
/\
- a0 + a1 + a3 - a4 + b4 + p0 >= -1
/\
- a1 + a3 - b2 + b3 + b4 + p0 >= -1
/\
a0 + a1 - a2 + b0 + b3 - p1 >= -1
/\
- a0 + a1 + a3 + a4 + b1 + p1 >= 0
/\
a1 + a2 + a4 - b0 - b4 + p1 >= -1
/\
a0 + a1 + a4 - b0 + b4 + p2 >= 0
/\
a1 - a2 + a4 - b0 + p0 + p2 >= -1
/\
a0 - a2 + a3 + b0 + p0 + p2 >= 0
/\
a0 - a2 + a4 - b1 + p0 + p2 >= -1
/\
a2 - a3 - a4 - b3 + p0 + p2 >= -2
/\
a0 + a3 + a4 - b3 + p1 + p2 >= 0
/\
a2 + a3 - a4 - b0 - b1 - b2 - b4 >= -4
/\
- a1 + a3 - a4 + b2 + b3 - b4 + p0 >= -2
/\
- a1 + a3 - a4 + b2 - b3 + b4 - p1 >= -3
/\
a2 + a3 + b0 - b1 + b4 + p0 + p1 >= 0
/\
- a0 + a1 - a2 - a3 + b0 - b4 + p2 >= -3
/\
- a2 + a3 - b0 - b1 - b2 + b4 + p2 >= -3
/\
a0 - a1 + a4 + b2 + b3 + b4 + p2 >= 0
/\
a1 + a4 + b0 + b4 - p0 + p1 + p2 >= 0
/\
- a2 + a3 - a4 + b0 - b1 - b2 - b4 + p2 >= -4
/\
a2 + a3 - a4 + b0 + b1 - b2 - b4 + p2 >= -2
/\
a2 + a3 - a4 - b0 + b1 - b2 + b4 + p2 >= -2
/\
a0 + a1 + a2 + b0 + b4 + p0 - p1 + p2 >= 0
/\
- a1 + a2 + a3 - a4 + b0 - b1 - b2 + b4 + p2 >= -3
/\
- a0 - a1 + a2 - a3 + a4 + b1 - b3 - b4 - p0 + p1 + p2 >= -5
/\
- a0 - a1 + a2 - a3 + a4 - b1 + b3 - b4 - p0 + p1 + p2 >= -5
/\
- a0 - a1 + a2 - a3 + a4 - b1 - b3 + b4 - p0 + p1 + p2 >= -5
/\
- a0 - a1 - a3 + a4 + b1 + b3 + b4 >= -2
;
predicate sbox_diff_star(var 0..1: a0, var 0..1: a1, var 0..1: a2, var 0..1: a3, var 0..1: a4,
var 0..1: b0, var 0..1: b1, var 0..1: b2, var 0..1: b3, var 0..1: b4,
var 0..1: d) =
a1 + a2 + a3 + a4 - b2 >= 0
/\
a1 - a2 + a3 + a4 + b2 >= 0
/\
- a1 + a2 + a3 + a4 + b2 >= 0
/\
a0 + a3 + a4 + b2 - b3 >= 0
/\
a0 + a3 + a4 - b2 + b3 >= 0
/\
- a0 - a3 - a4 + b2 + b3 >= -2
/\
a1 - a3 - a4 - b0 - b4 >= -3
/\
a1 + a3 + a4 - b0 - b4 >= -1
/\
a1 + a3 - a4 + b0 + b4 >= 0
/\
a1 - a3 + a4 + b0 + b4 >= 0
/\
- a0 - a1 - a2 - a3 - a4 - b1 >= -5
/\
a0 - a1 - a2 - a3 - a4 + b1 >= -3
/\
- a0 + a1 + a3 + b0 + b1 + b2 >= 0
/\
a0 + a1 + a2 - a3 + b0 - b3 >= -1
/\
- a1 - a2 - a3 - a4 - b1 - b3 >= -5
/\
a0 + a1 - a2 - a3 - a4 + b3 >= -2
/\
- a0 + a1 + a2 + a4 - b0 - b4 >= -2
/\
- a0 + a1 - a2 - a3 + b0 - b4 >= -3
/\
a0 + a1 - a2 + a3 + b0 - b4 >= -1
/\
- a0 + a1 + a2 - a4 + b0 - b4 >= -2
/\
a0 + a1 + a2 + a4 + b0 - b4 >= 0
/\
a0 + a1 + a2 + a3 + b1 - b4 >= 0
/\
a1 - a4 - b0 - b1 + b2 - b4 >= -3
/\
- a1 + a3 - b0 + b1 + b2 - b4 >= -2
/\
a0 - a1 + a4 - b2 + b3 - b4 >= -2
/\
- a0 - a1 - a4 + b2 + b3 - b4 >= -3
/\
- a0 + a1 - a2 + a3 - b0 + b4 >= -2
/\
- a0 + a1 + a2 - a4 - b0 + b4 >= -2
/\
- a1 - a2 + a3 + b0 + b1 + b4 >= -1
/\
a1 + a3 + b0 - b1 + b2 + b4 >= 0
/\
a1 + a3 - b0 + b1 + b2 + b4 >= 0
/\
a3 - a4 + b0 + b1 + b2 + b4 >= 0
/\
a0 + a1 - a3 - b0 - b3 + b4 >= -2
/\
a0 + a1 + a4 - b0 + b3 + b4 >= 0
/\
a0 + a1 - a4 + b0 + b3 + b4 >= 0
/\
- a0 + a3 - a4 - b2 + b3 + b4 >= -2
/\
- a2 + a3 - a4 + b0 - b1 - b2 - b4 >= -4
/\
- a2 + a3 - a4 - b0 + b1 - b2 - b4 >= -4
/\
- a0 - a1 + a3 - a4 - b2 - b3 - b4 >= -5
/\
- a2 + a3 - a4 - b0 - b1 - b2 + b4 >= -4
/\
- a0 - a1 + a3 - a4 + b2 - b3 + b4 >= -3
/\
a0 - a1 - a3 + a4 + b2 + b3 + b4 >= -1
/\
- a0 - a1 + a2 - a3 + a4 + b1 - b3 - b4 >= -4
/\
- a0 - a1 + a2 - a3 + a4 - b1 + b3 - b4 >= -4
/\
- a0 - a1 + a2 - a3 + a4 - b1 - b3 + b4 >= -4
/\
a2 + a3 - a4 - b0 - b1 - b2 - b4 >= -4
/\
a2 + a3 - a4 + b0 + b1 - b2 - b4 >= -2
/\
a2 + a3 - a4 + b0 - b1 - b2 + b4 >= -2
/\
a2 + a3 - a4 - b0 + b1 - b2 + b4 >= -2
/\
a0 - a1 + a4 + b2 - b3 - b4 >= -2
/\
- a0 - a1 - a3 + a4 + b1 + b3 + b4 >= -2
/\
a0 - a3 + a4 - b2 - b3 + b4 >= -2
/\
a0 - a1 + a2 - a3 - a4 + b3 >= -2
/\
- a1 + a3 + b0 - b1 + b2 - b4 >= -2
/\
- a1 + a3 - b0 - b1 + b2 + b4 >= -2
/\
- a0 - a3 - a4 - b2 - b3 >= -4
/\
- a0 - a1 - a2 - a3 + a4 + b1 >= -3
/\
- a1 - a2 + a3 + a4 - b2 >= -2
/\
a0 - a1 - a2 - a3 + a4 - b1 >= -3
/\
if exists([a0 == 1, a1 == 1, a2 == 1, a3 == 1, a4 == 1]) then d = 1
else d = 0 endif
;
predicate sbox_lin(var 0..1: a0, var 0..1: a1, var 0..1: a2, var 0..1: a3, var 0..1: a4,
var 0..1: b0, var 0..1: b1, var 0..1: b2, var 0..1: b3, var 0..1: b4,
var 0..1: p0, var 0..1: p1) =
- a0 - a3 - p1 >= -2
/\
- a1 - a3 - p1 >= -2
/\
- a2 + p0 + p1 >= 0
/\
- b2 + p0 + p1 >= 0
/\
- b3 + p0 + p1 >= 0
/\
- b4 + p0 + p1 >= 0
/\
a1 + a2 + a3 - p0 >= 0
/\
a0 + a1 + a4 - p0 >= 0
/\
a0 + a1 + b2 - p0 >= 0
/\
a2 - a3 + b2 + p0 >= 0
/\
- a0 + a4 + b3 + p0 >= 0
/\
a0 - a1 - a4 - p1 >= -2
/\
- a0 - a1 + a4 - p1 >= -2
/\
- a2 + a3 - b0 - p1 >= -2
/\
a0 - a4 - b2 - p1 >= -2
/\
a0 - a1 + a3 + a4 + b0 >= 0
/\
a0 + a1 - a2 - a3 - b2 >= -2
/\
- a0 - a1 - a4 - b1 - b2 >= -4
/\
a1 - a2 + a3 + b0 + b2 >= 0
/\
- a0 - a1 - a4 + b1 + b2 >= -2
/\
- a0 + a3 - a4 + b0 - p0 >= -2
/\
a0 + a3 - a4 + b3 + p0 >= 0
/\
- a0 + a1 + b0 - b2 - p1 >= -2
/\
a0 + a1 + b1 - b2 - p1 >= -1
/\
- a1 + a2 - a4 - b4 - p1 >= -3
/\
- a1 + a3 - a4 - b0 - b2 - b4 >= -4
/\
a1 + a3 + a4 - b0 - b2 - b4 >= -2
/\
a2 - b0 + b1 + b2 - b3 - b4 >= -2
/\
- a0 + a2 + a3 - a4 - b0 + b4 >= -2
/\
a0 + a2 + a3 + a4 - b0 + b4 >= 0
/\
- a1 + b0 + b1 + b2 + b3 + b4 >= 0
/\
- a1 + a2 + a3 - b0 - b4 - p0 >= -3
/\
a1 + a2 + b2 + b3 + b4 - p0 >= 0
/\
a1 + a3 - b0 - b3 - b4 + p0 >= -2
/\
a0 + a1 + a2 + a3 + a4 - p1 >= 0
/\
a1 + a2 + a4 - b0 + b2 - p1 >= -1
/\
- a0 - a4 + b0 + b2 - b4 - p1 >= -3
/\
a1 + a2 + b2 - b3 - b4 + p1 >= -1
/\
- a1 + a2 + a3 + b0 + b4 + p1 >= 0
/\
a3 + b0 - b1 + b2 + b4 + p1 >= 0
/\
- a1 - a2 - a3 - a4 - b1 - b2 + b3 >= -5
/\
a0 - a2 - a3 - a4 + b1 + b2 + b3 >= -2
/\
a0 + a2 + a4 + b1 - b2 + b3 - b4 >= -1
/\
a0 + a4 + b0 + b1 - b2 + b3 - b4 >= -1
/\
a0 + a2 - b0 - b1 + b2 + b3 - b4 >= -2
/\
- a0 - a1 - a2 + a3 - a4 + b1 + b4 >= -3
/\
a1 + a2 - a3 + a4 - b2 - b3 + b4 >= -2
/\
a0 + a2 + a4 + b1 - b2 - b3 + b4 >= -1
/\
a0 + a4 + b0 + b1 - b2 - b3 + b4 >= -1
/\
a1 + a2 + a3 + b0 + b4 + p0 - p1 >= 0
/\
- a1 - a2 + a3 + b0 - b2 - b4 + p1 >= -3
/\
- a1 - a2 + a3 - a4 + b2 + b4 + p1 >= -2
/\
a0 - a1 + a2 - a3 - a4 + b0 - b1 - b3 >= -4
/\
a0 - a1 + a2 - a3 - a4 - b0 + b1 - b3 >= -4
/\
- a0 - a1 + a2 - a3 + a4 + b0 - b1 - b4 >= -4
/\
- a0 - a1 - a3 + a4 + b0 - b1 + b2 - b4 >= -4
/\
a0 - a1 + a2 + a4 - b1 - b2 - b3 - b4 >= -4
/\
a0 - a3 + a4 + b0 - b1 + b2 + b3 - b4 >= -2
/\
- a0 - a1 + a2 - a3 - b0 - b1 - b2 + b4 >= -5
/\
- a0 - a2 + a4 - b0 + b1 - b2 - b3 + b4 >= -4
/\
a0 + a2 + a4 + b1 + b2 + b3 + b4 - p0 >= 0
/\
a0 + a4 + b0 - b1 - b2 - b3 - b4 + p1 >= -3
/\
a0 + a2 + a4 - b1 - b2 + b3 + b4 + p1 >= -1
/\
a0 + a4 + b0 - b1 - b2 + b3 + b4 + p1 >= -1
/\
- a1 + a2 - a3 + a4 - b0 - b1 + b2 - b3 + b4 >= -4
/\
a0 - a2 - a3 + a4 - b0 - b1 + b2 + b3 + b4 >= -3
/\
- a0 - a2 + a3 + a4 - b0 - b2 + b4 - p0 + p1 >= -4
/\
a0 - a1 - a2 - a3 - a4 + b1 - b2 - b3 - p0 + p1 >= -6
/\
a0 - a1 - a2 - a3 - a4 - b1 + b2 - b3 - p0 + p1 >= -6
/\
a0 - a1 + a2 - a3 - a4 - b0 - b1 + b3 - p0 + p1 >= -5
/\
a0 - a1 + a2 - a3 - a4 + b0 + b1 + b3 - p0 + p1 >= -3
/\
- a0 - a1 - a3 + a4 - b0 - b1 + b2 + b4 - p0 + p1 >= -5
/\
- a0 - a1 - a2 - a3 + a4 + b0 + b1 - b2 - b4 - p0 + p1 >= -6
/\
- a1 - a2 - a3 + a4 - b0 + b1 - b2 + b3 + b4 - p0 + p1 >= -5
/\
a0 - a2 - a3 + a4 - b0 - b1 - b2 - b3 + b4 >= -5
/\
a0 - a2 - a3 + a4 - b0 + b1 + b2 - b3 + b4 >= -3
/\
- a0 - a1 - a2 - a3 - b0 - b1 - b2 - b4 >= -7
/\
- a0 - a1 - a2 - a3 + b0 - b1 - b2 + b4 >= -5
/\
- a0 - a1 + a2 + a4 - b0 + b1 - b4 >= -3
/\
a1 - a2 - a3 + b0 + b3 + b4 >= -1
/\
- a2 - a3 - b0 + b1 + b2 + b3 - b4 >= -3
/\
- a0 - a1 + b0 + b1 + b2 + b4 >= -1
/\
a0 - a2 - a3 - b0 + b1 - b2 - b3 - b4 >= -5
/\
a0 - a2 - a3 - b0 - b1 + b2 - b3 - b4 >= -5
/\
a0 + a4 + b0 + b1 + b2 - b3 - b4 >= -1
/\
a1 - a2 - a3 - b0 - b3 + b4 >= -3
/\
a1 - a2 - a3 + b0 - b3 - b4 >= -3
/\
- a2 - a3 - b0 - b1 - b2 + b3 - b4 >= -5
/\
- a0 + a1 + a2 - a4 + b3 + b4 >= -1
/\
- a0 - a1 + a2 + a4 + b0 + b1 + b4 >= -1
/\
- a0 + a1 + a2 - a4 - b3 - b4 >= -3
/\
a0 + b0 - b1 + b2 - b3 + b4 >= -1
/\
a1 - a2 - a3 - b0 + b3 - b4 >= -3
/\
- a0 - a1 + a3 - b0 + b2 - b4 >= -3
/\
- a0 - a1 - b0 + b1 + b2 - b4 >= -3
/\
a1 + a2 + a4 - b2 + b3 - b4 >= -1
;
predicate sbox_lin_star(var 0..1: a0, var 0..1: a1, var 0..1: a2, var 0..1: a3, var 0..1: a4,
var 0..1: b0, var 0..1: b1, var 0..1: b2, var 0..1: b3, var 0..1: b4,
var 0..1: d) =
- a0 - a1 + a3 - a4 - b0 >= -3
/\
a0 + a1 - a2 - a3 - b2 >= -2
/\
- a0 - a1 - a4 - b1 - b2 >= -4
/\
a0 + a1 + a4 + b1 - b2 >= 0
/\
a0 + a1 + a2 - a3 + b2 >= 0
/\
a0 + a1 - a2 + a3 + b2 >= 0
/\
a1 - a2 + a3 + b0 + b2 >= 0
/\
- a0 - a1 - a4 + b1 + b2 >= -2
/\
a1 - a2 + a3 + a4 - b0 - b2 >= -2
/\
a0 + a1 + a3 + a4 + b2 - b3 >= 0
/\
a0 + a2 + a3 - a4 - b0 - b4 >= -2
/\
- a0 + a2 + a3 + a4 - b0 - b4 >= -2
/\
- a0 + a2 + a3 - a4 + b0 - b4 >= -2
/\
- a0 + a3 - a4 + b0 + b2 - b4 >= -2
/\
a1 + a2 + a3 - b2 - b3 - b4 >= -2
/\
a1 + a2 + a3 + a4 + b3 - b4 >= 0
/\
a1 + a2 + a3 + b0 + b3 - b4 >= 0
/\
a0 + a2 + a3 + a4 - b0 + b4 >= 0
/\
a0 + a2 + a3 - a4 + b0 + b4 >= 0
/\
a1 + a2 + a3 - a4 - b2 + b4 >= -1
/\
a2 + a3 + a4 + b0 - b2 + b4 >= 0
/\
- a0 + a1 + a2 + a3 + b2 + b4 >= 0
/\
a0 - a2 + a3 - a4 + b2 + b4 >= -1
/\
a3 + a4 + b0 - b1 + b2 + b4 >= 0